A gold cube was divided into 8 pieces by 3 cuts parallel to its sides as the picture shows. Four of the pieces were marked by stars. The marked pieces weighed 8, 8, 20, and 80 grams.
Find the weight of the original cube.
A gold cube was divided into 8 pieces by 3 cuts parallel to its sides as the picture shows. Four of the pieces were marked by stars. The marked pieces weighed 8, 8, 20, and 80 grams.
Find the weight of the original cube.
A rectangular place mat is slowly being pushed off a wooden table.
At what point is the mat just about to fall off the table?
At this scale you probably can't read the labels on the Google Maps' satellite image of the North pole. The big white blob is Greenland.
To the left we have Russia, in the middle we have Canada and the USA, to the right we have Europe.
Why is Greenland so big?
I have infinitely many apples. To be fair, this concept is far too hard for your puny human brain to understand.
Nevertheless, I am generous beyond your comprehension. I pass one fifth of my apples to Jane on my left, I pass less than two fifths but more than one fifth of my apples to Gerry on my right, and I keep the remaining apples for myself.
Who now has the greatest amount of apples?
by Leslie Green
A conventional oven (gas or electric) heats the air to the required set temperature, and the food is heated by the hot air.
A microwave oven is totally different. A microwave field is created in the oven cavity, and the food absorbs power directly from that electromagnetic field.
A particular frozen dessert has microwave cooking instructions printed on the cardboard packaging. For a 700 Watt microwave oven, the dessert takes 3 minutes to cook. This particular dessert is stated as serving 3 people, so you cut off one third of the dessert, replacing the remainder back in the freezer compartment at -18°C.
For how long should you cook your piece of the dessert in your 700 Watt microwave oven?
A Blogger, who incidentally was expelled from three schools, and never passed any exams, has revealed the cause of rainfall in his latest post:
"Don't take your umbrella to school or you will make it rain!"
The Blogger posts his experimental results, showing that when it rains there are always more open umbrellas than on sunny days.
Do you believe the analysis?
Every month Manny Pitt spends $1000 on his credit card, but only pays off the debt after it has accrued 2% interest. He pays off the debt and the interest at the same time so nothing carries forward to the next period.
Every month his girlfriend Jane spends $1000 on her credit card, but pays off the debt before any interest is payable. She pays off the debt completely so nothing carries forward to the next period.
After 10 months of these payments, how much more money does Jane have compared to Manny as a result of the credit card spending alone?
I make a steel sphere with a 1 mm wall thickness and fill it with hydrogen at atmospheric pressure. The dimensions are such that the sphere weighs 0.1 grams.
When my back is turned an evil super-villain throws the sphere at me at 50 km/hour.
What happens?
Part of an old army base has been repurposed as a dance hall. The floor of the dance hall is directly on top of a reinforced concrete slab which used to take the weight of a main battle tank, and is therefore adequately strong. However this concrete slab has been covered in pretty wood panelling which is not so strong.
Which static load would be most problematic for the floor?
In an upcoming election, each block votes for the green party, the yellow party, or the red party as shown in the picture. The region is to be divided into 5 districts of 7 connected blocks. For each district, the party with the most number of votes wins the district. Then the party who wins the most number of districts wins the election. No ties allowed in any districts.
Which party is able to win the election?
You have sufficient money that you can invest £500 and not be in financial difficulty if the investment fails.
If the investment succeeds you will get £1500 total returned to you. If the investment fails you lose all your money.
Having studied the situation carefully you have estimated that the probability of success is 40%.
How do you evaluate the investment?
I attached the frame of my bike (red circle) to a lamppost (blue circle) with a green flexible cable and three locks. I want to take my bike. I don't need the green cable.
Which lock should I open to separate my bike and the lamppost?
The picture shows a square pyramid with a 7x7 base and its quarter.
The "external" cubes of the pyramid are yellow and the "internal" cubes are grey. There are 35 grey and 49 yellow cubes in the pyramid.
Which is the smallest such pyramid that has more grey than yellow cubes?
The image shows the results of a (fictitious) study on a new cream used to treat a skin rash.
As usual, the participants in the trial were put into a control group and a treatment group.
At the end of the two week trial period the participants reported the change in the rash, which is summarised in the table.
Nobody in the trial, including the medical staff, knew which particular cream type they were using using, other than "type A" or "type B". This is standard "double blind" trial methodology.
Which cream was better? (Note that the "null cream" was a placebo, having no active components.)
HINT: Don't rush to an answer!
Adapted from a TEDx talk by Dan Kahan (via Veritasium).
When you spin the left spinner A, you have a 1/2 chance of winning $100.
When you spin the right spinner B with a 1/4 chance of winning $100 and did not win, you have the right for the second spin B.
Which of the options do you choose?
Two right-angled triangles share the same base. The base and side of the blue triangle have been drawn slightly in from the red triangle's edges to aid in visibility. In reality those line-segments are directly on top of each other.
The red triangle has been dimensioned in RED units. The blue triangle has been drawn by the rival blue team, and of course they use their own BLUE units.
What is the area of the yellow region when the area is based on BLUE units.
A road machine sweeps all roads in an area. We say that there are 12 two-direction roads shown in gray. Its task is to drive along ALL the roads in both directions, starting and ending at the bottom left corner.
It can drive along the same road in a direction more than once. It can drive down a road, and then drive back. Since all roads have the same length, it is possible to define an optimal path as one of minimal length.
Repeatedly driving over the same roads is inefficient. It must minimize the number of such repeated trips. A score of +6 means your path is 6 roads longer than the total number of roads in the network. Maneuvers at intersections are not counted.
What is the best possible score?
Inspired by Leslie Green's similar problem.
We can put 105 round pencils with a diameter of 1 cm into a box with the 10x10 cm cross-section.
We also have hexagonal pencils with the same sizes (the hexagon is on a circle with the diameter of 1 cm).
How many hexagonal pencils can we put in the same box?
Endo spinal sacrosis is a horrible condition from which 50% of patients die within 1 year. However, the Hollywood SuperStar private medical facility is pushing their new treatment which promises to halve the fatality rate, with published externally-audited statistics on a relatively large patient population which demonstrate that only 25% of their patients die within 1 year.
Is it guaranteed that this new treatment is a breakthrough in medical treatment?
The blue parts form a light rigid construction, the weight of which we will ignore for the purposes of this problem.
The green triangle represents a low-friction bearing or pivot, the friction of which we will ignore for the purpose of this problem.
The grey spheres represent equal masses subjected to a uniform gravitational field. The red lines represent nylon monofilament, such as fishing line.
Is this arrangement stable?
(HINT: Don't rush to an answer.)
The image shows the plan of part of a ventilation shaft. The black lines are the walls of the horizontal shaft. The blue circle is a hole of indeterminate depth into which you really don't want to fall. With the aid of a colleague on the other side, you have managed to set up some rigid planks across the gaping chasm. Sadly the planks are not long enough to easily do the job.
Is it safe to walk across the plank(s)?
Ann and Brian are both new to archery, and still pretty useless. They struggle to hit the target.
Brian hits the target once out of every four shots, on average.
Ann is slightly better. She hits the target once out of every three shots.
Every day they are allowed to shoot just one arrow each, with Brian taking the first shot. Whoever hits the target first wins.
If nobody hits the target with that single arrow, the contest is a draw for that day, and the process repeats the next day.
Who has the better chance of winning?
Four friends (Alan, Betty, Carl, and Diane) take an important test at school. No two friends get the same score.
No boy comes last in the test.
No girl comes first in the test.
Betty beats Carl.
Which is the correct ordering of test results when the highest score comes first?
John runs a fair game using a single standard 6-sided die. There are always 7 players, and every available number is picked by at least one player.
Each player pays the $1 fee to play the game.
Any player who has the winning number gets paid $5.
On average, who wins the game?
Consider the following experiment:
STEP 1A: Two balls, labelled 1 and 2 respectively, are placed into an urn.
STEP 1B: One ball is randomly chosen from the urn and then discarded.
STEP 2A: Two balls, labelled 3 and 4 respectively, are placed into the same urn as in step 1A.
STEP 2B: One ball is randomly chosen from the urn and then discarded.
Define PN(1) as the probability that ball 1 remains in the urn at the end of step NB.
Evaluate PN(2N).
Start at square S, and end at square X. Each step must be on the grid. Each step can be only one of UP or RIGHT.
To be clear, each step is only one grid square movement, and this cannot be a diagonal step.
Which square most effectively block paths to X?
The image shows a towing hawser for ships, dated around 1945.
As a thought experiment, we wish to compare the weights of a 1 inch diameter rope and a 10 inch diameter towing hawser (used for towing or mooring a ship), both having the same infinite length.
We suppose the rope and the hawser to be made of the same natural material, so that a 10 inch diameter hawser would have 102 = 100x the weight per unit length of a 1 inch diameter rope. The weight/length measure might be given in pounds per yard, or kilograms per metre.
What is the ratio of the weights, hawser/rope?
by Leslie Green
During this war, casualties have been high, so no more volunteers are coming forward. Desperate for more conscripts, the Chief of Staff enacts a new plan to stop conscript dodgers and cowards from doing their part. Men are rounded up and herded into a testing centre where 8 stations are set up to each detect a specific medical condition. These are labelled A to H, inclusive, but what the conditions are is TOP SECRET.
The (highly suspect) statistics are that 70% have A, 80% have B, 90% have C, 90% have D, 90% have E, 95% have F, 95% have G, and 98% have H.
The men (and the doctors) are happy to get a positive test result for any given condition as they think that excuses service. No such luck! A man is declared fit for duty unless he has all 8 conditions.
What is the lowest possible proportion of applicants who are unfit for service?
[HINT: You could try an easier problem first.]
Consider two solid uniform spheres just touching each other. There will be an attractive gravitational force between them because they have mass.
How much does the attractive force increase when the spheres, made of the same material, are doubled in size?
HINT: The equation shows Newton's Law of Universal Gravitation. G is the gravitational constant. m1 and m2 are two masses. r is the distance between their centres. The force of gravitation acts as if all the mass of a sphere is concentrated at the geometric centre of that sphere.
You are playing a game of chance with a computer. The computer generates a random natural number in the inclusive range 1 to 9.
Your task is to find the number by making guesses and receiving an answer in the form of a comparative operator:
>, =, or <.
What is the minimum average number of guesses required?
A blind darts player throws a single dart at a graduated scale some distance away. The metal spike of the dart is pointy at the end, but soon becomes a 1 mm diameter. The dart is thrown hard enough to enter the scale up to the non-tapering part of the dart.
The graduations on the scale are marked with integers at 20 mm intervals. The actual scale lines themselves are 1 mm thick. The darts player has an assistant so the dart goes into the scale and not into nearby people or objects. If the dart fails to hit the scale, the throw is repeated until it does hit.
What is the probability that the dart hits the thin line?
The delegates to the inter-galactic scientific conference all book into Hilbert's Hotel, which of course has infinitely many rooms.
Sadly, the number theorists all arrive late, and by that time the hotel is completely full. Fortunately Hilbert is on hand to resolve the problem. Number theorists are given VIP status so that all of the infinitely many number theorists are accommodated by kicking other delegates out of their rooms. The procedure is simple. As each number theorist arrives they are assigned a room number according to their arrival position. Thus the first person goes to room 1, the second goes to room 22, the third goes to room 32, and so on.
After all number theorists are settled in, what is the probability of randomly picking a room with a number theorist in it?
by Leslie Green
Gerry and Jane are at the foot of a conical-shaped mountain with a direct way to the peak of 8 km and the foot circle's diameter of 8 km. They walk from point A to the diametrically opposite point B by the shortest route.
Find the distance c from the foot of the mountain to the highest point of their route C.
You have a 3-digit padlock, and you have tried to find the code with five guesses (labelled A to E). For each guess, your instructor has 'marked' the correctness of your guess very carefully. You are told two numbers:
(1) The count of right digits which are in their correct positions.
(2) The count of right digits which are in incorrect positions.
If either of the counts is zero, your instructor may not bother to mention that.
Suppose the actual code is 591 and your guess was 195. Your instructor would report one right digit (9) in its correct position, and two right digits (1,5) in incorrect positions, although you should appreciate that the instructor will not say which specific digits are in correct or incorrect positions.
Can you give the correct code to open the padlock?
Interest on savings is typically considered as income, and is therefore taxable. In the UK (2023) you can earn £1000 of interest before you are liable for additional tax. You can earn £10,000 of interest before you have to explicitly declare it, since the tax collectors (HMRC) tap in to your bank and building society accounts to see what interest you are earning.
If you are receiving 5% interest on your savings, how much savings do you have to have in order to reach the £10,000 interest reporting limit?
The pirate Captain has 3 crew, and more than 3 gold coins in a chest. There are too many coins to give each pirate (crew) the same number of coins without leftovers. The Captain pockets (takes) a coin to correct the situation.
Suddenly, another pirate arrives, and the Captain is again 'forced' to overtly (obviously) pocket a coin, "Arrrhh!", to make the coins evenly divisible.
Another pirate arrives, and the Captain takes another coin to once again bring the coin-count into a harmonious state. "Arrrh, shiver m' timbers!"
What is the least number of coins that could have been present in the chest at the start of the proceedings?
This question relates to the rated slope up which an electric mobility scooter is specified to be able to safely drive. It is a question of understanding a technical specification, and how it relates to the real world.
These examples actually come from user manuals for mobility scooters.
Which is correct to specify the rated slope?
Diseased bunnies have arrived in your area. One bite and a human dies horribly within hours. They must be eradicated before they spread out and infect other bunnies.
There are 3 large burrows in the area, all in a line. We don't know how many bunnies there are in each burrow, but we do know that at night they all leave their burrow and go to a nearby burrow (but not a far away burrow).
This is a public health emergency, and yet cost is always a factor. One extermination team can only handle one burrow per day, at a cost of $5000/day. Two extermination teams cost $20,000/day (total) due to the logistical problems involved.
Three extermination teams cost $50,000/day (total) since they have to be flown in from far away.
What is the minimum cost to complete this vital mission?
There is a traditional kid's problem which is some variant of "Which is heavier, a ton of feathers or a ton of lead?"
We ask an adult question, "Which has more mass, a ton of feathers or a ton of lead?", although we will substitute expanded polystyrene (EPS, styrofoam) for feathers since we are more familiar with that everyday packing material.
Window glass is typically manufactured in optically flat sheets using the float-glass process, developed in the 1950's. Molten glass is poured onto a molten metal, typically tin, and since liquids naturally form flat surfaces, the glass floats on a flat layer of molten tin as the glass cools and solidifies. The glass is slowly cooled so as to prevent sudden temperature changes, but comes off the production line in a continuous sheet. It is therefore necessary to cut it whilst it is still moving. The idea is to cut it in such a way that the finished sheet is rectangular.
The glass has been coloured to make it easier to see in the drawing. The glass is moving left to right, as shown by the blue arrow. The left edge of the glass is shown as a strange curve. This is a drawing office technique suggesting that the glass extends further to the left, but we have truncated the length to allow it to fit on the page.
We are looking at a plan view, so the glass is actually in a horizontal plane. The glass cutter is held in a sliding mechanism (fixed to the ground) at one of the three angles shown as A, B, or C. The glass is therefore moving past the glass cutter.
Which is the correct angle for the glass cutter slide in order to get a rectangular sheet?
(The glass cutter starts from the top of the drawing.)
We are confident that many people believe that trigonometry could never have any use in their lives, and therefore should not be learned. Here we share a recent real-life story, starting with the trigonometry problem.
A road-worthy mobility scooter is specified to travel at 8 mph (3.6 m/s) on level ground. If it were travelling at 3.6 m/s up a 17° slope, what would its effective vertical speed be?
[Note: it is important to work in "metric" (SI) units throughout, or subsequent calculations will not work correctly.]
Gerry started trading in crypto. Once he gets back double the invested money, he donates $800 to a charity and invests all the rest again.
Gerry makes 4 cycles and he has no money at the end.
How many times more money does the charity receive if he gives all the money not after each cycle, but only at the end of the fourth cycle?
Granny can't walk as well as she used to, so she wants to look at buying a mobility scooter. The brochure for one model specifies the maximum gradient as 14%.
Having used an app on your phone, you have measured the worst nearby hill as having a slope of 12°. Given that 95% of adults have forgotten trigonometry within 5 years of leaving school*, you have been nominated as tech support for the scooter selection process.
How do you calculate the maximum slope for this scooter?
The inverse cosine function is written in 3 different ways: arccos(x), acos(x), and cos-1(x).
Likewise for the inverse sine and inverse tangent functions.
If arccos(x) = a, then x = cos(a).
In some near future an Artificial Intelligence (AI) has achieved an error rate of 1 per thousand questions. The Senator for an unnamed state proposes that the AI is now fit for teaching since it is more accurate than many human teachers.
Do you vote for the proposal, given that budgets are tight and the AI costs almost nothing to run.
Event A happens with a mean occurrence of 25 times per day. Event B also happens with a mean occurrence of 25 times per day. Events A and B are independent from each other.
A researcher claims that when she starts looking for events, on average Event A occurs before Event B. After she finds an event of a given type, she stops looking for several hours before she starts looking again.
Critique the research.
Two entrepreneurs visit an all-day coffee shop. Each arrives at a random time between 14:00 and 15:00. Each one of them stays for exactly 15 minutes.
What is the probability that the men will meet at the coffee shop?
Jane and Gerry cut a pizza by 4 cuts so that the intersection of the cuts does not go through the centre of the pizza O.
Gerry takes 4 brown sectors and Jane takes 4 light yellow sectors.
Guess who takes more pizza.
Don't use geometry, just your gut feelings.
In a city, 85% of taxis are yellow and 15% are white.
After a night accident, a witness confirmed that the car was a white taxi.
After a medical test, the experts discovered that the witness correctly reports the colour of a taxi at night only in 80% of the cases.
Estimate the probability that a white taxi was involved in the accident.
In 2022, 92.5% of Swiss train passengers arrived at their destination with no more than a three-minute delay.
According to the train timetable, Peter should arrive at his place of work 3 minutes before his shift starts.
Guesstimate how many times in 2022 he was at least one second late.
A binary search is a very efficient way of dividing up a range to quickly find something.
Examples include:
A) Searching a (paper) dictionary to find a given word.
B) Searching a road for lost keys.
C) Guessing the exact value of a whole number between 1 and 100.
A professional magician has had some special 6-sided dice made to his specification. Rather than being numbered 1 to 6, inclusive, they are numbered 1 to 5 as usual, but the "six" face has been numbered by some value between 1 and 5 inclusive.
