In the penal colony of Peena Munda, the governor likes to play a game every day.
100 prisoners are chosen at random, and given a large numbered plaque to put on their backs #1 to #100 inclusive).
They are led to a long straight corridor of closed doors, numbered 1 to 100 inclusive.

Prisoner #1 opens every door which has a number evenly divisible by 1, which is of course all doors.
Prisoner #2 closes every door which has a number evenly divisible by 2.
Prisoner #3 changes every door which has a number evenly divisible by 3, which is to say he closes open doors, and opens closed doors.
...
The task continues for all the prisoners. If a prisoner touches a door they are not supposed to, they are dealt with ... harshly.

How many prisoners change the state of exactly five doors?

Octavia and Septimus start a family, and in order to satisfy their cultural preferences call their children Primus, Secundus, and Tertius.

How many children are missing to complete the full set?

Three pirates Ardy, Bardy, and Cardy play a game.
There are 1,000 coins in a chest.
First Ardy takes 1, 3, 5, 7, or 9 coins.
Then Bardy takes 2, 4, 6, 8, or 10 coins.
Then Cardy takes 1 coin and Ardy starts again.

If one of them cannot take the allowed number of coins, then he skips his turn.

The winner is the pirate who takes the last coin.

The other two pirates give him at least a half of their coins.

What is the largest possible number of coins the winner gets if two other pirates play their best strategy?

At the end of the school year the Head of Mathematics becomes her alter ego, Lady Mathemagik. She seats five (trained) staff members in a circle, facing inwards. Each staff member has a large digit { 1, 2, 3, 4, 5 } on their chests, clearly visible to the staff member wearing it, and to everyone else. The digits on the seated staff members seem to have no obvious pattern around the circle.
Each of the five staff members wears a white cardboard hat, and students are randomly picked to randomly write a number between 1 and 5 (inclusive) onto a hat. Finally, each staff member has a single number on their hat which only they don’t know.

The students want to make it difficult for the staff members to guess their own hat-number. If at least one staff member correctly guesses their own hat-number, then Lady Mathemagik wins. If no staff members guess their own hat-number, Lady Mathemagik has to buy ice-creams for all the students. The staff members write down their guesses secretly so none of the five seated staff members knows the other guesses.

There is no cheating involved.

What is the expected outcome?

The numbers from 1 to 2048 are placed clockwise on a circle. We move around the circle clockwise erasing every other number until only one number remains.

If we start by erasing 1, what is the last remaining number on the circle?

In this solo game you add the numbers clockwise according to the chosen rule, but the number in any position can only be added once per game.

Suppose you chose the rule "start from 2, skip two". In this case you would add: 2 + (skip the 3 and 5) 6 + (skip the 4 and 8) 9 + ...
You can continue around the circle as many times as you like.

Given that this is a mental arithmetic problem, using a calculator invalidates your answer.
Likewise, writing anything down invalidates your answer.

Which rule will you choose to maximise the sum?

by Leslie Green

You have been challenged to a game of Last Coin by the arrogant head of the Nerds gang. You will take the first coin or coins from the heap, but you get to choose how many coins are in the heap. Once you have chosen the number of coins in the heap, the Head Nerd will then decide how many coins can be taken per turn. This will be one of (a) 1 to 3, (b) 1 to 4, or (c) 1 to 5.

To be clear, if the Head Nerd chooses option (b) you must take at least one coin, but not more than 4 when it is your turn. The last person to take a coin loses, and of course will suffer endless humiliation as a LOSER!

How many coins will you put in the heap?

HINT: It would be sensible to make sure you know how to win at the easy version of Last Coin.

Three cars start from the same location on a large open and flat prairie. The wind is a constant 50 mph from the South.

Nigel drives directly to North City, which is 60 miles due north from his present location, at a constant speed of 60 mph.

Simon drives directly to South City, which is 60 miles due south from his present location, at a constant speed of 60 mph.

Earl drives directly to East City, which is 60 miles due east from his present location, at a constant speed of 60 mph.

There are no obstacles like rivers, mountains, canyons, or anything which would prevent direct driving to the chosen destinations.

Who arrives first?

Democracy has come to the little kingdom of Zirconia, in which the king and each of the other 65 citizens has a salary of one zircon. The king can not vote, but he has power to suggest changes. Each person’s salary must be a whole number of zircons, and the salaries must sum to 66.