A trusted colleague has been given the task of throwing one of the dice multiple times, and then reporting the value on the top face. You are not allowed to inspect the faces of the chosen die, and your colleague is not allowed to tell you anything other than the top face for each throw.
How many times would the rigged die have to be thrown before you were confident which number had been duplicated on the die?
by Leslie Green
The image shows the insides of a UK 3-pin mains plug.
There are three wires: Brown (Live), Blue (Neutral), Green with Yellow stripe (Earth).
There are three terminals, one for each wire.
In how many different ways can the plug be wired-up incorrectly?
Three standard, six-sided dice are rolled.
Find the ratio: (Probability that the three numbers are different) / (Probability that the three numbers are the same).
The construction shown is made from free-standing cubes which are not attached to each other. It sits on a large extended plane.
What is the least number of cubes necessary to make this construction?
We arrange 16 squares with the side length of 1 meter inside a circle with the diameter of 5.66 meters.
Can you rearrange the squares and suggest a design with a smaller circle?
By how much can we decrease the diameter of the circle?
We have the sum of squared reciprocals of odd natural numbers being equated to three-quarters of the sum of squared reciprocals of all natural numbers.
Do you believe the result?
by Leslie Green
A careless worker has randomly cut an expensive Invar rod into three random lengths. That was the last stock of Invar available for several months.
The supervisor has to clear up the mess and quickly supply the customer with four rods, such that at least one rod is the average length of the other three.
Chemical scientists need a shape with the largest ratio of external surface to its largest size.
Consider a structure of nine glued 1-unit squares for simplicity.
The ratio of its perimeter (green) to the diagonal (red) is (4 x 3) / (3√2) ≈ 2.8.
If the length of a glued joint has to be at least one unit, what is the greatest ratio of the shape perimeter to its largest size possible for the 9-square structure?
A vast hoard of pirate treasure has been discovered, consisting only of 5 and 7 doubloon gold coins, with lots of each type.
You are not sure if the person counting the coins is honest.
Which reported sum could not be correct?
The graph represents the first test of a new weighing-scale using a set of calibration weights: 10 g, 50 g, 100 g, 200 g, 200 g. The calibration weights were specified as M1 accuracy meaning: 200 g ± 10 mg, 100 g ± 5 mg, 50 g ± 3 mg. The budget/economy weighing scale reads to 500 g and displays to 0.01 g. It's specification sheet states "accuracy 0.01 g", which is a pretty big claim for something costing £10 (GBP).
At these claimed levels of accuracy it is no good plotting (the actual reading of) 200.45 g for a 200.00 g calibration weight. You can't see the deviation. Here, only the deviation from the expected value has been plotted, that is 0.45 g. The scale is always reading high.
The slope of the best-fit line is given by the equation y = 0.0025x on the chart.
Using only the data presented here, what is a more representative specification for the weighing-scale?
FS = full-scale, R = reading, D = digits, refers to the least significant digits displayed.
by Leslie Green
Alien robots are invading a particular part of the Earth, shown as a 5x5 grid. Each symbol on the map represents a battalion of robots.
For reasons which are beyond the comprehension of your puny human mind, the invasion plan has a maximum of one robot symbol per square on the grid, an equal number of robot symbols in each column, and in at least one row there are exactly three robot symbols.
What is the maximum number of robot symbols that can fit on the grid, whilst still following the given rules?
A father, mother, and their two adult daughters have just finished a game of golf on a huge, and relatively featureless golf course. The youngest daughter heads off at 50 m/minute due south in her hand-me-down GolfPro (Basic) golf cart. The eldest daughter heads off due north at 100 m/minute in her GolfPro Super golf cart.
The mother immediately notices that the father doesn't have his hat, and they suppose that the hat is in one of the other golf carts. What to do? The father says they should go north after the eldest daughter or they will never be able to catch her. The mother says they should go south after the youngest as she is closest. The parents are in a GolfPro Special, which can power along at 150 m/minute, even with two adults on board.
They have no idea which daughter has the hat, so they suppose they will have to plan to meet with both daughters in order to guarantee a safe return for the hat. They take 2 minutes to think before enacting the agreed-upon plan.
Which is the optimum direction to start with?
There is an old wives' tale about domestic central heating: "It should be left on all the time to reduce energy costs", since the system "doesn't have to work so hard" on a cold building.
If there were any truth in this story, to which heating technology would it most correctly apply?
Strict equality is easily achieved with integer quantities. You can go to the shops and buy 3 apples. You are unlikely to go to the shops to get 3.14159 apples (unless you are taking joint cookery and mathematics, and are making apple PI).
Given that you are on this site, it is improbable that you have little interest in STEM (Science, Technology, Engineering, Mathematics) subjects. In this case you need to understand 'adult' language in order to not be treated as an idiot. You may ask for, or demand, an EXACT dimension, quantity, or result, and be upset when this is not (and cannot be) delivered.
Let's consider an example. Will a 2 mm rod fit into a 2 mm hole?
We show two bases of operation, (1) and (2). Each base has a fuel supply, A and B respectively. The fuel required to get from base (1) to base (2) is a. The fuel required to get from base (2) to base (1) is b. There is no assumption made that a and b are equal, since it could be an uphill journey in one case. The units for A, B, a, and b are all the same, for example gallons.
You are allowed to start at either base, but you must complete a round trip (to the other base and back). You can assume that the vehicle used has no fuel initially, and that its fuel capacity exceeds both A and B.
You are told that (A + B) > (a + b).
What is your conclusion?
A young girl plays a game with her grandfather in the Grand Hall. The hall has a regular pattern of grey circular tiles on a 10 foot grid. The grey circles have a radius of 1 foot. They take turns throwing a thin flat plastic disc which has a radius of 3 feet, blindfolded. If the disc hits a wall then the throw is taken again. The grandfather explains that this is equivalent to having an infinitely repeating pattern.
The grandfather gives the girl a choice: Does she want to score by touching (or covering) a grey circle, or does she prefer to score by not touching a grey circle?
She wants to win. What should she choose?
The image shows a freshly planted hazel tree from a reputable source.
Can I reasonably claim to have sustainably replaced a tree cut down elsewhere by this act?
In the 1880s the cost of sending a telegraph message was very high. You paid according to the number of words in the message. The telegraph operator would read the message and tap it out in Morse code over the telegraph wires. This had two problems: (1) it was expensive, and (2) an unethical telegraph operator could sell your secrets to his dodgy friends.
Both problems were solved by using a pair of identical code books, one for the sender and one for the recipient. If you send a four digit code you can send 10,000 unique messages which only you and your colleague can understand. Maybe 0102 means "Meet me at 11 o'clock by the clock tower, and bring the secret papers". With 10,000 messages you can include various different times by using up a small block of these code words for different times.
Why is this excellent scheme not used today?
The image shows a simplified design of a through-truss road bridge, crossing a river. To give a sense of scale, the top beams are higher than the height of large trucks. The trucks drive through the structural members, hence the name.
Suppose this bridge was designed and built more than 40 years ago with a safety factor of 3, by which we mean it is designed to withstand three times more than the maximum possible load. Further suppose that it has been well inspected and maintained. Would it be considered safe by modern standards?
A runway numbered 09 points east (90°), runway 18 is south (180°), runway 27 points west (270°) and runway 36 points to the north (360° rather than 0°).
There are two numbers at different ends of a runway. They have the same digits.
What is one of the numbers?
Solar power is of increasing importance in the transition from fossil fuels to renewable energy. It is therefore something you might like to learn a little bit more about.
The curve shows how the voltage output from a solar cell decreases as you draw current from it.
Given that the power output from the solar cell is the product of voltage and current, which point on the curve represents the maximum power output from the panel.
Here we wish to consider the idealised case of accelerating an electric car up to speed. In this simple case we neglect frictional and windage losses. Imagine, for example, towing the car up to speed either quickly or slowly with constant acceleration. Suppose for example we can either accelerate at some standard rate, or ten times more slowly.
Which method consumes more energy?
The pointy bit is the bullet, and the casing holds the bullet, primer, and propellent ("powder") together.
Suppose some (TOTAL IDIOT) lays such a live round on a flat stone near an open camp fire and the primer detonates. Further suppose that the relative weight of the bullet is 5x greater than the weight of the casing.
Describe the (highly theoretical, idealised) motion of the bullet and casing.
Jerry is living on the edge such that every week his wage is completely spent. Any adverse event will tip him over the edge into ruin. This could be a trip, a fall, a fault with the car, a fault with the central heating boiler, an illness, an accident, an unexpected bill, ... In life, stuff can happen, and you need to be resilient against the bad stuff.
Write the probability of an adverse event during any week as P, with any adverse event being independent from any other adverse event.
What is the chance that Jerry survives for a year without ruin?
Banks typically charge interest on loans, overdrafts, and credit cards on a monthly basis, probably because salaried staff also typically get paid monthly. You may then find the cryptic abbreviations AER (Annual Equivalent Rate) or EAR (Effective Annual Rate) on statements of Interest paid.
Suppose you pay a certain monthly interest rate, R, where 2.1% per month would be presented as R = 2.1
How would you convert R into the Effective Annual Rate using the same sort of number format? EAR uses compound interest.
(We would write an EAR of 24.3% as EAR = 24.3)
The little ferrite ring magnet is glued to the wooden rod. The wooden rod is just resting on the eraser, and the eraser is just resting on the table. The wooden rod and magnet pair can be balanced as shown between the two ferrite magnet blocks.
Which is the correct reason for the ring magnet being suspended in the air like this?
A single-lane road bridge is just wide enough for a standard car. We will say that a standard car is 2 m wide, 5 m long, and weighs 2 tonnes (2000 kg).
For the sake of comparison we will consider a standard person as being 50 kg, and fitting within a square of 0.33 m on a side when people are standing close together.
Using this simplified model we would like to compare the loading on a bridge of either cars, bumper to bumper, or people standing packed together.
Which situation gives the worst average load on the bridge?
The picture shows a map of a part of a country with its states colored in different colors.
What is the least number of colors, so that no two adjacent states have the same color?
It is difficult to derive a complete mathematical proof, try to guess.
A laboratory receives blood samples of 30 patients. None or only one of them is seriously ill. The laboratory needs a very small quantity of blood for a test. The test sets are extremely expensive.
What is the least possible number of test sets that can be used to find the sick person?
We suppose (without even a shred of evidence) that the peak electrical demand from the power grid will occur on winter evenings when people come home from work and start cooking. Suppose this is between the hours of 5 pm and 6 pm.
Which renewable power source is able to fulfil this requirement most reliably?
In computer programming it is useful to have randomly generated ID values. If each character position is a digit from 0 to 9 you only get ten values per character. If you use hexadecimal you get 16 values per character. If you use upper-case letters you get 26 values.
How many values do you get if a character position can be either a digit or an upper-case letter, given that the ID needs to be accurately human readable, and some letters and digits are too easy to misread?
Suppose the year was 1971 and you wanted to find out about dioxin.
How would you have gone about it?
Penelope and Hiram Hackenbacker go to the orphanage to choose a new addition to their family. There are 100 apparently suitable children, but one of them is known to be entirely unsuitable due to being a random compulsive liar. Sadly the records have been destroyed under suspicious circumstances.
It is known by hearsay (since the records have been destroyed) that the compulsive liar has a 50% probability of lying on a particular question when averaged over a large number of questions. The significance of the question is irrelevant since this is a random compulsion. The other children do not know who the liar is. The staff are contractually obliged to not cooperate since they are sworn not to harm the life-chances of any child.
The Hackenbackers wish to reduce the chance of picking a liar to below 1 in one million. They are allowed to question the children, but only YES/NO answers will be provided. All other children will answer truthfully.
Which method is both successful and uses the minimal number of questions?
A pair of identical twin sisters, Saintly and Sinnerly, even though they are only 8 years old, have very different characteristics. Saintly always answers truthfully. Sinnerly randomly lies half the time when measured over a long average.
You come across one of them, but are unsure which one. How many questions (with YES/NO answers) do you require to give you a better than 95% chance of having correctly identified the sister?
Why do concrete structures such as bridges and high rise buildings seem to fail with such alarming regularity?
The picture shows an abandoned bridge made from concrete.
In a modern 'first world' country, is it always reasonable to suppose that the drinking water is safe to drink, given that there have been no dramatic weather or seismic events recently?
Most people acknowledge that there is a climate crisis, but actually doing something about it is more problematic.
Which is the least valid reason?
Because of staff shortages, the school workshop class is being supervised by a mathematics teacher. Dick is living up to his name (being a nuisance) so the teacher gives him two random lengths of rod with this instruction: "Cut completely through one of the pieces such that one of the three pieces is equal to the mean length of the other two pieces."
How does he do this?
During the COVID-19 global pandemic of 2020/2021 (weak) efforts were made to prevent the spread of the virus by making it illegal to move from place to place without an important reason. Truck drivers were exempt from this rule because food, vaccines, and essential supplies, still needed to travel. This meant that the lockdown was very 'leaky', with lots of virus still spreading around from country to country, and county to county.
Since you are such a genius, how would you consider handling this problem for the inevitable next outbreak?
If you calculated the sum
s = (1/2) + (1/3) + (1/6)
by hand you would (hopefully) get 1. If you then subtracted 1 from the result you would get zero. If you did this set of operations on a calculator you would probably get the same result. The designer of the calculator has protected you from errors.
It is very helpful to be able to check calculations using a computer. Now there is nobody holding your hand, and protecting you!
If you know any computer language at all, it should not be difficult to read the examples shown. We assign values to variables x, y, and z, then sum them. The answer is less than 1, even though if you printed the sum in BASIC it might well display as 1. When you print the difference in BASIC between 1 and the sum you get 1E-16. When you do the same thing in C you get a difference of 1.
Speculate as to why.
The image shows the cross-section of two identically shaped solid circular cylinders, with tapered holes along their axes shown to an exaggerated scale.
The low density (pink) material is held under the water (blue) by a fishing line anchored below the image. The high density (green) material is supported by a fishing line anchored to some superstructure above the image. The tension in the fishing lines is the same.
Which statement is incorrect?
by Leslie Green
Self-styled 'citizen scientist', Thickus Tooplancks, has revealed to his many social media followers that the Archimedes principle is false. There is in fact no Archimedean upthrust when you lower a brick into a bucket of water.
Thickus weighs a bucket 3/4 filled with water: 5 kg. The brick weighs 1 kg. The bucket is on a digital kitchen scale.
When the brick is put into the water, none of the water overflows, and the weight is 6 kg. Clearly the brick is not any lighter in the water.
Why?
Suppose it was reported in the media that somebody 'dropped dead' 15 minutes after receiving a dose of COVID-19 vaccine.
Is it wise to then be hesitant about having that vaccine, given that more than 15 million people have already received a first dose?
A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 10 minutes of walking. The second door leads to a tunnel that will return him to the similar mine with 3 doors after 20 minutes of travel. The third door leads to a tunnel that will return him to the similar mine after 30 minutes. The miner has no idea which door is the safe exit, but he knows that the exit is in 10-minute walk.
What is the expected time until he reaches safety?
We wish to traverse this table from 6 at the top, to 46 on the right, collecting the box contents on the way. We can only move down or right at each step. There are plenty of ways to traverse the table, and each path gives a guaranteed collection of integers, collected in strictly increasing order. But look near the top, to box 8. We can go right then down to 12, or we can go down then right to 12. Both paths collect a 10 from a different location. The paths are distinct, but the resultant collection of integers (an array) is not distinct.
The diagonal pair of equal values, shown in grey, is messing up our simple counting method.
How many of these diagonal pairs are there?
by Leslie Green
The fake-psychic and her assistant have baffled the current batch of incorruptible but naïve investigators with their psychic mind-reading parlour trick.
There is a 2 x 2 board with comically-large discs (1 foot / 30 cm dia) which are red on one side and blue on the other. The assistant investigator sets up a random pattern of the four discs, and overtly hides a little slip of paper under one of these discs.
Having seen where the slip of paper has been hidden, the assistant psychic turns over one of the discs in order to couple his psychic energy into the Kirlian-aura surrounding the experiment. The assistant investigator nudges the flipped disc to ensure the position has not been set in some secret prearranged way. All of this happens on a glass table in an otherwise empty and fully transparent sound-proof booth.
The pair exit the booth and go into a nearby windowless room where the assistant investigator presses a button to signal her boss. On being signalled, the investigator takes the psychic to the outside of the glass booth. After 5 minutes of strenuous psychic effort, and with much subtle manipulation of the vortex energy, the psychic reveals the number of the square under which the slip of paper has been hidden. This is confirmed (or not) by the assistant investigator in due course.
The investigators have eliminated all possible additional communication routes between the two groups, and after 100% success on 100 of these double-blind experiments, they are convinced that the psychics, who are being paid handsomely for their contribution to science, are the real deal.
How have they faked it?
Fully charged battery packs are in short supply, so the quartermaster has been putting near-empty packs back in the storage lockers to make it seem as if there are plenty available. It means soldiers are randomly being sent out on missions with faulty kit. Your friend is first out on a mission tomorrow, and will get to randomly choose between one of the two storage lockers, and to pick a battery pack at random from that locker. He will not be able to detect a duff battery pack, and you cannot communicate with him, or mark the good packs in any way. You also can't throw out the duff packs as the quartermaster will notice.
Right now you don't know how many packs there are, and how many of these are duff. You can sneak into the storage shed tonight with a battery pack tester and move battery packs around between the two lockers, but you cannot tell your friend what you have done (and you have made no earlier plans with him).
What is your plan to give your friend the best chance to get a charged pack?
Angela, Bethany, and Carol all do the same type of work, and to the same exacting quality standard. Their work rates are independent of who they work with, and are constant throughout any particular project.
Working together, Angela and Bethany complete a project in 1/3 of a day.
Working together, Angela and Carol complete an equivalent project in 1/4 of a day.
Working together, Bethany and Carol complete an equivalent project in 1/5 of a day.
If all three work together, how long will the project take?
Your mission is to take at least one vehicle through the wild taiga forest (but two are available, if needed) to urgently deliver some medicine. All the choppers are grounded. Abandoned vehicles can be safely recovered later when the panic is over.
The undergrowth is almost impenetrable, so even the formidable Taiga Beast uses all its fuel in 60 miles of smashing through the forest. However, any secondary Beast in a convey only uses half as much fuel, since it is not doing the whole fuel-hungry taiga-crunching thing.
There are no fuel cans left on base, but you can transfer fuel directly from one Beast to the other.
How far can you get?
We’re Oscar Mike in 10!
Alex and Betty share what their grandfather left them:
1. $400,000 in cash.
2. A country house with a forest. Alex values the house at its market price of $500,000. Betty loves the neighborhood, so she values the house at $560,000.
3. A vacation house that Alex can use for fishing. Betty values the vacation house at $200,000, while Alex at $240,000.
4. A portrait of the grandfather, which Alex values at zero, while Betty values at $20,000.
What would be the total value of the wealth to fairly share?
A particular casting process has a very poor quality result, such that 50% of the castings are defective in some way. They get manually inspected and 90% of the defective castings are thrown into the reject bin.
What is the final defect rate?
There is a climate crisis. Too much 'greenhouse gas' is going into the atmosphere, causing climate change. Something needs to be done, but all proposals are not all equally good. You need to do your own research to evaluate what is the right course of action for you, rather than believing the claims of salesmen. The modern world is no place for a technophobe (a person who fears, dislikes, or avoids technology).
Suppose your old gas boiler is on its last legs and you have read that a ground-sourced heat pump is the green alternative. Which statement is the least misleading?
Billions of dollars of technology, and some dumb sap still has to parachute in to a tropical rainforest and manually reboot each of the 40 nodes in the defence grid. Not only that, but once started, they have to be rebooted in a vaguely clockwise sequence as shown on the map. And then you have to return to the starting point to complete the start-up sequence.
Each of the (all unmanned) outposts with a green number against it has a vehicle, and that number of fuel cartridges. One cartridge will get you to the next outpost with no energy left over. The outposts are far too far apart to make it on foot.
There are 40 outposts, and 40 fuel cartridges, so the mission is possible. There are 15 outposts with fuel, so you have a 1 in 15 chance if you pick randomly. This is not some sort of kid's puzzle where you get to pick one of 4 options!
Pick from the 15, and make sure you are right. Rebooting the defence grid is already Plan B. There is no Plan C.
Listen up! Stuff just got real. You and your vehicle are being dropped in to 1 of 17 outposts with enough doses of vaccine for each. Fuel is a problem though. Your armoured vehicle can just travel to the next outpost on 1 tiny fuel pellet. The fuel capacity of your vehicle is hundreds of fuel pellets, but the zombie outbreak means fuel is in short supply. You will be dropped with no fuel, and have to collect it as you go.
You need 17 pellets to cover the whole trip from wherever you land all the way back around to the drop-site outpost, the outposts being arranged in a sort of circular pattern. You know that the total fuel available at all the outposts is exactly 17 pellets, but it is not evenly distributed between the outposts.
If you know the fuel distribution, and pick the drop site, can you guarantee to make it?
The BioNTech-Pfizer COVID-19 vaccine (against the SARS-CoV-2 virus) needs to be transported at -70°C. This is done by packing the vaccine in insulated boxes with dry ice (solid carbon dioxide), assuring a temperature below -78°C as long as the dry ice remains.
For a given volume ratio of dry ice to vaccine, compare the transport storage times for individual cubical boxes of a small size, or twice (2x side length) as big as that, all else being equal.
In the UK, December 2020, up to 2 million university students were potentially going home for Xmas during the middle of the COVID-19 pandemic. A new fast COVID-19 test had been developed to spot the causative virus, SARS-CoV-2. These lateral flow tests had the characteristics shown in the image. At the time, around 1% of the population had the virus within the community.
The scientific advice, given on prime-time news channels, was that a pair of negative tests meant it was safe to go home, as a negative test meant a 99.75% chance of not having the virus.
Was this true, and good advice?
A computer-controlled automatic paint sprayer is required to spray the orange bar with a uniform thickness coat of paint, starting from one end as shown. The back rectangular outline represents a set of small paint nozzles fine-tuned such that there is a uniform thickness of paint applied over that rectangular area.
Describe the motion of the spray head required to coat the bar uniformly as required.
In one type of typical central heating system, water is heated in a boiler, circulated around the house using pipes, and gives off heat at particular places using 'radiators'. (We put radiators in quotes because of the three primary heat transfer mechanisms - conduction, convection, and radiation - a domestic central heating 'radiator' dominantly uses natural convection to heat the room.)
Obviously the water in the pipes needs to be hotter than the desired air temperature in the room, or heating (of the room) cannot occur.
What is the strongest (but still accurate) statement you can make about the transfer of heat to the air in the room.
In order to fit the answers within the confines of the buttons, we have resorted to the short form dT to mean the temperature difference between the water and the air.
WARNING! If you correctly decode this message within 1 minute you may be automatically recruited by the CIA/MI6 as a potential cryptanalyst.
What does it mean?
A disgruntled postal worker has 21 letters to put into 21 sorting bins. If correctly sorted, there should be one letter per bin. Being disgruntled, he is careless of which letter goes where.
What is the probability that exactly one letter is placed in the wrong bin?
The BioNTech-Pfizer COVID-19 vaccine-candidate (BNT162b2) interim results from 8 Nov 2020 showed 38,955 participants in a placebo controlled double-blind trial.
94 participants became "evaluable", which we presume to mean they showed COVID-19 symptoms. The analysis presented was that the vaccine efficacy rate was above 90%.
What is the maximum number of (genuinely) vaccinated people who showed COVID-19 symptoms?
A piece of string is 1 m long. It is cut into two pieces, such that one piece is twice as long as the other.
We then repeat this process:
(1) Find the longest piece of string in the whole collection of pieces of string
(2) Cut it into two pieces, with one piece twice as long as the other.
After five cuts, how long is the longest remaining piece of string?
A novice designer has drawn the part shown, with no idea about how it can be made as a prototype in aluminium. Typically you make such a part by milling from a solid piece.
An end-mill is a tool similar in some respects to a drill bit, but with a flat end rather than a pointy end. It is spun at high speed and makes a circular hole like a normal drill. The clever thing is that having 'drilled' down (plunge cut) you can now drag the end-mill sideways. If the z axis is the into-the-page direction, you can slide the end-mill in the x and y axes to make slots and pockets in the material.
If the pocket is a 50 mm x 50 mm square, roughly how much material is left behind when using a 10 mm diameter end-mill to mill out the pocket?
How many of these shapes can be drawn with a wide felt-tipped pen, given that we should not lift the pen from the paper, we should not substantially redraw over existing lines, and we don't look at the result too closely?
This disc has a hole in it, the diameter of the hole being sqrt(10) times smaller than the outer diameter of the overall disc. The hole center is offset from the disc center by one quarter of the disc's diameter.
By symmetry, the Center of Gravity of the part is on the marked horizontal axis. How far from the disc center is it?
In the image we have a weight of 1 g at a distance of 1 cm from the pivot point. Then we have 2 g, 2 cm from the pivot point. We continue in this fashion until we reach 300 g, 300 cm from the pivot point.
As it stands, the left side is pushed down. By sliding the weights along so that the Nth weight from the right is on the pivot point, the load is fairly well balanced.
What is N?
The strong and rigid blue beam rocks freely about the grey triangular pivot point. The heavy and rigid green bar has uniform density and cross-section. It is freely suspended by the light red ropes as shown.
A weight is required at either A or B to balance the beam in the horizontal position shown.
Given that the weight of the green bar is M units, what weight is required, and at what position?
What do you consider to be the most important characteristic for the project leader of a large public-aware cutting-edge project?
We consider mission failure of our new rocket system solely based on the failure of the control system, which has been assessed as having a probability of failure of 1/1000 on an individual mission. Since we consider this as unacceptably poor, we put in an identical system as a backup. Switching between the two systems is considered to be error-free.
What is the new probability of mission failure?
This is a strange looking drawing. We have a concept, or an idea, drawn as a C in a bubble. We then push or pull on this concept with virtual vectors. The idea we are trying to convey is that the boss has an idea which she thinks should progress in a certain direction. Your comments could be considered as being in one of the four marked directions.
Which direction should be considered as being least helpful?
A particular comic book super-hero is actually a guy with a regular physique inside a high-tech metal suit which gives him the ability to fly, lift heavy objects, and get smashed into concrete at high speed without being injured.
We suppose that he can lift 10x more than a normal person, can fly at 100 m/s (224 mph). If smashed into concrete at 20 m/s, the penetration dent in the concrete is estimated to be 10 cm deep.
Neglecting technological considerations, is there any reason why some of this is fundamentally not possible?
A train is carrying a full load of birds in open wire cages. The weight of the birds is considerably more than the unladen weight of the train.
In the event of an emergency stop situation, the driver has been told to sound the horn prior to braking so the birds take flight. The theory is that their inertial load is entirely removed so the train can brake much faster.
Comment on this plan.
Suppose that 400 technicians are required to replace 400 ordinary electricity meters with 400 smart meters in 400 homes in 200 days.
All else being equal, how many technicians are required to replace 40,000 ordinary electricity meters with 40,000 smart meters in 40,000 homes in 400 days?
Obesity (excessive weight) is common-place amongst both children and adults in developed countries.
Which theory best explains why?
The image shows information from two separate websites. The FACT CHECKED page shows the total 2017 revenue of Pfizer as $52,540 billion.
The red box is a snip from an entirely unrelated website.
(GDP = Gross Domestic Product)
Comment on the figures.
A hollow copper pipe is shown in brown. A hollow rubber tube (green) fairly easily slips over the copper pipe. However, a mistake has been made in the assembly, as a washer needed to be fitted over the rubber tube before it was pushed onto the copper pipe. The rubber tube cannot now be pulled off.
Why not?
The image shows a simple experiment, demonstrating that a beam with a rectangular cross-section is much more rigid when the force is applied parallel to the longer face than when the force is perpendicular to the long face.
Your architect has specified a steel beam which has a depth, d, which is twice the breadth, b. The builder has fitted it incorrectly, so the load goes through the weak face. When you mention it to the surly builder, who is much larger than you, and who will lose his bonus if he reworks the beam position, he tries to convince you that it will make very little difference.
What is the true situation?
WARNING: Do not try bending any beams like this without using eye protection. Certain rulers are known to shatter and eject bits of plastic into nearby eyes.
The image, which is not exactly to scale, is intended to show the cross-section of proposed support towers for a wind turbine. The first tower is solid, and the second tower is hollow, but with twice the external diameter. The idea is to reduce the weight whilst increasing the rigidity. Both towers use the same material and are the same height.
What is the internal diameter, d, of the hollow tower, given that we require a factor of two reduction in weight?
We work in multiples of the diameter, D, of the solid tower.
The following game has been designed to train traders of shares and FOREX (FOReign EXchange).
The dealer shuffles the 6 playing cards shown, and places them face down such that neither of the two players knows which card is where.
The dealer shows one card only to Player 1, then keeps that card separate from the rest.
Player 1 can now pick a card from the remaining set for a fee of $1. If the card chosen is red he gets $2. He can alternatively let player 2 choose.
The above situation constitutes 1 game.
Traders use a special metric to determine profit.
PROFIT FACTOR = (gross profit) / (gross loss)
If the profit factor for a large group of trades is 1 then no profit has been made.
What is the expected Profit Factor for Player 1 over a large number of games (supposing that Player 1 is adequately smart)?
Which of these examples best represents amplification:
A Victorian ear trumpet, which was a horn that tapered down to a small earpiece and which amplified sound so that a partially deaf person could hear.
A pulley system attached to the ceiling which amplifies your strength so that you can lift ten times as much weight as you could do without.
A gear system on a bicycle which amplifies the force from your legs so you can go faster.
You are having a bad day. First the 1.5 mm drill you were using sheared off, leaving the broken part jammed in the metal.
In a second attempt, having drilled a 2.5 mm hole, the self-tapping screw went in only so far before the head sheared off. (The matching screw on the other side went in fine.)
You go on to a forum asking for help to remove the screw.
Pick the best answer.
You can easily imagine a wall built of bricks, where the wall has a constant thickness. At some height the wall cannot be built any higher because the bricks at the bottom will be crushed by the weight above.
Starting from some standard size of brick, can the wall be built taller using smaller bricks of the same material, or larger bricks of the same material?
Imagine that smaller or larger in this context is a linear factor of 1.5 in all directions.
A piece of wood with a square cross-sectional width, W, sits flat on a square shallow tray with an internal side length of S.
What is the maximum length of the piece of wood?
An inventor and his two junior partners wish to sell their latest invention to a businessman who is widely believed to be honest.
The inventor comes up with the following formula to value the invention:
"Find the lowest value, below which you would feel that you were robbing us.
Find the highest value, above which you would feel you would be robbing yourself.
Give us the mean of these two extreme values."
The businessman agrees that this is fair, but then asks if the inventors wish to receive the arithmetic mean or the geometric mean.
Which should the inventors take in order to maximise their gain?
Suppose you have just tested positive for a highly contagious virus such as COVID-19. The virus spreads directly by breathing in virus particles breathed out by an infected person, or indirectly by touching a virus-contaminated surface then typically transferring the virus to your nose, eyes, or mouth.
What is the safe course of action?
The image shows some 47 nF ±5% capacitors. 20 were bought, measured, then exactly matched using a computer program.
The 4-digit capacitance meter used only displayed 3 digits since it was set to read 200 nF full scale, but each reading was perfectly stable, without any noise or drift.
N5 + N6 was 47.1 + 47.4 but read 94.6 when the pair were connected together.
N12 + N15 was 47.4 + 47.1 but read 94.4 when the pair were connected together.
We want an optimally matched pair. What do we do?
by Leslie Green
The image shows 10 capacitors, with their measured values on the tape.
From this 10 it is required to select 4 values, say A, B, C, D, such that the maximum ratio between
A, B, and (C+D)/2 is minimised. To be clear we end up with three values which have to be as close together as possible.
The values are: 100.4, 100.3, 100.1, 100.5, 100.7, 100.3, 100.7, 100.8, 100.5, 100.5
What is the best that can be done with these components alone?
There are 10 new resistors shown in the image. They are specified as being 15.80 K ±0.1%, where 1 K is 1000 ohms.
All 10 of them measure as exactly 15.65 K with the multi-meter shown.
What can you say with certainty?
A capacitor is a type of electronic component. Five of them are shown in the picture.
The manufacturer cannot make them with an exact value, so they are marked as being in the range ±5% of the stated value.
You have an accurate capacitance meter to measure them with, and you wish to select a pair which are as close as possible in value.
Assuming that you are infeasibly unlucky, how close will the best match be?
The little child has painted her hand in bright non-washable red paint. She then opens father's expensive text book at three distinct random positions and places her handprint on one of the visible pages each time. The paint dries quickly so the pages are not stuck together by the paint, and the paint is sufficiently thin that the pages are not visibly distorted.
Father comes along and sees her, and the paint, with his precious book nearby. He opens the book at three distinct positions, these positions being randomly chosen.
Estimate the probability that he finds any paint marks inside the book, given that it has roughly 200 pages.
Some materials are "magnetic" in the sense that they are attracted by magnets.
Iron and steel are the most common magnetic materials.
Copper, aluminium, brass, and most other metals, are not attracted to magnets.
Which material will not affect a magnet based on the magnetic interaction alone?
by Leslie Green
You have found a secret message, and your spies tell you that it has been encrypted using the code table shown, where each plain-text letter is represented by three cipher-text letters. Your spies have also found out that the word "SECRET" has been encrypted as "TUV FAB DAB QLK GFH XSW"
Which plain-text corresponds to the cipher text "PKN HCF SYX"?
A physics student has decided (for no clearly defined reason) that she does not want to eat the same pair of meals on successive days. For example if she ate cod & chips yesterday, and pizza & beans today, she does not wish to repeat that pattern, in any order. She has only 5 meal choices.
The first day of her plan is "day 1".
On what day does the last possible meal on her plan get eaten?
Two slices of bread either side of a fried egg is an egg sandwich. Two slices of bread either side of grilled sausages is a sausage sandwich.
By (exaggerated) extension, put two slices of bread on the ground at diametrically opposite sides of the earth and you have an Earth sandwich (according to university student Etienne Naude at least).
Etienne placed his bread on the ground in Auckland, New Zealand, at Latitude -36.86377°; Longitude 174.90031° (Jan 2020).
Where did Etienne's confederate place his bread in order to make the sandwich complete?
You will probably have seen rocking chairs, but the glider chair (shown in the picture) with its pivots and linkage arms is not nearly as common. Nevertheless we can ask the same question of both.
Is it possible to produce the rocking motion without having your feet (or any other connection) on the ground?
The car park by the railway station has 100 places, a daily parking fee of $20, and a fine for unpaid parking of $1000. The current daily takings from the ticket machine are $1600, despite the fact that the car park is always full. The town council considers re-instating a parking warden at $20/hour.
Pick the best scheme, and its likely outcome after a month or so of operation:
The Travelling Salesman Problem is a famous problem in computational complexity. The idea is to find an optimal path for a salesman to visit each location just once, starting and ending at the same point. In the image shown, the locations (nodes) are connected to either the closest point (shown as a red link) or the second closest point (shown as a green link). This is computationally trivial. The hard part is finding the shortest path which visits each location once, since every single possible map has to be tried.
On a particular computer, using a particular program, the following simulation times were reported:
11 points: 0 seconds
12 points: 3 seconds
Estimate the reported simulation time for 13 points.
by Leslie Green
My life is hard. Every day I have to walk 1 mile to the village center, collect water, and return home. The return trip is often slower because of the increased load.
Each day I need 6 buckets of water which I can do in one of three ways:
6 round-trips with 1 bucket; 3 round trips with 2 buckets; 2 round trips with 3 buckets. I cannot carry more than 3 buckets of water.
With one full bucket I can maintain my normal walking pace. Empty buckets do not change my walking pace.
Assuming that I can maintain my pace from trip to trip, how much slower can I walk with 3 buckets, and yet still be faster than doing 6 round trips with one bucket?
From what point do you get the largest sum of the shortest distances to each of the three sides of the right triangle?
There has been an accident on a major road, and the road has been blocked for so long that a 6 km tail-back of stationary cars has been created. We shall assume that an average car is 4.5 m long, and that on average drivers leave a gap of 1.5 m between cars when stuck in a traffic jam. We shall further assume that it takes a driver 6 seconds to start moving when the car in front starts moving, and a further 6 seconds to slowly creep up to 20 mph.
How long after the road has been cleared does it take for the last car in the traffic jam to start moving?
You have crash-landed in a flat featureless radioactive desert at night in the middle of a sandstorm. You have a sketched-map (not to scale) of this desert (as shown in the image) but you have no idea of your whereabouts within this desert. You must get out of this radioactive wilderness quickly!
Your only directional device is a hand-held inertial navigation unit. Whichever direction you decide to take can be set as "0", and you can then walk in this straight-line direction. You also know exactly how far you have walked along each path. You can make exact left or right turns (only) relative to whatever direction you are heading in. Even if you are within 1 meter of the boundary you will not know it.