Each suggestion is voted on, and carried if there are more votes for than against.
Each voter can be counted on to vote
“YES” if his or her salary is to be increased,
“NO” if decreased, and
otherwise not to bother voting.

What is the maximum salary the king can obtain for himself with the democratic system?

This puzzle was devised by Johan Wastlund inspired by historical events in Sweden.
The text of the puzzle is from the Mathematical Puzzles, 2021 by Peter Winkler.

There are three light bulbs, labelled A, B, and C.
There are three switches, labelled (1), (2), and (3).

Each switch connects to exactly one light bulb, and each light bulb connects to exactly one switch. Sadly the information about which switch connects to which light bulb has been lost by the incompetent electrician who did the work.

The equally incompetent site foreman makes the following three statements, but only one of these statements is actually correct.

Switch (1) controls light bulb B.
Switch (2) controls either light bulb A or light bulb C.
Switch (3) controls either light bulb A or light bulb B.

Which switches connect to which light bulbs?

We can place eleven green L-shapes on a 10x10 board so that no shapes have a common point. The corners of the green shapes coincide with the nodes of the 10x10 grid.

How many L-shapes can we place on a 17x17 board?

In ancient times the Roman Gods used humans as playthings. On one such occasion Diana, Goddess of the Hunt, lined up infinitely many young women, all sequentially labelled with natural numbers starting from 1. Mars, God War, placed infinitely many young men facing the line of women, and again all were sequentially numbered starting from 1. Woman 1 faced Man 1, Woman 2 faced Man 2, and so on.

Mars was particularly grumpy on that day and insisted that having looked into the future of mankind, he had discovered that man would eventually realise that Infinity was only to be understood by the immortal Gods. Diana spoke is hushed tones: "There are as many woman as men on this field."

Mars thundered out his reply as all the odd-numbered men vanished from that plane of existence. "Hah! Your mind is weak, huntress Diana. Lo, there are still as many men as women, since there are infinitely many of both."

The skies darkened, lightning flashed, and the very fabric of space-time was threatened with oblivion as these gods disputed with each other, Diana knowing she had twice as many women as Mars had men.

Fortunately, Minerva, Goddess of Wisdom, coalesced out of nothingness. "Please let me help you to resolve this problem." Mars and Diana agreed to this intervention, albeit reluctantly. "I will remove humans from their lines in pairs so that you may see if the lines have equality in any sense"

With a subtle waft of her arm, Minerva caused man 2 to touch woman 2, man 4 touched woman 4, and so on in rapidly accelerated fashion, and as they touched they phased out of space-time.

What was the final result?

by Leslie Green

Hilbert’s Hotel has infinitely many single rooms, all of which are occupied by number theorists. Suddenly, and unexpectedly, infinitely many philosophers arrive. Hilbert, being a genius mathematician, issues a loudspeaker announcement for all current guests to move out of their current room to the next available odd-numbered room. Since there are as many odd-numbered rooms as any-numbered rooms, Hilbert has freed-up infinitely many even-numbered rooms for his new guests.

Since all the rooms are currently occupied by number theorists, each knows immediately where they need to go to find their new room.

Given that a number theorist is in room H, to which room number do they then go?

by Leslie Green

1000 people stand in a large circle in a football stadium, all facing towards the centre of the circle. Each person starts with $2000 in nice fresh $1 bills (notes).

The people are sequentially numbered from 1 to 1000 by first randomly picking a person to be "1", then incrementing this number for each successive person to their left. Thus person 2 is to the left of person 1, and so on.

Each person gives an amount in dollars equal to their number to the person on their left. Thus person 21 gives 21 dollars to the person on their left, and so on.

Once everyone has given and received their share of money, who has the most money?

1000 people stand in a large circle in a football stadium, all facing towards the centre of the circle. Each person starts with $1000 in nice fresh $1 bills (notes).

Each person waits an arbitrary time before handing a random amount of money (but not more than $300) to the person on their left.

We wait until everyone has given money to the person on their left.

What is the total amount of money held by the people in the circle?

Granny Grasshopper has tragically lost the spring in her step. Being extremely old, but wise, she compensates by making a catapult which she takes wherever she goes.

Having fallen down a slippery 20 foot deep well, she arranges her escape by catapulting herself (and her trusty catapult) straight up the well walls. She attains a height of 5 feet for each jump, but slides back down 4 feet almost immediately. She can only make one such jump each day, as winding up the elastic is very tiring.