Using an optimal strategy, but being as unlucky as possible, estimate how far you have to walk to escape from this wilderness?
Adapted from the Bellman Forest problem. by Leslie Green
On an island, every 16th mathematician is a philosopher.
Every 4th philosopher is a mathematician.
Every 2nd philosopher is a writer.
Every 4th writer is a philosopher.
Can you find the relation of the numbers of mathematicians (M), philosophers (P), and writers (W)?
A committee is in deadlock over the choice between three equally-supported important and mutually-exclusive plans. The intention is to put three balls in a bag and pick one at random. The Red ball means plan A; the White ball means plan B; and the Blue ball means plan C.
Precautions taken:
(1) The balls have the same diameter.
(2) The balls have the same mass.
(3) The man pulling the ball from the bag is totally unclothed apart from swimming trucks.
(4) The bag is opaque, as seen by holding it up to the sun.
(5) After the ball is picked, the bag contents are emptied out to show that only two remain, and are of the expected colors.
Is this now guaranteed to be a random choice?
A large corporation has made good profits this year, so a $1,000,000 bonus pot is shared between the managers.
The top manager is the CEO.
Below her there are 5 managers, each of whom gets 1/10th as much as their manager, but never more than that.
This pattern repeats for at least 20 levels of management.
Any bonus is always truncated down to the nearest dollar, so if the calculated bonus was $3.95 they would get $3.00 exactly.
Estimate how much the CEO gets.
A drinks dispenser has three labelled buttons: JUICE, SODA, and LUCKY. If you press LUCKY you are supposed to get either juice or soda with equal probability.
Each button is wired to exactly one of these functions, but each function is not necessarily wired to exactly one button.
Assuming that you are clever, but unlucky, how many button presses are required to establish exactly which button does which function?
If the probability of meeting a wolf in the forest in 3 hours is about 0.1, what is the probability of meeting a wolf in the forest in 1 hour?
Farmer John has 4 piglets in a pen. The piglets kept ‘escaping’ between each weighing, so we don’t know which piglet was included in each weighing, although we are confident that each piglet has a distinct weight.
One of the piglets weighs 3.3 kg.
Two of the piglets together weigh 7 kg.
Three of the piglets together weigh 10.1 kg.
All four piglets together weigh 13.6 kg.
What is the weight of the heaviest piglet?
by Leslie Green
Our pond is 6 m by 8 m and has been infested with an invasive species of pond-weed. This weed grows in a radially symmetric pattern (circular) from its starting point, but stops growing just where it touches the banks of the pond. Starting from the exact center of the pond, the infestation has only just completely covered the pond by the end of the 15th day.
If unchecked, the weed would have doubled the covered area every day.
What area was covered at the end of the 13th day?
by Leslie Green
One train is three times longer than the other train. The trains, running in opposite directions, and each at a speed of 60 km/h, completely cross each other in 6 seconds.
What is the length of the long train?
We suppose that a bathroom accumulates grime at a rate of one unit per week if it is not cleaned. We further suppose that this process is linear so that after one year it has accumulated 52 units of grime.
The new tenants have inherited 1 year’s worth of grime, but cannot confront the idea of cleaning it fully. However, they do feel able to clean off half of the remaining grime each week, always cleaning at the start of the week (which is also the day they arrived). You should consider grime to be a continuous rather than a quantised phenomenon.
How long does it take to get the grime down to less than 3 units?
Having moved to a region with a more temperate climate, you are more used to driving in snowy conditions than the locals. You therefore drive a car with full winter tires, and an anti-lock braking system, because that is normal where you are from. The locals seldom see heavy snow, and are less well equipped.
Are you therefore now very safe driving in this new environment in snowy weather?
The King's army marches at 5 mph for 12 hours each day, from 6 am to 6 pm, the other 12 hours being spent sleeping, cooking, and eating. At 6 pm every day a messenger is dispatched back to the King's castle, from which the army started its march. The messenger rides at an average speed of 10 mph all the way back.
How often do the messengers arrive back at the castle?
I wish to drive through the city, starting from the red point on the map, and returning to this same point. I will never drive down the same road more than once, although I am willing to revisit intersections. I am only able to make right turns.
What is the maximum number of right turns I could make?
According to a theorem by Bachet, any weight up to 13 kg (in 1 kg steps) can be found using balance scales, and weights of 1, 3, 9, and 27 kg.
Given that each of these weights is inaccurate by up to ±1% of its weight, which amount of grain to be measured could have the largest error?
A full jar of compote (stewed fruits) weighs 1050 grams, and the same jar half full weighs 650 grams.
What is the weight of the empty jar?
On the right, in yellow, we have Santa's sledge (sled). The reindeer are nowhere to be seen. On the left in red we have the Tooth Fairy's sledge, having run out of pixie dust. Both sledges are sitting on a flat and level frozen lake. Between the two sledges, but attached to Santa's sledge, is a gas-operated pneumatic ram, which when activated will push the two sledges apart with great force.
To be very clear, the pneumatic ram acts by pushing between the sledges; there is no gaseous release, as found for example in a jet engine.
Given that momentum is conserved, and that the total weight (mass) of Santa's sledge is M, which of the available selections of weight for the Tooth Fairy's sledge imparts the maximum kinetic energy to that sledge?
(Remember that Kinetic Energy (KE) is proportional to the product of the mass and the square of the velocity.)
by Leslie Green
A 100 kg novice male ice skater (M) is travelling in direction A.
A 50 kg slightly less inexperienced female ice skater is travelling at twice the speed in direction B.
They collide in the center of the chart, and grab hold of each other to prevent themselves from falling over.
Given that momentum is conserved, in which direction do they then travel?
by Leslie Green
The image shows a rainwater barrel with a diverter taken from the gutter downpipe.
Look at the circled region; the tap, which screws into the side of the barrel, is probably at an angle of 30° to the vertical. The tap came with the barrel, and is of an all plastic design. It has been tightened up as much a possible, and ended up in this annoying position. There is a rubber sealing washer between the tank and the tap. There is no locknut or position adjustment of any kind.
How should I fix this problem?
by Leslie Green
A careless pharmacist is putting multi-mineral tablets into a bottle when he is interrupted by a rush order for some powerful medicinal tablets. He is working in dim light and his eyesight is poor because he forgot his glasses today. In the rush, he knocks over the medicinal tablets so they get mixed in with the multi-mineral tablets.
There are 50 multi-mineral tablets and 50 medicinal tablets all mixed up. The pharmacist can't tell the difference between them, so he randomly puts the tablets into the two separate bottles.
Estimate the probability that he does this correctly.
by Leslie Green
Two empty cube-shaped cardboard boxes are balanced perfectly as shown. The side length of the large one is double that of the small one.
The distance from the center-line of the large box to the balance point is 1 m. What is the distance between the balance point and the center-line of the small box, given that the thicker walls of the large box have double the mass per unit area compared to the thinner walls of the smaller box?
When a uranium atom captures a slow neutron it disintegrates into smaller atoms and releases energy. This is an example of nuclear fission.
When hydrogen atoms combine to form helium, energy is also released. This is an example of nuclear fusion.
How can it be that making a molecule smaller (fission) and bigger (fusion) both release energy?
by Leslie Green
The image comes from the assembly instructions for a flat-packed self-assembly wardrobe (after minor editing, and the addition of some colour).
The instructions say "Square up the unit by making sure the measurement x to x equals y to y."
The width was 75 cm, and the height was 180 cm. We estimate the measurement uncertainty to be ±1 mm for each measurement (neglecting any error in the tape measure itself).
Estimate how far off-square it could be when following their instructions.
(Defined in terms of the lateral deviation of the top relative to the bottom.)
As a thought experiment, we wonder about accidentally dropping a tin of paint onto a hard surface.
If the tin was almost full, or almost empty, would there be any difference in the likelihood of the lid coming off?
(We assume that the lid opening forces are equal in both cases.)
by Leslie Green
We suppose that the probability of a particular aircraft sensor failing during a flight is 1 in 1,000,000 (1E6). This seems too low, so we include two identical sensors.
What is the probability that both fail during the same flight?
by Leslie Green
Untidy grass and weeds is apparently a first-world problem. This is a scan of an advertising leaflet from 2019, edited to remove the more obvious branding information. All three string trimmers ('weed wackers') are by the same manufacturer. The two on the left are mains powered, whilst the one on the right is battery powered.
Using the information given, and your powerful intellect, what could you reasonably conclude?
by Leslie Green
You go to work by train every day so you buy an annual ticket costing $6000. Your fixed monthly expenditures on rent, food, and repayments for the sofa, TV, and so on, account for 95% of your net monthly income of $6000.
If the train ticket price increases by 10%, what is the effect on your disposable income (the left over part every month) when averaged over the year?
by Leslie Green
Heidi is a petrol-head, a total speed-freak. Whilst tuning up the quad-turbos on her hot rod, she wonders how much thrust her dream rocket-powered car would need in order to go faster than if she just pushed it off a cliff. She imagines that she could get the gross weight of the car, rocket, fuel, and herself down to 1000 kg.
Earth's gravitational acceleration is around 9.8 m/s2.
by Leslie Green
Bettie the Bug comes from a long line of red bugs, although she had a grandmother who was mostly blue. The grandmother came from a long line of blue bugs, although she had a grandfather who was a red bug.
Given that for these bugs, genetic material comes half from one parent, and half from the other, what value will Bettie calculate as her genetic blueness?
by Leslie Green
A year ago you did a last-time-buy on a discontinued component, and now you have 10,000 of these in stock with a cost price of $100,000. (Each unit uses one of these components.)
A new (fully qualified and tested) design is now available and saves $500 per unit. The best price you can get for the discontinued components is $1 each. It is around 1 month to the end of the financial year.
What is your best course of action?
by Leslie Green
The blue ring is the patrol area for the cat, C, represented by the green dot. The central red dot represents the position at which the mouse, M, will emerge from its safe hiding place. We consider the cat and the mouse as dimensionless points for simplicity.
The cat has a uniform probability of being anywhere within its (blue) patrol region when the mouse randomly appears. The red ring is the danger zone for the mouse. If the mouse appears within the confines of the red ring it is munched (eaten).
Given that the blue ring has a diameter of D and the red ring has a diameter of d, what is the probability that the mouse gets munched?
by Leslie Green
We will state this problem in units of pounds sterling (GBP) since we have ready access to data in this currency.
There is a big push for energy efficiency in appliances, especially for the 'new' condensing gas boiler technology. Companies who are pushing you to change quote figures of hundreds of pounds a year cost savings. Let's do the calculation ourselves.
Take the old gas boiler efficiency as 70% and the condensing gas boiler efficiency as 90%. Take the gas bill as £1500 per year, of which one third is due to the fixed standing charge.
What is the annual cost saving due to the higher efficiency?
by Leslie Green
The image shows a motorway with three lanes going around a bend. Imagine that cars are heading up the motorway and turning left. Since all drivers want to feel that they are making adequate progress, imagine that three cars all pass the blue line at the same time. They adjust their speeds so that they all pass the red line at the same time.
In which lane will the driver be at more risk of skidding off the road?
(HINT: make sure you answer the question being asked.)
by Leslie Green
The two (red) pulley wheels have frictionless bearings attached to a rigid plate. The single piece of strong sticky (green) string is light and inelastic. The heavy (blue) mass is tied on to the (green) string at its two upper corners. The (blue) mass is initially held stationary in the position shown, hanging downwards.
What happens when the (blue) mass is released?
by Leslie Green
The light pulley (RED) has frictionless bearings and is mounted to a rigid plate. The string (BLUE) is strong and inelastic. The masses are hanging down from the pulley.
What happens when both masses (GREEN and CYAN) are released from the stationary position shown?
by Leslie Green
Gerry's truck loses 25% of the straw packs every time it bumps into another car.
If Gerry started today with 256 packs and has bumped 3 times into other cars, how many straw packs are left in his truck?
Having recently replaced the worn-out tires (tyres) on your modern front-wheel-drive car, you notice that the fuel efficiency (mpg) has dropped a little.
This is difficult to understand since you drive long distances at the same constant speed and the new tyres are similar to the old ones.
What might be the reason?
by Leslie Green
Having just found out that a vital ingredient required for dog biscuit production has become very expensive over-night, you predict that the share price of the ACME Premium Dog Biscuit Corporation is likely to plummet.
How can you profit by this?
by Leslie Green
Sometimes I hold bad people.
Sometimes I am the smallest unit of a living organism.
Other times I make your torch work.
What am I?
Consider the image text, showing a (real) prime-time TV advertisement.
Comment accurately on the claim.
by Leslie Green
Why do tall glass bottles fall over when placed upright on the conveyor belt at a supermarket checkout till?
(Pick the best answer.)
by Leslie Green
A train crosses a man on a platform in 9 seconds.
The train crosses the 210-meter platform in 30 seconds.
What is the length of the train?
Which rope has the largest force inside?
The rings, R, should be considered as smooth and frictionless.
A magician tells you that if you correctly guess the exact value of money in an envelope then he will give you the money.
The envelope contains bills (notes) totalling between $1 and $10 (inclusive). After each guess he will tell you if your guess is too high or too low, or he will give you the money in the envelope. You can try only 3 times.
If you choose the best approach, what sum of money do you expect to get from the generous man?
Towards the top right of the image is a gas-fired central heating vent.
What is best name for the stuff coming out of the vent?
by Leslie Green
What happens if somebody leaves the refrigerator door open in the kitchen?
Pick the least correct answer.
by Leslie Green
You might call this arrangement a 'budget wardrobe', typical of student accommodation. The top rail is tall and thin, so it needs two little feet to prevent it tipping over forwards or backwards. The feet are screwed to the rail, but not attached to the chest of drawers.
If the large blue weight, W, is too big compared to the smaller weight w then the rail will presumably tip over to the left.
What is the minimum size of the red weight (w) required for stability?
(Neglect the weight of the rail.)
Mary lives in an old stately home with a long straight staircase. She wants the dog toy (a rubber bone) in her hand upstairs, but can't be bothered to walk up and down the stairs to do so. Being mathematically inclined, she wants to calculate the landing speed of the toy if she throws it up the stairs in an optimum manner.
The stairs go up a vertical distance of 5 m (above the height from which she throws) and are roughly at an angle of 42° from the horizontal.
Taking gravitational acceleration as 10 m/s2, and neglecting wind resistance, estimate the slowest possible theoretical landing speed for an optimal throw.
The picture shows a section of a road network. Cars only enter at the left, and only exit at the right. There are no stopping places, and accidents or breakdowns have a negligible effect in this area. This section of the road network is not very busy.
Formulate a rule for the traffic flow.
This is my trusted 25 year old Tandy (Radio Shack) EC-4017 10-digit scientific calculator.
The display shows the result of 2 raised to the power 31 using the x to the y button.
Comment on the display.
by Leslie Green
Rational numbers apparently have a good public image. They are somehow perceived as being nice well-behaved numbers, formed from the ratio of two whole numbers.
Irrational numbers just seem a bit crazy, just because they can't be expressed as the ratio of two whole numbers.
Rational numbers can be expressed as decimals which either end in zeros, or have a finite sequence of repeating digits. Typically, however, nobody mentions that the length of such a repeating sequence could extend to thousands of digits (or more!). Irrational numbers have a genuinely non-repeating decimal expansion.
Are there more or less irrational numbers compared to rational numbers?
by Leslie Green
We wish to pack flat discs into rectangular trays and to achieve the highest possible packing density for just two rows. We can adjust the rectangular outline of the tray to fit whatever arrangement we choose.
How do we achieve our goal?
by Leslie Green
Computers typically work in binary, and then convert back to decimal when interfacing to humans.
If you wish to store a fairly large positive whole number N in a computer (in binary), how is the minimum number of bytes required related to the number of decimal digits in N? Don't worry about special cases.
(1 byte = 8 bits)
by Leslie Green
We wish to estimate how the water flow out of a bath changes with the height of the remaining water. To do this we equate the potential energy of the water in the bath to the kinetic energy of the outgoing water.
We vaguely remember that potential energy is proportional to height, whereas kinetic energy is proportional to the square of the velocity.
Using only this model, how does the outflow depend on the height?
by Leslie Green
Calculate how much faster a two-hour journey will be for Gerry if the 120 km/h speed limit is increased by 25% in the country.
He usually drives 120 km at the maximum speed.
Jennifer is admiring a brooch in a jewellery shop window. The brooch has brilliant vibrant colors (colours) which change according to how she moves her head.
Which is the correct word to describe what she sees?
The left side of the image shows a faulty (leaking) gate valve with a flange adaptor to fit a central heating water pump. The right side of the image shows the repaired assembly (using new parts) for context.
The replacement gates valves looked similar to the faulty ones, including the cut-outs in the clamping nuts, shown with red circles surrounding them. The parts came without instructions of any kind.
What is the purpose of the cut-outs?
by Leslie Green
Daylight falls on a broadband optical sensor and produces a power reading. If a particular thin planar optical device is placed in front of the sensor, the power reading drops to 1/3rd of its initial reading.
What reading should you expect if you put another one of these optical devices (in the same orientation) in front of the first one?
You have wired up a 12 V light bulb at the end of your 100 m garden using wire rated for 1 A. The light bulb takes 1 A because it is a high efficiency type.
You power the circuit from a 12 V supply designed specifically for this light bulb.
Why is the bulb so dim?
by Leslie Green
In the atomic model taught at school you have matter composed of electrons, neutrons, and protons.
Which substance is it most difficult for electrons to travel through?
by Leslie Green
Having just completed a small painting job, you now need to clean the paint brush. The recommended method is to 'paint' some old magazine or newspaper until the brush is fairly clean before washing the brush in water. The idea is to put as little (wet) paint down the waste water drain as possible since it is a bit poisonous when wet, but ok when dry.
Having removed the staples from an old TV listings magazine, you can see two pages from the one sheet. Having 'painted' one side, you can't then lay that wet side down anywhere because the paint will transfer, ruining the worktop.
How many pages do you get to 'paint' from this one sheet?
by Leslie Green
The image shows the side view of a solid cylinder on a gently sloping (brown) ramp.
The red upper half of the cylinder has 4x the density of the blue lower half. The slope is just sufficient to compensate for the rolling resistance so the cylinder can continue rolling without limit. The cylinder has been rolling down the slope for several revolutions now.
Describe its motion.
by Leslie Green
Battery manufacturers tell you to not leave batteries in toys, torches (flashlights), or appliances for storage, or for extended periods. They typically do not say why.
Why do you think they make this recommendation?
by Leslie Green
In an automated checkout till at a supermarket, goods are scanned by the customer and placed on the out-going counter. This counter is actually a weigh-scale so it verifies that the bar code and item weight agree. Having finished scanning your items you then pay for them before bagging them up and taking them away.
What is the best point for the automated till to say "good bye" in a polite and friendly voice?
by Leslie Green
A particular satellite costs $350M to make and $350M to launch. A deal is possible such that the launch date can be pushed forward by 1 year, but the launch vehicle is of a very new design and has very little launch data to back it up. Because the launch vehicle owners want to get more launch data, they offer to launch the satellite for $20M. Statistics are hard to come by, and unreliable, but you estimate the chance of mission success is 9%.
Is it worth the risk?
by Leslie Green
A 4 kg adult cat is floating on a 3.9 kg wooden block in a paddling pool filled with water. The water depth is 10 cm.
The surface of the wooden block is covered in a non-slip rubber mat so the cat can get a good grip on it. This cat is an experienced jumper, but is unused to water.
The cat decides it is now close enough to the edge of the pool and jumps. What is the most likely outcome?
by Leslie Green
The voltage (potential difference) across the terminals of an ordinary 12 V battery found in most cars never exceeds 15 V, and typically is in the range 13.8 V to 14.4 V.
Typically safety standards refer to supplies whose voltages are guaranteed to be below 60 V (DC) as being safe from the point of view of electric shock.
Which statement is both correct and the most comprehensive we can be sure of?
by Leslie Green
Passwords protect very important things like bank accounts, and yet people formulate them very poorly.
When told to make the password at least 10 characters long they make them exactly ten characters long.
When told to include at least one digit they include exactly one digit.
When told to include at least one special character ($%£€&*+=#^?/!®) they include exactly one special character.
You then get the situation where the person picks a dictionary word (a word found in a dictionary) then tags on a digit and a special character.
Compare the security of a proper 10 character password to the word "password" with random capitalisation followed by a number and a special character in any order.
(For simplicity use one of ten special characters.)
by Leslie Green
Many computer and mobile systems lock you out if you enter your password incorrectly 3 times in a row.
Given that this is the case, why do you need a password with more than a few thousand possibilities?
by Leslie Green
It is reported on the news that survival rates for particular aggressive cancers are worse in your country than in at least one nearby country. The survival rates are in the range 7% to 12%, so that is not good anyway.
Which statistic would you now like to know to understand this news better?
by Leslie Green
ABCD corporation organizes a corporate meeting. They want to invite different employees of the corporation from different locations:
Washington DC - 50 people,
New York - 200 people,
Denver - 50 people,
Los Angeles - 20 people,
and 20 people from different cities of the USA.
What is the best place for the meeting to minimize the travel expenses?
When offered the mains extension lead shown by an elderly relative for the purpose of assisting in pressure-washing their patio, what would be the most appropriate response?
by Leslie Green
As part of a school project, your team has made a quad-rotor lifting platform with directional thrust control. The grey box is the container. The red blobs are electric motor nacelles (pods) which can rotate about an axis into the plane of the drawing (at right angles to your computer monitor's screen). The blue lines are the rotor blades.
You have realised that as the thrust direction moves away from the vertical, the effective lifting force will be reduced, causing the quad-rotor to rapidly descend. In order to make it more controllable you therefore want to automatically increase the thrust when the pods tilt.
What thrust should you use, given that the initial steady thrust per rotor is T when the pods are vertical, and the pod angle relative to the vertical is A.
by Leslie Green
We wish to consider the effect of component tolerance on the product we are designing.
Which component has the greatest effect on the specification of our new product?
(HINT: don't make a hasty choice.)
by Leslie Green
The picture shows a mains extension reel for use in the UK market. These things are disproportionately likely to be the cause of domestic fires.
Why?
(Pick the most comprehensive correct answer.)
by Leslie Green
The photo shows a 4-way extension socket rated at 13 A, 230 V, and for the UK market. It seems to be well made.
Which operational use is most likely to cause a safety problem.
by Leslie Green
Peter's newish car has a reversing camera built-in. The reversing camera has a nice big display, giving a very high resolution image.
When reversing out of a parking space in a large open-air car park, which should he look at?
by Leslie Green
A gas-fired central heating boiler is a safety critical system. On one particular design, if the pilot light doesn't light, the boiler doesn't fire up. Then, if the pilot light is on, the fan is turned on. If the fan doesn't work the boiler doesn't fire up. Then, if the fan is working the main burner will light.
The fan working is sensed by a pressure switch. The pressure differential not only shows that the fan motor is running, but also shows that the fan blades are present, and that the flue is not blocked.
The pressure switch has changeover contacts. Why might this be the case?
by Leslie Green
Please don't experiment with gas-fired boiler circuitry. It needs to be touched only by fully trained and certified technicians.
Leslie Green asks:
Given machines of equal performance, on what basis might you argue that there is hardly any difference between the world's best motorcycle racer and the world's worst motor cycle racer?
by Leslie Green
Disaster has struck the widget production line. Dowels and dowel holes on metalwork supplied by a new vendor are horribly out of spec. The corporate trouble-shooter has been called in because at least some widgets are desperately required.
The dowels have been accurately measured and any that are within the range 10.00 mm to 10.10 mm have been kept; the rest of the batch have been returned. Likewise, the dowel holes have been inspected, and only parts with dowel holes in the range 10.05 mm to 10.15 mm have been kept. The selection process has been so severe that the initial Gaussian distribution of sizes is effectively now uniform on the remaining parts.
If the dowel is the same size or smaller than the dowel hole we call that a fit.
What percentage of the selected pairs now fit together?
by Leslie Green
Having done extensive calculations and simulations, you know that a honeycomb composite strut will be exposed to a peak load of 10 kN. The mechanisms of stability loss of the strut under compression can be different. A batch of 1000 struts has arrived, and on a large sample you have established the breaking point as having a mean value of 10 kN, with a standard deviation of 1kN, the distribution being approximately Gaussian.
What is the probability of an in-service failure?
by Leslie Green
The image shows a valve on a domestic hot water radiator. This valve has a drain-off connection to which the blue hose pipe has been attached.
The 1/2" diameter hose pipe is a very poor fit to the drain-off point, such that it would not stay in place without something pushing the pipe up into position. Nevertheless, when the drain valve was opened, and water flowed nicely out of the other end of the pipe, no water came from the loose pipe-to-valve joint.
Given that water typically leaks from any loose joint with great enthusiasm, what is the best explanation for why this did not happen?
by Leslie Green
During a summer day, everything looks bright and crisp. During a winter's day, things look dull and not so sharp.
What is the best explanation for this?
by Leslie Green
Regardless of the eye color (colour), the middle of a human eye is always black.
How can you see through it?
by Leslie Green
Leslie Green asks:
When charged particles are accelerated (or decelerated) they emit radiation; the greater the acceleration the greater the radiated energy.
Why is a particle accelerator like a synchrotron so large in diameter?
It is a common experience that operating old water shut-off valves (isolators) can cause them to leak.
By 'leak' we mean that water comes out of the valve itself, rather than the valve failing to shut-off completely.
Why?
(Pick the most likely answer)
by Leslie Green
There is a tiny fraction of the world's population who apparently believe that the Earth is flat, not spherical.
In one experiment to 'prove' their point, a believer takes a spirit level on an aeroplane (airplane) and sees that the spirit level stays level during most of the flight. He expresses the opinion in his video that the nose of the plane would need to be dipped down if the Earth's surface was truly curved. (It is hard to understand if the video maker is actually a Grade A nut-job or just a troll.)
Why does the spirit level stay level?
by Leslie Green
You will often hear figures of 80%, 90%, or more quoted for the efficiency of electric motors.
You may be (temporarily) disturbed when you hear that there are load and speed conditions which cause this efficiency to be 0%.
For some random electric motor of any design, how many such load/speed conditions will always be present?
(HINT: you need to think about this from first principles.)
by Leslie Green
It is very upsetting to see scammers posting videos on popular upload sites for the purpose of cheating people out of their hard-earned money. In a particular scam, two 12V batteries are wired in series. One battery drives a motor, the motor driving a generator that recharges the batteries, .. at least that is the claim for this "over-unity" (free energy) machine. They do also state that the other battery has an extra output of the generator charging it.
The video clearly shows the series battery voltage rising with time, which is the scammers’ evidence that excess power is being generated. As usual with these scams, the video is full of "vacuum energy", "quantum physics" and other misplaced terms used to baffle the innocent.
How is it possible for the series voltage of the batteries to rise without over-unity (excess) power being generated?
by Leslie Green
Life is always presenting us with choices and opportunities.
John is fairly well off and settled in his life. He has been presented with three opportunities all at once, only one of which he can choose, and such an opportunity is unlikely to present itself again.
A) He can get $11 with 95% certainty for no cost, and no risk.
B) He can get $10,000 with 0.1% certainty for no cost, and no risk.
C) He can get $10,000,000 with 1 chance in 1000, but he has to pay $5000 to participate.
Which option should he choose?
by Leslie Green
The picture shows a cross-sectional view of some pipes and tanks filled with cold water. There are no strange temperature or gravitational effects going on. Just imagine that this is in a house.
The water has been stationary for some time, with the valve shown by the large X closed.
What happens when the valve X is opened?
by Leslie Green
For a typical lead-acid battery used on a car there are 6 cells, wired in series. A heavily discharged cell gives a voltage of 1.8V on no-load, whereas a fully charged cell has a no-load voltage of 2.1 V.
Typical car batteries do not allow access to the terminals of the individual cells. All you can measure is the voltage between the main battery terminals.
You measure the voltage between the battery terminals with the ignition and lights switched off using an accurate multi-meter. The reading is 10.1 V.
What is your initial diagnosis?
by Leslie Green
Leslie Green asks:
The image shows a typical cordless-drill battery pack, and the 20 individual cells which were contained within. All the cells are connected in series so they all pass the same current. But when one or more cells become weaker than the rest, they cannot sustain the high current and get damaged, causing the whole pack to fail.
Compare the failure of one cell to the failure of a pack containing 20 cells.
Attention: please don't pull apart working equipment just to see what's inside. There are plenty of videos available on the web to see the insides of practically anything.
Leslie Green asks:
The image shows a wooden block held by three rubber (elastic) bands, two on the left and one on the right. The block would ordinarily stay where it is, but using our (calibrated) fingers we can push it to the left or the right.
It will always return to the position shown when released. We never push it so hard that a band becomes loose or breaks.
Which curve represents the restoring force?
(HINT: Be careful, think it through!)
Leslie Green asks:
The little wooden block is held in position by two rubber (elastic) bands.
It is apparently not moving.
Why?
(Pick the most accurate answer).
Leslie Green asks:
In comedy sketches, it is not uncommon for somebody to fall backwards from an upright position, whilst remaining relatively rigid.
Without worrying about the real-world consequences of this, consider it scientifically.
For your latest project you want to screw the plywood top onto the workbench. 4 screws evenly spaced along the 24" (24 inch) short side should do the trick.
What is the correct spacing between the screws?
(This is practical mathematics. You are not allowed to use a calculator or even a pencil/pen and paper.)
by Leslie Green
A manufacturing facility in a backward and corrupt country is discharging toxic heavy metals into the nearby river. The local regulations stipulate that the heavy metal concentration in the outlet water must be less than 0.1mg/L. The plant manager knows that the level is actually more like 1mg/L, discharging at a rate of 1L/minute.
What rate of clean river water does the plant manager need to mix with the outlet water to bring the toxic discharge rate below the regulatory limit?
by Leslie Green
The drawing shows one elevation of a solid square block. All dimensions are in mm. A datum surface, A, has been defined on the bottom face. The top face has a parallelism tolerance defined relative to the bottom surface. What this means is that two parallel planes are defined with the separation shown in the geometric tolerancing box. The top surface has to fit between these parallel planes.
Assuming that the part meets the specification, what is the maximum angular tilt of the top surface?
(We are using angular units of milli-radians, mrads)
by Leslie Green
In an experiment the ratio X is evaluated as 1.10 with an estimated uncertainty of ±1%.
The result is evaluated from the equation
R = 1 - (1/X).
What is the uncertainty in R?
Inflation is the rate at which the general level of prices for goods and services is rising.
For example, if the annual inflation rate is 5%, then an apple that costs 1 piastre now will cost 1.05 piastres in one year.
An apple costs 1 piastre today.
If the annual inflation rate is 120% and the bank interest rate is 10% in a country, how much money do I need to put into my bank now to be able to buy an apple in 5 years?
Leslie Green asks:
The quoted text shown to the right is from a specialist concrete expert website.
Assuming the example given is correct, which is the correct formula?
(Use J for the distance between joints, and D for the depth of the concrete slab.)
Leslie Green asks:
The image shows an actual screen snip concerning an order from a large online retailer.
Can you deduce what promotion has been applied?
(VAT = Value Added Tax)
£ means British pound
We suppose that there are 5 programs on your computer that like pushing updates every month. We further suppose that each of them, during the update process, has some extra software which will get loaded by default if you are not awake enough to spot it. Now suppose that you are randomly inattentive, such that you spot the unwanted tick boxes only 99% of the time.
What is the probability that during the course of a year you will unintentionally download at least some unwanted software?
by Leslie Green
In the design of multi-stage rockets, electrically-triggered explosive bolts can be used to separate the stages. If the bolt fails to break, the previous stage is still held in place -- and that is typically catastrophic. If the bolt breaks too early then that again can be catastrophic. Having made the part as reliable as possible, further improvement can be made by using series and parallel combinations.
For a series connection of N explosive bolts, if separation failure occurs randomly with probability S for a single bolt, the series assembly fails to separate with probability SN. However, premature separation is N times more likely.
For a parallel connection of N explosive bolts (or explosive bolt series-assemblies) separation failure is N times more likely. Premature separation is less likely by the premature separation probability, P, being raised to the power N.
On a single mission, compare the probability of mission failure with a series chain of two bolts to the probability of mission failure with a single bolt.
(We take the probability of premature separation as 0.0001, and the probability of separation failure as 0.001 for a single bolt.)
by Leslie Green
Jane randomly chooses a pen from a collection of white, yellow, and cyan pens. There is an equal probability of choosing any particular color.
John randomly chooses a piece of paper from a collection of white, yellow, and cyan sheets. The probability of choosing paper of a particular color is proportional to the number of sheets available. There are equal amounts of yellow and cyan, but twice as much white as yellow.
Jennifer takes the pen and paper and writes a note.
What is the probability that the writing will be visible, given that writing using a pen of the same color as the paper will not be visible?
by Leslie Green
Leslie Green asks:
Not so long ago, but before petrol engines, and before small steam engines, cargo was transported down canals in barges, pulled by horses or even people.
Why would it have been advantageous to use a long rope?
The picture shows "Barge Haulers on the Volga", an 1873 oil-on-canvas painting by the Russian artist Ilya Repin.
A toy car is released from stationary on a smooth planar slope which is angled relative to the horizontal, the sine of the angle being 0.2 (an angle of about 11.5°).
Taking the acceleration due to gravity as being 10m/s2, and assuming frictionless wheel bearings and negligible air resistance, describe the motion of the car (which was pointing directly downhill when released).
by Leslie Green
The grey block is extremely heavy and rigid, with a perfectly horizontal frictionless top surface. The blue inelastic string over the frictionless pulley transfers force without loss from the small weight to the big weight. The big weight has 3x the mass of the small weight.
Initially, the masses are held in place by some unspecified device, with the string taut.
Describe the motion of the small mass when the masses are released.
Take the gravitational constant as 10N/kg.
by Leslie Green
Leslie Green asks:
Nerdy McGeekface throws a baseball as hard as he can, but unfortunately hits Beefcake O'Reilly square in the back.
Neglecting any subsequent activity, who feels the most impact?
Leslie Green asks:
A man wishes to throw a ball such that it stays in the air for as long as possible.
The speed at which he throws the ball is some fixed value. The ground around him is featureless and level as far as the eye can see.
What is his optimum strategy?
(Neglect wind resistance and spin.)
Leslie Green asks:
A particle has a velocity which increases in direct proportion to its distance from its origin (the zero reference point for distance).
Describe its acceleration characteristic.
Here we show the side view of a kitchen worktop with a grain of uncooked rice (not to scale) ready to be brushed into the blue bin below.
What is the optimal speed with which to brush the rice?
(Take the gravitational force as 10 N/kg)
by Leslie Green
Your school work has finally paid off and you get to design your own electric dragster. The electric motor is so massively powerful that the torque output to the road wheels has to be limited to a certain fixed value to prevent the wheels from spinning.
What is your prediction of the real-world performance of this new design?
by Leslie Green
Leslie Green asks:
In a typical lab setup for measuring frictional force, a block (blue) is pulled by a thin string or line (green) by a weight (yellow). The pulley wheel (red) is typically considered to be free to rotate without friction in the bearing.
What is the effect of some small friction in the pulley wheel bearing?
Leslie Green asks:
The 0 - 60 mph (≈100 km/h) test is frequently used to compare cars. Suppose you took a car and pushed it off a cliff.
Neglecting wind resistance, how long would it take to complete this 0 to 60 mph test?
Take gravity as 10 N/kg and 60 mph as 30 m/s (it is actually 26.82 m/s, an inconvenient value for hand calculations).
Leslie Green asks:
The iPhone has a smooth underside resting on the smooth level surface. It has a mass of 130g. The coefficient of static friction between the phone and the surface is 0.2 exactly. I horizontally pull the phone by the attached white cable with a force of 0.2 N.
What is the magnitude of the frictional force?
1kg = 1000g
Your boss has told you to buy a more accurate (and therefore more expensive) thermocouple probe to more accurately measure the room temperature using a thermocouple meter (digital thermometer).
Comment on this requirement.
by Leslie Green
By the time you have completed the calculations, and set the thrust magnitude, the 1000 kg lunar lander will be heading directly for the moon's surface at a speed of 50 m/s and just 250 m from the surface.
How much thrust is needed to give a perfect landing (neglecting the weight change due to the fuel expended)?
(The gravitational acceleration on the surface of the moon is 1.6 m/s2).
by Leslie Green
John, the physics student, wishes to dry his wet rectangular towel on a radiator. He wonders how much has to overhang at the back to make the towel stable.
He is reluctant to put too much of the towel behind the radiator as this is the unknown territory of stale pizza, dead bugs, cobwebs, and other indescribable debris.