At the start of day 1 she is exactly at the bottom of the well. On which day does she escape?

You have been given a set of 4 weights, namely {2, 3, 5, 7} kg respectively. You have also been given a very sensitive pair of balance scales

Each of the weights has the same relative accuracy, for example ±1.5% of the stated value.

You have realised that any weight from 1 to 15 kilograms can be measured in a single balance operation. However, which measured weight will have the greatest (worst case) relative inaccuracy, that is the greatest possible ±percentage error?

Linda received her Doctoral degree in Ecology two years ago. She has been a member of the campaign for nuclear disarmament since her undergraduate days. She is vegan, blonde, and loves her two cocker spaniel puppies.

Which is more probable?

Gerry and Jane visit the home of an eccentric and reclusive billionaire. When the visit naturally comes to an end, the butler shows them to a special exit room. First he explains the rules. Gerry and Jane will enter the room separately, with only one in the room at any time. They will leave by another door. In the room will be two closed boxes, each randomly labelled by a distinct capital letter of the English alphabet. (To be clear, both cannot be labelled by the same letter.)

In each box is a ticket labelled either "Gerry" or "Jane". Whoever goes in first opens just one box. If they find their name they show it to the under-butler who then restores the box and ticket to their original locations before escorting them out of the room.

If Gerry or Jane communicate in any way after the first of them enters the room they are disqualified and leave with nothing. If both Gerry and Jane find tickets with their name on, they receive £100,000 each. Otherwise they both leave with nothing. The butler will arbitrarily decide who enters the room first, at which point no further communication is allowed between Gerry and Jane.

What is the probability that both Jane and Gerry leave with the prize, given that they are both very clever, and they discuss their strategy beforehand?

The director of an old prison decides to give 4 prisoners a slim chance to survive. He arranges 4 numbered boxes [1 - 4] in a room. Under each box is a random unique ticket in the inclusive range 1 to 4. The 4 prisoners are also numbered 1 to 4, inclusive. Unknown to the director, one of the 4 prisoners chosen is literally a screaming genius at Statistics and Number Theory. He briefs the other 3 prisoners and they follow his simple plan. The prisoners enter the room one at a time in a random order, then pick two boxes to see the hidden ticket numbers. The prisoners leave the room in exactly the same state as they entered it, and by another door, so they do not interact with the other prisoners.

If and only if all four prisoners find their respective numbers do they get early release.

What is the probability of this outcome?

Try the earlier problem first to ensure you can appreciate this epically hard problem.

Income tax is apparently straightforward. Typically you are allowed a certain amount of income before you pay any tax at all. Your personal tax allowance might be say £13,000 per year. If you earned say £14,000 you might pay 20% tax on the difference between your earnings and your allowance. In this example it would be
(20 / 100)x(14,000 - 13,000) = £1000 / 5 = £200.
If you earn an extra £1000, you pay an extra £1000 / 5 = £200 in tax. The 20% value is known as a marginal tax rate. If you earn a lot more, maybe you fall into the 40% tax band. In that case any additional income attracts a marginal tax rate of 40%.

If the marginal tax rate exceeds 100% you lose money if you earn more.

Is it ever possible that earning more money means you take home less money according to government rules.

How does your government covertly steal from you?

An ancient English puzzle

"Three fishermen lay down to sleep, not having counted or divided their catch. In the night, one of them woke up, and (not quite trusting the others) decided to divide the pile of fish and take his share. But the number of fish wasn't equally divisible by three. However, he found he could throw one fish away, then take exactly a third. Later, the second and third fishermen woke in turn, and each went through the same process. What is the minimum number of fish in the catch that fulfils these conditions?"

English quantum theorist Paul Dirac postulated not only antiparticles but also "antifish". He made a joke that the correct answer is -2 fish (or two antifish).

However, the old puzzle has a positive integer solution. Which one?

Ahmed has hidden two single-digit numbers in the binary string shown.

The first number is 7.

What is the second number?

Which is the most dangerous word in mathematics?

On an distant island, all men are clean-shaven, and nobody visits this island. We describe three sets:

A   = { men who shave themselves }
B   = { men who are shaved by the barber }
C   = A + B = { all men on the island }

The "+" sign, when acting on sets, is a convenient short-hand notation for the union of the sets.