He models the situation as shown, where the radiator thickness has been exaggerated. He supposes that the towel only rests against the top of the radiator, shown in red. He further supposes that the length of this red section is negligible compared to the length, L, of the towel. He guesses that the coefficient of static friction for this situation is 0.50
What is the length of the long part of the towel in the limiting case?
by Leslie Green
Cathy is all alone in the spaceship, 1km from the mother-ship. The (hideous) fate of the other crew members has been withheld. Cathy's spaceship is totally out of fuel and is dead in space relative to the mother-ship.
Cathy has one chance. She has created a loose bundle of equipment and waste which she estimates to be 1000kg of mass. If she can eject this mass from the airlock with enough speed she should reach the mother-ship before her air runs out. She estimates the current mass of her spaceship as 100,000kg.
Cathy cannot throw the 1000kg mass, but by pushing it and running on the ridged floor in the air-lock, she is confident she can eject the mass at 5m/s.
What is her estimate of the time to reach the mother-ship?
by Leslie Green
Leslie Green asks:
A satellite in geostationary orbit is essentially weightless. It needs to move about its axis and to nearby positions at the same orbital radius.
Which is the best (correct) statement?
A group of student friends went to an open-air rock concert.
They evenly distributed themselves in the available cars.
Half way to the concert, two cars broke down so it was necessary for each of the remaining cars to carry one more student.
The drivers of the broken cars also went to the concert.
When they started for home town after the concert there were three persons more in each car than when they started out in the morning because two drivers individually drove to other locations.
How many students attended the concert?
The value of a computer depreciates evenly over five years for tax purposes (every year it is worth less by the same amount, so that it is worth $0 after five years).
The value of a table depreciates evenly over ten years.
If I bought the computer for $2,200 and the table for $500, how much will they be worth after 3 years?
An inexpensive hand-held laser range finder projects a red laser beam onto a wall a few meters away, and returns an accurate distance reading with 1 mm resolution.
Assuming this is a simple time-of-flight measurement of a laser pulse, what is the necessary resolution of the time measurement?
The speed of light is 3 108 m/s (300,000 km/s).
by Leslie Green
A certain premium digital meter has a stated DC volts accuracy specification of ±[0.15% of reading + 2 counts].
On the 6 V range it will display 6.000 V, in other words the last digit is changing in steps of 1 mV.
You have correctly set the zero reading and you are measuring a voltage of 50 mV.
What is the specified accuracy of your reading, neglecting any other considerations?
by Leslie Green
Leslie Green asks:
Consider the following statement:
"A hovering helicopter has no vertical or horizontal motion. Given that mechanical power requirement is force x velocity the mechanical power required to do so is zero so the helicopter engine is just idling when hovering."
Leslie Green asks:
Which material in this list is the most strongly anisotropic?
School problems often deal with uniform isotropic materials: iso- as a prefix has the sense of sameness. Isotropic materials have properties which are the same, independent of direction. The an- (or sometimes just a-) negates the following items such as anaerobic, anastigmatic.
Leslie Green asks:
Analyse the graph shown.
What is the strongest statement you can be sure of from this data?
Image credit: Quora
Leslie Green asks:
The image shows an electrical power meter set to read in units of kWh. Given that at the start of the test the last three digits were 9.45, and 30 minutes later the last three digits were 9.50, what was the average power consumption during that period?
(The image shows the last three digits as 9.48 as the picture was taken during the test.)
You have 4 electronic weighing scales. You test all 4 with the same 50.000kg reference weight at different times of the day, and on different days, to establish their accuracy.
The results are shown below.
You want to pick the best scale to use in your new coffee bean filling station because you know that you have to supply at least as much coffee as you put on the label, but any amount more than that is just wasted as far as your profit is concerned. The bags are marked as 50kg.
Which scale should you pick?
by Leslie Green
In the recycling of cars, the body of the car gets shredded by powerful machines and the resulting chunks of material first have the ferrous metal (primarily iron and steel) removed by means of permanent magnets. The next step takes the non-ferrous metals (aluminum, brass, etc) and throws these off the conveyor belt using the force from an alternating magnetic field (an eddy current separator). The non-metallic waste passes through the eddy current separator a second time to improve the separation.
If we suppose that 10% of the non-ferrous material remains after the first pass, how much remains after the second pass?
by Leslie Green
A manned space mission to Mars requires a reliable electrical power generation system. For this reason three of the same type of power generators are taken, each of which is sufficient to supply the requirements of the mission. The probability of failure of any one of the power systems during the flight is estimated to be 1 in 100,000.
Estimate the probability of a failed mission in terms of just the electrical power supply.
by Leslie Green
The sinking of the unsinkable RMS Titanic in 1912 (after it hit an iceberg on its maiden voyage) is a well known engineering disaster story. As with many such disaster stories it was not just one thing that caused the disaster, but a whole catalog of errors.
The sides of the ship were 1 inch thick mild-steel plates, overlapped and hot-riveted together. In this case the iron (or steel) rivets were heated until they were red hot (actually glowing red) then pushed into holes in the steel plates and hammered over. As the rivets cooled, they contracted, making really strong joints.
We suppose that one faulty riveted joint is just about acceptable, but that two right next to each other is bad. Suppose we have 4 rivets in a row, and the probability of a faulty rivet is 1 in 1,000. Estimate the probability of two faulty rivets right next to each other.
by Leslie Green
The picture shows two temperature/humidity meters of the same type, bought at the same time and from the same place. The specification in the manual that comes with the meters says that the temperature resolution is 0.1°C and the refresh rate is 10 seconds. The website selling the meters claims the accuracy is ±1°C.
The meter on the left has been brought into the room and left to stabilise for about 2 minutes. (If it had been put on the right the reading on that meter would still have been low.)
Why are the readings so different? Which answer is the most guaranteed?
(The picture has been edited to remove branding information, and that is all.)
by Leslie Green
Americium-241 is a radioactive isotope with a half-life of 432.2 years. It is widely used in domestic smoke alarms in the form of Americium-241 dioxide.
Estimate how much of the Americium has decayed at the end of the typical 10 year service life of a smoke alarm.
by Leslie Green
There is a new and dangerous strain of bacteria in town. You have a total of 10 anti-biotics and supplements available, but you have been told that they are all individually ineffective. You hope that some combination of them may be effective, and you also hope that when testing, the relative amounts of each constituent is not critical.
Fortunately you have a fully automated sample handling facility at your disposal, so you can make as many tests as you want, and the computerized system can tell if the result is a success without your intervention. It is the nature of such tests that the sample needs to be incubated for 24 hours to see if it has been successful.
How long will it take to fully test your idea?
by Leslie Green
John lives in a fairly well-insulated fairly old brick-built house in a central European location. In the winter the weather gets quite cold at night compared to the day.
Peter lives in a much newer brick-built house in the same village as John. The locations of the two houses in terms of local climate and wind are similar.
John turns off the heating between the hours of midnight and 6am to save money on his heating bill. Peter claims that John is wasting money on his heating bill because John has to keep heating up the walls every day, rather than leaving them at a more even temperature. Peter has evidence that his scheme of leaving the heating on all night is better since his heating bill is 10% lower that John's; their houses are similar sizes, and kept at the same sort of temperature.
Which scheme saves the most on energy costs?
by Leslie Green
PCR (Polymerase Chain Reaction) is a technique used to make copies of a particular section of DNA. It is therefore also referred to as DNA amplification.
A PCR machine heats the sample (along with various other ingredients) up to 96°C to split the DNA strands, cools it to around 60°C to allow binding to the separated DNA strands, then heats it up to 72°C to complete the copying process. Because DNA is a double helix, and each side is copied, each PCR cycle roughly doubles the amount of the DNA section present.
If we start with one DNA chain, how much would we expect after 30 PCR cycles?
by Leslie Green
It is easy to show that making beams long in the direction of the load makes them disproportionally stiffer.
We have a simple wooden rod with a square cross-section. Let's suppose it has a 2cm x 2 cm cross-section and is 1m long for the sake of clarity. One piece of this wood is not stiff enough for our purposes, so, knowing the stiffness equation, we stack three pieces up vertically. The three rods together now have a cross-section of 2cm x 6cm with the rods stacked upwards (the direction of the load), not sideways.
How much stiffer is this new assembly compared to the original rod?
The problem was suggested by Leslie Green.
WARNING: Do not try bending any beams like this without using eye protection. Certain rulers are known to shatter and eject bits of plastic into nearby eyes.
In one of the many different types of domestic heating systems, hot water is pumped around the house and heats rooms via "radiators".
The fact that the dominant heat transfer mechanism is actually convection, rather than radiation, is easily demonstrated by putting your hand just above the radiator and just below. It is of course much hotter above the radiator, showing that convection is dominant.
Why is it necessary (and recommended) to turn up the water temperature in the boiler and radiators during the cold winter months?
Pick the most accurate answer.
Leslie Green asks:
The sketch to the right suggests the outline of a wire-frame bird-cage. The wire is thin, with large gaps, so the bird can easily see in all directions. The cage is supported by a single cord (red). The size of the bird is very small compared to the size of the cage.
As an experiment we hang the red cord from an electronic force sensor to measure the weight of the bird plus the cage.
What happens to the measured weight when the bird flies around?
Leslie Green asks:
Desmond von Dummkopf is showing off to his (so-called) friends again. He is going to ride his unicycle over the strong concrete structure shown.
(5 m/s is roughly 11 mph.)
Does he have any chance of making the jump?
(We only consider the jump, not the ability to hold the unicycle upright during and after the landing --- if he should succeed in getting to the other side.)
g = 10m/s2
Leslie Green asks:
This is a UK mains socket, nominally rated at 230 V AC, 13 A (3 kW)
When is it most likely to catch fire?
A rich father has 71 camels. He promises at least 1/2 of the camels to the oldest son, at least 1/3 to the middle son, and at least 1/9 to the youngest son.
He wants to give several camels to his daughter.
What is the maximum number of camels he can give to his daughter and still keep his promises?
In a Math country, a young man dates a girl until they decide to marry or to split.
If they split, he dates another girl.
The girl does the same.
There is 50% chance that the man is ready to marry. The same probability applies to the girl.
Estimate the number of girls a young man dates before he marries one.
By the way, the expected number is the same as the expected number of boys a young girl meets before her marriage.
You spin a wheel and it randomly lands on a money sector or END. If you land on a money sector you get the money and spin the wheel again. You keep receiving money until you land on END.
How much do you expect to win on average?
Drawing A shows a massive black beam easily supporting the load of two heavy red beams by means of the blue tension rods and the green nuts. The blue rods are very strong, while the nuts are the weakest elements of the design. It is not your design, but you have been given computer simulations of this design and it all looks fine, ... in theory.
The contractor hired to build the design hates it. The green nuts have to be spun all the way up the blue threaded tie rods, which is very time consuming and therefore expensive. Also both red bars have to be assembled at the same time and that is difficult. The contractor has come up with design B which uses shorter tie rods of the same design but uses the same nuts. They have calculations to show that the upper red beams will not be damaged by offsetting the lower tie bars slightly.
Will you approve the change?
Author: Leslie Green
Leslie Green asks:
The image shows a wooden door with a brush-type draft excluder ready to be cut to fit on the bottom of the door. The door hinge is on the left in this picture.
The instructions that come with the part just say to cut it to size, but where should we cut it?
Every day over a period of 10 days ten boys each sent a love letter to their unique and special girlfriends.
John always writes to Rosy, Pete always writes to Jade, Steve always writes to Celeste, and so on.
The postman put the letters into ten different boxes without looking at the name of the recipient.
What is the expected number of letters that reached the correct recipient?
You wish to pour juice into a glass from a container with a rectangular cross-section (as shown).
We can define the orientation of the container by saying that one of the 4 marked edges is uppermost.
Which edge should be uppermost to give the best control when pouring from a fairly full container?
Author: Leslie Green
Aplusclick website had about 120,000 visits per month in 2017. At such websites, the visitors click on ads in 30 cases out of 1000 exposed pages on average.
A dominant Internet search leader pays as little as $0.10 per click.
Aplusclick team thinks that any ads disturb visitors. The team excludes advertising on the website.
What was the annual revenue the team missed in 2017 by excluding the ads?
How should you split a 125-dollar check (bill) without tax in a restaurant among 5 friends?
Remember to add 8% tax, to tip 15% after the tax and round to dollars.
Americans currently average 6.8 hours of sleep at night, down more than an hour from 1942.
On average, Americans spend just 90 minutes eating during the day.
American adults are watching television five hours per day on average.
The average man employed full-time works 8.4 hours per work day.
Bob, a typical American, shaves himself and spends the necessary time in the bathroom, amounting to 18 minutes per day. He goes to work and back in 1 hour.
How much time does he have left for his family, leisure, Internet, and helping his beloved wife Mary on a day when he works?
A spaceship is going to fly from the Earth to Mars. The project has been sabotaged by politicians so that the flight starts when the planets are in the worst possible orientation for the flight.
What is the distance the spaceship has to travel?
Assume circular orbits with a common center (centre) about the Sun. Assume both orbits are in the same plane. Don't worry about matching orbital velocities.
Radius of the orbit of Mars = 230 million km.
Radius of the orbit of Earth = 150 million km.
Author: Leslie Green
Whilst the Earth gets all of its heat from the Sun, roughly how much of the Sun's heat does the Earth get?
Don't over-complicate the question. Just think about how much of the Sun's radiated power is intercepted by the Earth.
Sun's diameter = 1.4 million km
Earth's diameter = 13,000 km
Distance to the Sun = 150 million km
Author: Leslie Green
Suppose you were travelling directly from the Earth to the Moon. You can imagine that as you got further from the Earth, the pull of Earth's gravity would be reducing, whilst the pull from the Moon's gravity would be increasing. At some point these two forces would be equal and opposite.
At what fraction of the distance from the Earth to the Moon would this equilibrium point be?
For the purpose of this question, assume that only the Earth and the Moon exist, and that they are stationary. Take the mass of the Moon to be 1/100th the mass of the Earth.
And we will remind you that the gravitational force on a small object is proportional to the mass of the astronomical body, and inversely proportional to the square of the distance to that body.
Author: Leslie Green
A Physics teacher is on a camping trip with his young nephew. Seizing the opportunity to impart some Physics to this youth, the teacher sketches out a graph of how well-stirred water in an open pan might respond to being placed in the middle of a well-established large fire, built using locally sourced wood.
Which graph did he draw?
(Note: you must be able to justify your answer to get full credit on this question.)
Author: Leslie Green
Rechargeable battery technology is familiar to most people with phones and other portable devices. But larger batteries to power cars and homes are less well understood.
A rechargeable battery can be modelled as an ideal rechargeable battery in series with a parasitic resistance. This resistance dissipates power dependent on the square of the charging current. Now given that electric cars typically have a range of only 200 miles, it is quite tiresome to stop every 150 miles and recharge it for 300 minutes (5 hours). It would be much nicer to charge it in 30 minutes, or even 3 minutes, to make it more like a gasoline (petrol) powered car. But is there a downside to this?
We are going to assume for simplicity that a rechargeable battery is charged by a certain number of ampere-hours. If you double the current (measured in amperes) you halve the charging time.
Consider what happens to the energy loss in the parasitic resistance in the battery if instead of charging in 300 minutes you use a hyper-charger and do it in 3 minutes (for the same battery type).
Author: Leslie Green
Leslie Green tells a story and asks:
The year is 2045. Most new cars now have electric drive trains. One particular new innovation is the DriverTron with its Insanity Mode. In this mode the car automatically applies a constant maximum power to the wheels at all speeds until it reaches 100mph or until the brakes are applied. Needless to say this mode has been banned in all jurisdictions apart from two States in the USA!
If we neglect wind resistance and bearing loss, what equation represents the velocity v of the car in Insanity Mode at speeds less than 100mph?
t is the time from starting at zero speed, and k is a constant.
Lenny throws a baseball to Kenny who is 20m away. The ball arrives in 500ms (0.5s).
Neglecting wind resistance and taking g = 10m/s2, how fast is the ball thrown, given that it is thrown from and received at the same height?
Although you may use different symbols, these formulae may provide some reminders:
s =ut + (1/2)at2 ; v2 - u2 = 2as ; v = u + at
Author: Leslie Green
Desmond the Dragon has been naughty. He has been burning cottages with his fiery breath and eating sheep between meals.
The villagers have decided this must stop, so they have tied him up with a tungsten cable which can hold twice Desmond's weight.
Is this an adequate precaution?
Author: Leslie Green
You have been told to suspend the blue object from one of the red mounting points. You have done some calculations and determined that one mounting point is only just inadequate to support that much load. The rope is easily strong enough, with plenty of margin, it is just the mounting point that is problematic.
You boss really really wants the blue object suspended, NOW. He is a very important person and tells you to use two mounts as it will halve the load. The angle of the mounting ropes to the horizontal is 25°.
What is your response? (And remember, if anything goes wrong you were the engineer, and your boss is just a marketing expert.)
Just in case it is relevant: sin(25°) = 0.422; cos(25°)=0.906; tan(25°)=0.466
Author: Leslie Green
Leslie Green asks:
I have 8 books with weights of 1001g, 1003g, 1005g, 1007g, 1011g, 1013g, 1017g, and 1019g, where the weights are all accurate to better than ±0.1g.
I have a set of balance scales which will balance provided the imbalance is less than ±0.5g.
I weigh four books at a time, two on each side of the balance.
How many unique sets of 4 books will balance?
(Swapping books from the left pan to the right pan does not constitute another set.)
Friday the 13th is considered an unlucky day in Western superstition.
What is the least number of Friday the 13ths that can occur in a calendar year?
A village lies in a mountain valley, with valley walls so high that the sun is blocked out for 6 months of the year. People can get very depressed when they live without sunshine for months on end.
At a council meeting it has been suggested that a mirror is installed on the valley wall to reflect sunlight onto the village below.
What would be the most useful response to the suggestion?
Author: Leslie Green
An elite special forces soldier weighing 100 kg (with full kit) is being winched up to a steadily hovering helicopter at a constant acceleration of 10 m/s2.
What is the steady-state load on the winch cable?
NOTE: take the gravitational constant as 10 N/kg or 10 m/s2
Author: Leslie Green
Joe, the ice-cream van man, has observed in his 20 years on the job, that almost all children like ice-cream.
In order to prove this he asks 999 of his child customers if they like ice-cream.
All but two say they like ice-cream.
What is the biggest problem with his assertion that around 99.9% of children like ice-cream?
Author: Leslie Green
A horse trough has a rectangular cross-section and is continuously being filled from a tap. The trough has two similar holes in it: one at the bottom, and one half way up. The leak-rate from a hole is proportional to the square root of the water height.
The tap is adjusted so that the water rises to one fifth the height of the trough as a steady-state condition.
The tap is now adjusted to double the flow rate. What is the new steady-state height of water in the trough?
Author: Leslie Green
The safety label on this charger for power tool battery packs says 'Rest charger 15 minutes between charges'.
Why is there such a requirement?
Author : Leslie Green
Leslie Green asks:
In a popular type of train, a Diesel engine drives an electric generator, which in turns powers the electric motors that drive the wheels.
Given that every step in this chain loses power, why is such an apparently complicated system used?
Hint: If you drive or have been frequently driven in a 'stick-shift' (manual gearbox) car you will have a distinct advantage in answering this question.
As a green-minded driver, Gerry has a dilemma.
He wants to buy a new eco model with emissions of 80 CO2 g/km instead of his current car that produces 120 g/km.
An article in the Telegraph says that production of a new car generates about 720kg for every £1,000 you spend on it.
The new car costs £20,000 and he usually drives 10,000 km every year.
How many years does he need to drive to make the purchase ecologically reasonable?
Gerry has one hundred identical books. He puts some of them in a stack on the edge of a table.
What is the maximum horizontal distance that the edge of the top book can get from the edge of the table without falling over?
The blade of an engine is 1 meter long.
If it spins at a full rate of 2400 revolutions per minute (RPM), how many kilometers will the blade tip travel in one hour?
1 kilometer = 1000 meters
You have been offered a game of chance by an eccentric multi-billionaire.
He will toss a coin repeatedly until it comes up heads. If heads appears on the first throw he will pay you $2. If it appears on the second throw, you receive $4; if on the third, you receive $8 and so on, doubling each time.
You know that this is the famous St. Petersburg paradox, with an expectation value of infinity, so his requirement for you to pay him $63 to play the game seems fair. And yet you hesitate …
What is the chance of you winning at this game?
Author: Leslie Green
Jane has always been a bit of an oddball. Rather than buying stocks and selling them later, she likes to sell stocks she doesn't have, hoping to buy them back later.
How does she make a profit?
Author: Leslie Green
You may have heard people use the expression "give it 110% effort" or similar.
What can you say about this?
Author: Leslie Green
A girl takes 3 identical ropes, mixes the ends, and asks her boy-friend to tie three pairs of knots on the top.
There is a superstitious tradition in their region that if the result is a ring with all three ropes, then their relationship will be long.
According to this tradition, what are chances of them getting a long relationship?
Leslie Green asks :
Two ordinary cars with gasoline (petrol) engines are being compared. At their maximum power outputs, one produces twice the power as the other when measured at the road wheels.
What can you say with certainty?
Leslie Green asks :
You try to work out the problem shown on the right by entering the values into a cheap hand-held calculator.
What answer do you get?
Why would anyone design something as apparently stupid as a penny-farthing bicycle?
And why would people buy it?
Author: Leslie Green
Many houses in Switzerland use domestic air-sourced heat pumps for central heating.
A heat pump is a marvellous device. You can put 1kW (kilo-watt) of electrical power in and get 3kW of heat out. Engineers dislike calling this 300% efficient, so they use the term Coefficient of Performance (COP) instead. COP=3 means 3kW out for 1kW in.
Different manufacturers produce units with different performances, so under the same conditions one unit can have a COP of 3 and another can have a COP of 4.
Given the same conditions, how much more electricity will the COP=3 system require than the COP=4 system?
Author: Leslie Green
Four million people paid to watch the movie "Green Coat" in the first month the film was released on iTunes. Some paid $5 to rent the movie, and the rest paid $15 to buy it.
The total payments made were 30 million dollars.
The movie star Jake receives 20 cents from the rented film and 80 cents from the film that was bought.
What is Jake's revenue in the first month?
A woman tries to hide her age from an inquisitive boy by telling him that her age is between 31 and 61 inclusive, but that the boy can only have 4 guesses at her age. The guess will be answered truthfully with higher, lower, or correct.
In the worst (unlucky) case, but with intelligent guesses, how far from the correct age will the boy be?
Author: Leslie Green
An eccentric retired professor of Mathematics has created a walled-off area in his garden in the shape of an isosceles triangle. He tethers a goat to the apex of the triangle with a rope of the correct length so that the goat can graze over exactly half of the garden.
Neglecting practical considerations like the size of the goat compared to the garden, how long is the rope?
(This question comes in two variants: you can either approximate the answer or you can work out the complete answer - an only slightly more involved process.)
Author: Leslie Green
A thin-rimmed hollow sphere (a black ball) and a homogeneous solid sphere (a white ball) both have the same weight and the same diameter.
They are both dropped simultaneously from a height of 3 feet (around 90cm) onto a glass table below. Both balls have the same material and the same finish on their outer surfaces.
Which has the most kinetic energy at the moment of impact?
Author: Leslie Green
Carlo Iznop has an interesting business model for his investment firm. Each client deposits exactly $1000. He gives them 10% return on their investment after 1 year. This profit must be withdrawn. In the first year he had 30 clients. 2 out of 3 withdrew their money at the end of the year. At that time, word spread about this fantastic investment opportunity giving double the normal rate of return, so the total number of investors increased. Over the years his fame spread and the business boomed.
In the fourth year suspicions were raised about the consistently high profits and the police were called.
What is the most probable outcome?
Author: Leslie Green
The yellow point is the center of mass of the shaft.
If we cut it into two parts along the vertical line through the point, which part is heavier?
Leslie Green asks:
Hopefully you realise that wood comes from trees. In fact the bulk of a tree is wood, along with a relatively small amount of leaves.
Where does all this wood come from?
(Pick the best answer, as some of the answers may be partially true.)
The heavy yellow weight is hung from the rigid red post. The blue plate is firmly attached to the ground with stakes (shown as thin vertical lines).
The joint at A is able to rotate freely.
What can you say with certainty about part B.
Author: Leslie Green
Peter has understood from his school work that an airplane's speed is relative to the wind, and not directly to the ground, so that if the wind is going in the same direction as the airplane, the airplane goes that much faster relative to the ground.
He therefore decides that it is a good idea to take off with the wind going in the same direction as his new model airplane is going to take off.
Comment on this plan.
Author: Leslie Green
Much of the brilliance of aircraft design comes in optimising the ratio Lift/Drag for a range of flight speeds and conditions. In level flight the weight of the aircraft is equal to the Lift, but the retarding force, the Drag, is smaller by a factor of 10 or more. This means that the engine can produce 10x less thrust than would be needed to lift the aircraft straight up. Another key specification for a design is the Thrust/Weight ratio, a pretty self-evident measure.
Analyse and digest this technical information before picking an answer.
Author: Leslie Green
A modern sailing vessel, powered only by the wind, is at the center of the image (plan view).
In which direction(s) can it travel?
Author: Leslie Green
You wish to establish how close to vertical a wall is. You have a spirit level and you can see that the bubble is not perfectly centred around its calibration marks when placed flat on the wall. Rotating the spirit level 180° about the vertical axis shows that the spirit level itself is essentially perfectly calibrated. In order to get the bubble centred you need to move the bottom of the spirit level about 1mm away from the wall. The spirit level is 30 cm long.
What is the angular deviation of the wall from vertical?
Note: you do not have a calculator or trig tables to hand.
Author: Leslie Green
There is a long distance hot-air balloon race. The winner will be the team that gets the furthest from the starting point. Each team consists of one or more team members. Each team member has a standardised weight by carrying ballast to make up to 150kg of load per person.
Gas for the burner is provided with one standard bottle per person. The spherical hot air balloons use the same materials, but the balloons have a constant volume of hot air per person.
There are four teams: which one has the best chance, everything else being equal.
Author: Leslie Green
For reasons best known to himself, John decides to get off the train before it comes to a complete stop. He jumps with both feet together in order to land well, although the height of the step is only a foot (30cm) off the ground. He makes sure to land facing in the direction of motion of the train. The train is moving at about 5 mph.
What is the most likely outcome?
Author: Leslie Green
John, now aged 18, goes out to work for the first time. Since he is working, and an adult, he can eat whatever he wants. He likes sugary drinks, chocolate, and crisps. He also gets less exercise than he used to at school. As a result of this lifestyle, his food calorie intake averages out at 100 calories more than he needs every day. The human body typically stores excess calories as fat.
Given that a rough estimate of calories per pound of human body fat is 3500, estimate John’s weight increase by the time he is 28, all other factors being equal.
Author: Leslie Green
Kitty has one thousand beads, 777 of which glass and the rest are precious stones; 333 of them have a red tint.
Estimate the probability that one randomly choosen bead will be a precious stone with a red tint.
When representing negative whole numbers in binary for computers it is convenient to use the two’s complement form.
For example we could represent +3 as 0000 0011 in binary.
To get -3 in the two’s complement form we first invert the bits (change a ‘1’ to a ‘0’, and vice versa) and then we add 1.
Which is the correct two’s complement binary representation of -3?
Author: Leslie Green
It is a serious mistake to think that computers can solve any numerical problem almost instantaneously. Whilst addition and subtraction are very fast, multiplication can take twice as long as that, and division can take ten times as long as addition.
In this example the values subscripted by N are to be calculated hundreds of thousands of times. The unsubscripted values are constants.
Rearrange the equation to minimise the computational time. We are after the form of the equation; the values of the constants vary as necessary to make the result correct.
Author: Leslie Green
A motorboat is in the middle of a fast flowing river heading directly for some rocks that also happen to be in the middle of the river. Its engine failed 5 minutes ago so the boat is just being pulled along by the flow. Seeing the rocks in plenty of time the skipper pushes the rudder hard over to the left. The river channel is safe either side of the rocks.
What happens next?
Author: Leslie Green
John got back to his car too late and now he is locked into an outdoor car park for the night. There is an escape path, but it involves driving down a fairly steep grassy slope. He has correctly drawn a diagram of the problem, but can’t quite finish it off. The axle of the wheels is roughly in line with the underside of the car. Hitting the underside of the car on the corner of the slope will mean a tow truck will be needed and there may be costly damage done as well.
The faint line above the ground is the path that the car axle travels as it rolls along the ground. This is called the locus of the axle.
What is the correct equation for the limiting case where the car base just touches the corner?
Author: Leslie Green
A secret service of a democratic country has 256 spies on payroll in a neighbor country. It is predicted that the network will increase the number of its spies by 25% each year.
How many spies will the secret service have in four years?
Leslie Green asks:
In a singles tennis tournament of 64 players, the winner of each game goes forward to the next round. Two sisters are both excellent tennis players.
Given that each sister wins all their matches until they meet each other, what is the probability that they meet each other at the final?
Leslie Green asks:
Jane sees the following text written on the blackboard in the classroom, evidently left over from a previous lesson.
X = X + 1
Which statement is true?
Gerry drives through the streets from crossroad A to crossroad B. Some of the streets are closed.
If he can only drive East or North, what is the number of different possible ways to get to crossroad B?
In a company, the heights of all employees were measured. The average height of women was 168 cm, and the average height of men was 178 cm. Cindy was the tallest – her height was 180 cm. Zach was the shortest – his height was 152 cm.
A new employee came the next day. The average height of the women and the average height of the men did not change.
Among four conclusions below, only one can be drawn from this information with certainty. Which one?
Three girls can complete writing four notebooks in five weeks.
Four boys can complete writing five notebooks in six weeks.
How many notebooks can 10 boys and 11 girls complete in 12 weeks?
A 1 ton car is heading due North at 60 mph. A 2 ton truck is heading due East at 30mph. There is sheet ice all over the intersection and the truck cannot stop. The truck smashes into the side of the car and the pair forms one tangled mess of bent metal. (Fortunately the drivers were wearing seatbelts and the air bags did their job. Nobody was injured.)
At what speed does the mangled mess initially travel?
(Hint: conserve momentum not energy).
Author: Leslie Green
This incident occurred in deep space. A space ship had been blasting its rockets at full power for several hours, such that the on-board accelerometers recorded an acceleration of 1g, the Earth normal gravitational acceleration of around 10m/s2. The Doppler Space Radar showed another space ship on a direct collision course so the Captain immediately cut off the engines. At this instant the other ship was 1000km away and a collision would happen in 1 hour if nothing was done.
What can you say with certainty about the speed of the other space ship?
Author: Leslie Green
Pop Quiz Paradox
A teacher announces to his class that there will be a quiz one day during the next week. The teacher gives the definition that they would not know when they came in to class that the quiz was going to be given that day.
The brightest student says that the quiz cannot be on Friday because they will know the day on Friday. With the same technique, she eliminates Thursday, then Wednesday, Tuesday and Monday.
"You cannot possibly give us a pop quiz next week," she says.
When does the teacher give the pop quiz?
I know the paradox from Charles Carter Wald. Probably, Martin Gardner described the quiz for the first time in The Colossal Book of Mathematics.
Three identical airplanes (aeroplanes) set out on a vital mission. The lead plane is carrying a secret message which needs to be delivered by hand. Each plane has a full fuel tank and a 1200 mile range. The planes can transfer fuel in mid-air; this process loses no fuel and happens almost instantly.
How far can the lead plane get?
(Note that a plane with no fuel can still land safely.)
Author: Leslie Green
I have 100kg of potatoes, which are 99 percent water. I dry them until they are 98 percent water.
How much do they weigh now?
I insert 2 pins into a white paper covered board and connect them with a thread.
I take a pencil and draw a shape, always keeping the thread strained.
What is the shape?
Sixty percent of people living on the left side of the Long Street vote for the Left Party and forty percent don’t vote. Forty percent of people living on right side vote for the Right Party and sixty percent don’t vote. There are sixty percent more voters on the right side than on the left side.
If there are 780 voters on the Long Street, how many more people voted for the Right Party than for the Left Party?
Sixty percent of people living on the left side of the Long Street vote for the Left Party and forty percent don’t vote. Forty percent of people living on right side vote for the Right Party and sixty percent don’t vote. There are sixty percent more voters on the right side than on the left side.
Who wins the election?
Eleven players are standing in a soccer pitch such that the distance between each pair is distinct. Each player has a ball, and when the trainer blows a whistle, each player kicks his ball to the nearest player.
Which is not necessary true?
A dealer has 40 brand new cars to sell. He paid $40,000 per car. He hopes that half of the new cars will sell at the original price he sets, 40% of the remainder will sell later when their price is reduced by 10%, and the rest of the cars will sell in a clearance sale at only 80% of the original price. In order to make an expected profit, the total sales revenue must be $6,000 on average for each car.
What original price should the dealer set?
Several students (N) randomly choose their places on a bench.
What are the chances for Gerry to be sitting next to Jane?
If I start walking at my normal speed from home to my granny's house I will be 10 minutes late.
If I run at twice my normal walking speed I will be 30 minutes too early.
After how much time do I need to switch from walking to running to arrive on time?
Jenny is a computer scientist and is shy about her age. On her birthday she encodes her age in binary in a row of 8 candles. Her boyfriend, who is sitting on the opposite side of the cake, is trying to work out her age from the pattern of lit candles.
Knowing that her boyfriend is fluent in binary, Jenny encodes the pattern correctly, but does not reveal if a lit candle represents a “1” or a “0” for that bit position. She also does not reveal if she has written the binary number either left to right increasing (standard notation) or the other way around.
Given that Jenny is 27 years old, which age cannot reasonably be read by the boyfriend?
Author: Leslie Green
A cellar square floor is to be tiled in the way shown in the picture.
If the cellar size is extremely large, what is the percent of the green tiles?
The knight jumps either 2 squares vertically and 1 horizontally, or 2 squares horizontally and 1 vertically, as shown in the picture.
What is the minimum number of moves required for the knight to attack any square of the board from the current position?
Edgar chooses one of the double sheets of "The Daily Dog News".
The two internal page numbers of the sheet are 18 and 31.
How many pages are there in the newspaper?
A 1-liter bottle contains a mixture that is 1/4 antifreeze, 3/4 water.
A 2-liter bottle contains a mixture that is 1/3 antifreeze, 2/3 water.
I pour the contents of the two bottles into my car's empty radiator.
What is the percentage of water in the radiator?
Note: Three liters of the liquid is not enough for a vehicle's radiator.
A peasant bought a cow and a horse.
Next day, he sold them for $1,800 each, making a loss of 20% on the cow and a profit of 20% on the horse. What did he make?
The photograph courtesy of Roland Sauter
A taxi driver called John has 6-week annual holidays.
If the average fare is $17 and he usually manages to make 70 trips per week, what is his total income (before tax and expenses) for the whole year?
Leslie Green asks:
"A professional design standard requires that free standing equipment must not tip over if subjected to a force equivalent to one fifth of its weight applied at the worst possible point. The latest design has failed the test.
What can the design team do to fix the problem?"
Leslie Green asks:
Suppose the light bulb in your fridge uses 1 Watt when on and you pay 10 cents = $0.1 per kWh.
Estimate the cost due to a faulty door switch, which keeps the light on over a 10 year period even with the fridge door closed.
(1 kWh is 1000 Watts for 1 hour)
Leslie Green asks:
Electrical energy can be measured in Joules, but for household use it is typically measured in kilowatt-hours (kWh). If a 100 Watt bulb is switched on for 8 hours every day for a year, roughly how much energy is consumed?
J = 1000 x kW x S
where J = Joules, kW = kilowatts, and S = seconds
If $1,000 is invested in an account at 10% annual interest, how long will it take the account balance to grow to $1,600?
You may have heard the expression "The shortest distance between two points is a straight line."
I have drawn a straight line on the map, directly along one of the grid lines of the map. Is this the shortest path for a ship to travel? (Neglecting winds, tides and so on.)
(NOTE: this is for a ship, not a submarine!)
Author: Leslie Green
The train driver knows there is a Granny on board so he wants to give her a comfortable journey and to make sure she doesn’t spill her tea. When he is braking, which curve should he follow?
(HINT: remember Newton’s Second Law, often expressed as Force = Mass x Acceleration)
Author: Leslie Green
Gerry promises to be at Jane's house at 19:00.
If he bikes at 15mph, he arrives 1 hour earlier.
If Gerry bikes at 10mph, he is 1 hour late.
What is the speed to be exactly on time?
Before his death, a rich man signed his will.
"If my wife gives birth to a boy, she will get a third of $42,000,000 and my son will get two thirds of the money.
If my wife gives birth to a girl my wife will get two thirds of the money and my daughter will get one third."
After the rich man died, his beloved wife gave birth to twins: a boy and a girl.
What is the fair share of the money for the lady?
There is a fault with the cruise control on Hank's car such that the speed continuously and linearly increases with time.
When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph.
For how many miles did he drive above the state speed limit of 70 mph?