The barber is a man, and he is the only barber on the island.

In which set is the barber?

by Leslie Green

My cell has four solid walls, a solid floor, and a solid ceiling. There is one door in one of the four walls, but it is solid steel, closed, locked, and leaves no gap. I have no tools and no ladder. I am an ordinary human being. Each wall sits on the solid floor without a gap. The ceiling rests on all the walls without a gap.

How then can I just walk out of my cell without breaking anything, without opening the door, and without assistance?

How is it possible that two plus two equals one?

Given that I only like people with beards, do I like girls?

The average annual income of a family is 240 thousand dollars in Richland and 100 thousand dollars in Poorland.

The move of 4 families from Richland to Poorland increases the average incomes in the two villages by 40 thousand dollars.

How many families live in the two villages?

In a particular prison, the warden wishes to eliminate smart people by letting them go if they pass his test. In full sight of a prisoner, a guard hides a key under one of three upside down soup bowls, arranged neatly left to right on a table against the wall. On the top of each soup bowl is a fair coin, placed either heads up or heads down at the whim of the guard. The prisoner is allowed (but not required) to flip exactly one coin (that is change it from heads to tails, or vice versa).

The prisoner and his cell mate are aware of the test, and formulate a plan of action. They want to find the key and therefore be released. The only information available to the cell mate is the head/tails states of the coins when he is taken to the room containing the table and inverted soup bowls.

Can they guarantee to be released?

You might like to try the easier question first.

A professor and his spouse are at a party. There are exactly five married couples at the party, and nobody else.

The people shake hands with at most some of the others, but not with their own spouse, and also not with themselves.

At the end of the party the professor asked all the other guests at the party (including his spouse) how many different people they shook hands with. Each person tells him a different answer (meaning, if one person said “five,” no one else said “five”).

With how many people does the professor shake hands?

Source: Steve Miller's Math Riddles

When is it ok to tell somebody they are doing a lousy job and tell them to resign?

Athena, the ancient Greek goddess of strategy, is playing a game with Zeus and Ares. She creates an infinite line of posts, sequentially marked with all integer values. All posts with values which are evenly divisible by 101 are coloured blue, whilst all others are coloured grey. On every post there sits a pigeon.

Athena speaks: "My Lords, there are as many pigeons as posts. They correspond directly, and therefore are equally numerous."
Ares, god of war, smites all pigeons on grey posts, leaving nothing on the grey posts. He bellows rather than speaking, being the god of war, "There are still infinitely many pigeons, so they still correspond directly to the posts. Equinumerosity between posts and pigeons is preserved."

Athena is visibly displeased that her construction has been violated. "No Ares, dog of war, disciple of can't-count-Cantor, and idiot of Mount Olympus. Less than 1% of my pigeons have survived your wrath. The fact that most posts are unoccupied proves that there are less pigeons than there were a few seconds ago."

What would you say if you were able to speak without Ares smiting you, and without fear of being downvoted on social media?

by Leslie Green

77 standard dice show a total score of 270.

What is the total of the 77 numbers which are hidden underneath each of the dice?

The UK prime minister Boris Johnson was forced to resign as Conservative party leader in July 2022, not for a huge lie (such as saying you are "helping" a country you have just invaded) but for a succession of smaller lies which meant he could not be trusted.

8 politicians made a bid to replace him, and on the first round of voting the results were as shown.

What percentage of the Conservative MPs wanted Rishi Sunak (the former Chancellor in Boris Johnson's government) as the new leader?

This tweet by a famous person (July 2022) came with the conclusion that inflation was running at 67.2%.

Suppose your monthly food bill went up 10%, monthly fuel cost went up by 20%, and your rent went up by 5%. For simplicity let's say this was your total expenditure, with rent being 50% of your outgoings, fuel being 25%, and food being 25%.

By how much did your outgoings for the month increase?

Each question mark represents a distinct number picked from 6 consecutive integers.
The sum of the numbers in the red triangle is 39.
The sum of the numbers in the blue square is 46.
The sum of the numbers in the yellow circle is 85.

What is the value in both the square and the circle?

adapted from Scottish Mathematical Council (2008)

Jane puts 15 playing cards on a table, face down. Gerry is blindfolded so he cannot see the flip-states of the cards, but he can feel where they are. Jane arbitrarily flips 4 cards face up.