Author: Leslie Green
Jane and Gerry compete in a best-of-five match.
If Gerry plays so that his girl-friend has a 75% chance of winning any particular game, what is the likelihood that she will win the match?
Leslie Green asks
"Suppose a particular nuclear waste material has a half-life of 100 years.
What could you do to reduce the radioactivity of the material itself to less than 7% of its current value?
(The half-life of a radioactive material is the time it takes, on average, for half of it to change into something else by spontaneous radioactive decay.)"
Leslie Green asks:
My neighbor John has invented a perpetual motion machine. It pumps water with no apparent power input and can even pump water up over a 2m fence.
How would you categorise this invention?
Shops have sales all the time to attract your business. Let the buyer beware! Not all sales are as good as others.
Given the same branded goods being sold, and the same quality of after-sales service, which shop offers the best value, given that two weeks ago all had the same price.
The problem was suggested by Leslie Green
A train runs from Lausanne to Zurich at 100 km/hour without any stops and another one from Zurich to Lausanne at 80 km/hour.
What is the distance between the trains 15 minutes after they meet each other.
There are 10 coins and 5 of them are fake.
The real gold coins have the same weight, while all fake coins have different weights and each fake coin is lighter than a real one.
How many times do you use a balance to find all real coins?
Find the best strategy and the least possible number of weighings in the worst case.
Now that I have made a vast fortune from my patented premium dog biscuits, I can afford to build the luxury mansion of my dreams.
I thought my design requirement was very clear: The water for the walk-through shower can be turned ON and OFF from both ends of the room.
The plumber doesn't understand so I have drawn him a plan. I had a few attempts before I got it right!
Which is the correct drawing?
The problem was suggested by Leslie Green
Eugenia wants to make a simple bridge for her dog. Currently he has to run through a tiny stream in the back garden and then walks mud into the house. Since Eugenia’s dad owns and runs a machine shop, she can easily get a single sheet of steel, aluminum or wood to bridge the stream.
The length suits the size of the stream, the width suits the size of the dog, and the weight will be as much as she can carry.
The strength of a plain sheet is proportional to the relative strength of the material, its width and the cube of its thickness.
Which of the available materials makes the strongest bridge?
Suggested by Leslie Green
A girl walked along a level road and up a hill from home and back.
Her pace on the level is 4 km an hour, uphill 3 km, and downhill 6 km.
How much time does it take if the total distance is 20 km?
Inspired by Lewis Carroll's Tangled Tale
A worker prepared a block for a pyramid in a day.
If every day the number of workers doubled, how many days did they need to prepare 1000 blocks?
The diagram shows some of the results of a six-person contest. There are two matches left for everybody.
An arrow pointing from one player to another signifies that the first player defeated the second player in the match. For example, player A defeated player C in their match.
Which player has the strongest opponents left to play?
When Pinocchio lies, his nose gets twice as long.
When he tells the truth, his nose gets 1 cm shorter.
His nose was 1 cm long in the morning, and it is 100 cm long in the evening.
What is the least possible number of times he opened his mouth today?
There are 17 parallels and 12 meridians on a globe.
Into how many areas is the surface of the globe divided?
If you have a piece of paper that is 0.1mm = 0.01cm thick, how many times will you have to fold it in half in order for it to become as tall as me?
I am 163cm tall.
The manager of a company planned to distribute a $101 bonus to each employee from the company fund, but the fund contained $1 less than what was needed.
The manager gave each employee a $100 bonus and kept the remaining $199 in the company fund.
What was the amount of money in the company fund before any bonuses were paid?
A district sends, on average, a total of 205 million gallons of water each month to its 20,000 residential customers.
In the winter months, it sends an average of 100 million gallons per month.
Assuming the winter period lasts three months, what is the average number of gallons sent to each person in each nonwinter month?
Gerry gave half of his money to Jane.
Next day, she gave half of all her wealth to Gerry.
After the last exchange they each have exactly as much as they originally started with.
Who is richer?
Gerry drives from his home to Jane's house at 50 mph.
How fast must he make the return trip via the same route such that the average speed of the entire trip becomes 40 mph?
I decided to use a password that includes 4 different numbers, with each number smaller than the previous one.
How many options do I have?
The limousine service makes 100 trips Center-Aeroport every day, each of which costs $50.
The company estimates that the number of trips decreases by 5 trips per day for each $5 increase in the fare and vice versa.
What fare is the most profitable for the company?
The water from an open swimming pool evaporates at a rate of 5 gallons per hour in the shade and 15 gallons per hour in the sun.
If the pool loses 8,400 gallons in June and there were no clouds, what is the average duration of night during that month?
While eating out Gerry and Jim together tipped their server $7 in total.
Gerry tipped 20% of his bill and Jim 15% of his bill.
Jim's bill was twice as much as Gerry's one.
They don't remember how much they paid.
Can you find the total including the tip?
James bought a new car for $30,000.
Each year it depreciates (loses value) at a rate of 20%. The maintenance costs $3,000 the first year and it increases by 20% each year.
When does the maintenance cost exceed the value of the car?
Texas experiences wide temperature fluctuations within a single day.
On January 1st, the temperature drops from 62° to 30° in Austin, and from 24° to 13° in New York.
What is the range within which the temperature difference between these two cities must be?
Gerry plans to swim a total of at least 1000 laps in January.
He swims every day except weekends (Saturday and Sunday).
He wants to increase the number of laps each day by one.
What is the least number of laps he must complete on the first weekday if January 1st is Monday?
You have a large supply of 5kg and 12kg weights.
Six 5kg weights and one 12kg weight have an average weight of 6kg.
What is the minimum number of weights that have an average weight of 7kg?
What is the total surface area of 100 identical cubes which together have a volume of 51,200 cubic units?
This is a typical SAT question.
What is the minimum monthly salary you negotiate with your boss if you need
$1,800 per month to pay your house loan and its expenses,
$200 per week for yourself, and
$18,000 per year for leisure and saving?
PS: You pay 20% tax.
A space rocket with 4 engines blew off one of them after one-month of flight.
It continued the flight at three-quarters of its former speed, which brought it to the destination two months late.
How long did the trip last?
A pirate boat is floating on a lake.
The pirates throw a heavy chest with gold coins overboard.
What happens with the water in the lake with respect to the shore?
An entrepreneur sells 1,250 tickets for a show, with 20% of them sold with 20% off.
How much does a normal ticket cost if the total income must be $120,000?
I see the reflection of the mountain peak 10 meters from me.
I know that the difference between the height of the lake and that of the peak is 1,800 meters.
How far from me is the mountain if I am 1.5 meters tall?
You pay $6 to enter a game.
You roll 2 dice.
You then have two choices: you can cash out and get paid the dollar amount of the roll, or you can pay $1 to roll the dice again.
What is your final gain per game if you play many times and choose the strategy to stop once you get more than 6 on two dice?
One by one, 40 cars enter a company parking lot with 40 assigned places.
The first driver forgets his place number and takes a random place.
The remaining drivers take their assigned place, if available, or take a random place.
What’s the probability that the last driver ends up in his original assigned place?
Inspired by Peter Winkler's airplane problem.
There are 190 people in a company including the project managers and CEO.
Project managers successively take out half of the total number of people to start a project until only the CEO is left.
How many project managers are there in the company?
Management support accelerates the execution of a project by 10% during the first year of the project.
The same support delays the project execution by 10% during the second year.
Will the project execution take more or less time with management support?
Mary and John manage two different projects and they have a common budget.
Mary spent half of the budget, then John spent half of what was left, then Mary spent half of what was left and so on until the last cent.
What proportion of the budget did Mary spend?
A father has left 47 donkeys for you to distribute to his three sons;
1/2 (23.5) should go to his eldest son,
1/3 (15.667) to the middle one, and
1/8 (5.875) to the youngest.
You arrive with your own donkeys.
You can add or take some donkeys from the herd.
How do you divide them so that all sons are happy with your decision?
How many donkeys does the eldest son get?
Inspired by Malba Tahan “The man who counted.”
When I put 10 cubes of ice into my glass and they melt, the water doubles its volume and completely fills the 160 ml glass.
Estimate the size of a cube.
1 ml = 10mm x 10 mm x 10 mm
Mrs. Brown bought 98 boxes of strawberries at $9 per box but 25 % of them were found to be rotten.
She sells the non‐rotten strawberries.
What should the selling price per box be so that her profit is 50%?
Two years ago, I bought a cottage for $800,000.
I was short of money and borrowed 12.5% of the money from a bank with 10% annual interest.
I sold it after two years and its value increased by 5% in that period.
What is my profit after I paid the bank back?
PS: I did not pay any tax for such a small transaction.
Twenty theater tickets for the first row were randomly distributed among 20 students.
What is the probability that John gets a seat next to his true love Jenny?
Water is pumped into an empty swimming pool at a linearly increasing rate, which was zero at the beginning.
If the pool is half full in 10 hours, how much extra time is needed to completely fill it?
Anna was planning to buy 10 souvenirs from the honesty shop.
She wanted some badges, which cost $5 each, and some Swiss chocolate bars, which cost $4 each.
After checking her wallet she decided to put 20% of the badges back, only at this late stage neglecting her initial plan to buy 10 souvenirs.
How much money did she spend in the shop?
Two proofreading programs, A and B, discovered 30 and 40 errors, respectively.
There are 10 errors in common.
Estimate the number of errors that are still undetected.
If John walks up an escalator at a rate of one step per second, 12 steps take him to the top.
If John goes up at two steps per second, he reaches the top in 20 steps.
How many steps are there on the escalator?
Inspired by A. Dunn, Mathematical Baffles, Dover Publications, 1980, p 17
The size of a swimming pool is 25 x 12.5 meters.
How many people must jump into the pool so that the water rises by 1 meter?
John and Mary have square tiles with sizes 5 cm by 5 cm, 10 cm by 10 cm, and 15 cm by 15 cm.
Their kitchen floor is a rectangle with a 230 cm by 440 cm surface.
What is the minimum number of tiles required to completely cover the floor surface without cutting the tiles?
John and Mary have square tiles with sizes 2 cm by 2 cm, 4 cm by 4 cm, 6 cm by 6 cm and so on.
Their kitchen is a rectangular room with a 224 cm by 288 cm surface.
What is the minimum number of tiles of the same size required to completely cover the surface without cutting the tiles?
In a game, Anna and Bill take 1, 2, or 3 coins on each turn.
The player to take the last coin from the pile wins.
If Anna goes first and there are 40 coins on the table, how many coins should she take to guarantee that she would win?
Wikipedia says that the equator is about 40,075 kilometers (24,901 miles) long; 78.7% is across water and 21.3% is over land.
A yard was defined as the distance from finger tip to nose with the arms straight out to the sides.
Allowing 5 inches for a palm grip at each side we have 2(36-5) = 62 inches or 1.6m.
Estimate the number of people who holding hands in a human chain cover the length of the equator.
Job interview logic puzzle in a bank.
Which of these pieces of information would be most useful for estimating the number of people who travel in a train with 20 passenger coaches?
The speed of a bus is 25 miles per hour (mph) including stops and 40 mph excluding the stops.
For how much time does the bus stop in a 48-minute trip?
Every year, a man who had a million dollars at the beginning gives away half of his money to his son, and after that gives 20% of what he has left to charity.
How much does he end up with at the end of the 5th year?
Four people arrive at a river at night. There is a narrow bridge, but it can only hold two people at a time. They have one torch and, because it is night, the torch has to be used when crossing the bridge.
Person A can cross the bridge in one minute, B in two minutes, C in five minutes, and D in eight minutes.
What is the shortest time they all can get across the bridge?
Source : Wikipedia
A red-line subway train arrives at a station every 5 minutes and stays for 30 seconds.
A blue-line train arrives on the other side of the platform every 2.5 minutes and also stays for 30 seconds.
I randomly arrive at the station and always choose the train which arrives first, or any of the two if both trains are present.
What is the largest possible percentage of time that I take the red-line train?
The Lucas problem.
François Édouard Anatole Lucas (1842-1891) was a French mathematician.
Every day at noon, a ship leaves Le Havre for New York and another ship leaves New York for Le Havre. The trip lasts 7 days and 7 nights.
How many ships will a ship leaving Le Havre today meet at sea?
The diagram shows a road network.
All cars drive in one direction from A to F.
The numbers represent the maximum flow rate in vehicles per hour.
Engineers want to construct a new road with a flow rate of 100 vehicles per hour. Drivers randomly choose the road at crossroads.
What new road decreases the capacity of the network (the number of vehicles at point F)?
I gave $120 to Brett, Cindy, and Daniel.
Together Brett and Cindy received twice as much as Daniel.
Together Daniel and Cindy received three times as much as Brett.
Who received the most money?
"Letter frequency. Accurate average letter frequencies can only be gleaned by analyzing a large amount of representative text." (source: Wikipedia)
What is the most common letter in the English alphabet?
John's dad pays him $2 for each correct answer he gives in his math homework and fines him $8 for each incorrect answer.
Today, John received nothing after doing 50 problems.
How many problems did John answer correctly?
A goldfish needs 1000 cubic inches of water to live in.
An aquarium is 20 inches in diameter.
How many goldfish can live in this aquarium?
I am selling my car. People come to my garage one at a time and make bids to buy it. I make an immediate decision whether to accept or reject an offer after receiving it. I decide to reject the first N offers, mark the highest price P, and accept the first offer that is greater than P.
What value of N do you recommend to me if I expect that 270 people will make a bid?
If N is small, I can accept a small amount of money.
If N is large, I can reject the best offer.
You have 240 golden bricks identical in size and appearance but one is lighter than the others.
How many times do you use a balance scale to find the odd brick?
Choose the best advice for a strategic decision to the top management of a bank.
The profits stated are NET, in other words costs have already been deducted.
Anna can complete a project in 40 days.
Bill can complete the project in 50 days.
Cindy can complete the project in 60 days.
Daniel can complete the project in 120 days.
Their daily wages are proportional to their performance.
Find 3 people who together complete the project in exactly 20 days.
The Legend of Carthage: Queen Dido and her followers arrived in North Africa.
The locals told them that they could have the coastal area that an ox hide would cover.
She cut the hide into a series of thin strips, joined them together, and formed a coastal shape.
The ox-hide enclosed area was known as Carthage.
If you had a 10 km long strip, which shape (rectangle, triangle, semi-circle, or semi-ellipse) would you choose to maximize the enclosed area?
The diagram illustrates five villages, A, B, C, D, and E, with the distances between them in miles.
A postman must travel from A through each of the other villages exactly once and then back to A.
Identify the shortest possible route.
The Chinese postman problem: a postman wants to travel along each road in his quarter and come back to the Start.
Find the minimal length of his route.
The traveling salesman problem: a salesman has to visit 9 towns and return home.
What is the shortest available route between the towns?
Any route chosen must lie on the paths shown. Paths with unmarked distances should be calculated from the geometry of map.
It costs $24 to paint a cube, the cost being proportional to the surface area.
Before painting, it was cut in two pieces by a plane.
What is maximum possible cost to paint these two pieces?
A city hall floor is to be tiled in the following pattern.
The hall measures 121 tiles x 61 tiles.
How many green tiles do we need?
You have eight dimes; seven are real and one is fake.
All the real ones weigh the same, and the fake weighs less than the real ones.
How many times do you use a balance scale to find the fake dime?
A bicyclist pedals downhill at 20 km/h (kilometers per hour) and uphill at 10 km/h.
It takes 4 hours to travel from a Swiss mountain village to another village.
The return trip takes 5 hours.
Find the distance between the two villages.
9 farmers grow 9 apples trees in 9 years.
How many years would it take for 999 farmers to grow 99 apple trees?
Jane wants to buy a car.
The fuel efficiency is two times more important to her than the price of the car.
Help her to choose a car.
1 mpg (mile per gallon) = 0.425 kilometers per liter
The photograph courtesy of Roland Sauter
A carnival game offers you the opportunity to bet $1 on a number from 1 through 6 on a single roll of the two dice.
If your number comes up on one die, you win $2 and keep the $1 you bet.
If it appears on both dice, you win $5 and keep the $1 you bet.
Only if the number does not appear on either die do you lose your $1 bet.
Does this game favor you or the carnival?
Source: Mathematics Teacher, NCTM Journal
A space rocket has a critical fuel-control system protected by 4 fuel micro-controllers running together. If all 4 micro-controllers fail during the flight, the rocket will crash. The probability that any one micro-controller will fail is 1/5.
If the rocket has lifted off, what is the probability that it will not crash due to the fuel-control system?
If eight deciliters of Swiss apple juice, priced at $4.00 per liter, are combined with twelve deciliters of Italian apple juice, priced at $3.00 per liter, what is the price of the resulting mixture?
Jim sells 80 percent of pineapples he had and throws away 10 percent of the remainder. On the next day, he sells two-thirds of the remainder and throws away the rest.
What percent of his pineapples does Jim discard?
If 9 workers can build 9 houses in 9 months,
then how many months would it take 11 workers to build 11 houses?
A taxi company charges $1.75 for the first kilometer and 25 cents for each additional kilometer.
What is the maximum distance someone could travel with seven dollars?
Tom is choosing between two brands of AA batteries for his pocket flashlight. A package of two Brand A-Super batteries costs $4.99, and a package of two Brand B-Cool batteries costs $3.29. The runtime of the A-Super package is 2.5 hours. The runtime of the B-Cool package is 100 minutes.
Help him choose.
John works 4-hour shifts at a call center.
He earns $4 per hour and $4 per call.
How many calls does he need to receive per shift to earn a total of exactly $400 in 4 shifts?
How many arrows are there in the picture?
Which of these equations has been re-arranged incorrectly?
Estimate the probability that it will rain some time today, based on the local weather forecast.
Between which two countries is the walking distance the smallest?
How many matchsticks are there in the picture?
Which is the most critical of these critical infrastructure items for an urban area?
Why does it matter where your food originates from, after all rice is rice? 
How should you go about becoming an excellent baker?
What is an unsatisfactory method of acquiring land?
In which direction should the heavy door on a pressurised cargo airplane open?
What is the message hidden in the cipher-text?
An irresistible force acts on an immovable object. 
Which item should be considered as the most harmful to the environment? 
What is the Mach number?
Leslie Green asks: 
Which two geometric shapes form the astronomically correct crescent shape?
Analyse the proverb. 
Which lottery is better, assuming each has the same cost to play?
I add blue blocks one by one on the top. 
What is average diameter of the logs if the height of the pile is 5.33m?
Estimate how many symbols you can write on your body if one symbol requires one square centimeter.
Which point cuts the rope into two equal parts?