Gerry separates the cards into two unequally sized groups of his choosing, and optionally flips any number of cards whilst still blindfolded. If Gerry manages to have the same number of face-up cards in both groups he wins.

Gerry wins this same game repeatedly, suggesting that his method is not random chance.

What is the number of cards in the smallest group?

Is it ever possible to complete infinitely many tasks/actions?

    Leslie Green

Suppose you wish to walk a distance of 100 metres.

According to mathematicians, (1) there are infinitely many points required to make up any distance.

Having walked 50 metres, you realise that you have travelled half the distance to your goal.

According to mathematicians, (2) half infinity is still infinity.

Therefore it is evident that you are still exactly where you started because there are still infinitely many points to cross, which is exactly the position you started in.

What should you conclude?

by Leslie Green

In the French Presidential election of 2022, Macron is reported to have received 58.5% of the votes and Le Pen 41.5%.

Is it reasonable to conclude that Macron had a substantial majority and was convincingly the people's choice?

It is often claimed that politicians are motivated by self-interest.

Is it true that this also occurs with voters, even in fair elections?

It might reasonably be argued that politicians act:

(1)   firstly, in their best interests
(2)   secondly, in the interest of their supporters and friends
(3)   thirdly, in the interest of their country
(4) fourthly, in support of truth, morality and decency, provided there is no conflict with earlier items on the list.

Given this cynical world view, how can you ever trust politicians?

The picture shows a subway map.
A team of inspectors verifies passengers' tickets at a station on a line, or all lines through it if there are many lines.
Then they randomly choose the next neighboring station and move there to make their inspection.

How many times is the probability of meeting the team of inspectors at station C greater than at station B?

What word or phrase is most closely related to resilience for a company?

by Leslie Green

Growing food in deserts seems like the ideal situation for large scale farming companies, since the land is really cheap. The only problem is where to get the water from.

Imagine we solve the water supply issue.

But is there another problem?

You have two parents, four grandparents, eight great grandparents, and so on, although probably not all of them will be currently living.

If you look back over time to your more distant ancestors what can you say?

Note that we are not looking at the accumulation of ancestors, just the most distant 'rank' at any point in time.
For example we might first consider the count of grandparents, then we might look at the count of great grandparents.

A column of king's troops was moving at a uniform pace. A lieutenant was sent from the head of the column, to check the rear of the column and return. He moved at a uniform pace and arrived back at the head of the column when it had just covered its own length.

How much further distance did the lieutenant ride than the troops?

(Express the answer as a ratio.)

Which species best represents Darwinian natural selection?

In a far away land, the ruler, His Divinity El Presidenti Lord High MucketyMuck is loved by his people so much that anyone who does not applaud his arrival sufficiently enthusiastically is sent to the Ice Mines of the Desert region for quiet contemplation and rehabilitation. It is only necessary to present the ice mine supervisor with one bucket of ice at midday to be released.

The arrogant new prisoner immediately figures out a plan of action and is able to present the supervisor with the required bucket of ice in a blazing 40°C of midday heat the next day. He is immediately released by the mine supervisor (who honours his commitment in full) but the newly released prisoner soon dies.

Why?

Marketing people like to use (and misuse) words to get you to buy their products.

One favourite is "natural" with the associated idea that natural means 'good' and artificial means 'bad'.

Which of these natural items is guaranteed to be good for you?

You go to a carnival and are offered a game. You get to pick three standard 6-sided dice from a large bin of dice. You get to choose a number between 1 and 6 before rolling the dice. The carny never touches the dice, so the game cannot be rigged. If your number comes up you get $1. If it doesn't come up you pay him $1. Clearly the odds are 50:50 so you decline to play.

He sweetens the deal. If you throw at least two of the number you picked he gives you $2. ($1 for exactly one of your number, and $2 for at least two of your number.) With your best poker face you decline again.

The carny is clearly desperate. He says his boss is watching, and if he doesn't get a player soon he will be let go. You hold your nerve and he sweetens the deal again. If you throw three of the number you chose you get $3. ($1 for one number, $2 for two numbers, $3 for three numbers). You play the game.

Which game was the first break-even for you?

Adapted from Numbers & Proofs by Allenby.

During a military exercise four missiles are launched from the corners of a square field at the same time. Each missile aims directly at the other missile in a clockwise direction without any intelligent algorithm. The side length of the square field is 30 miles. A missile flies 2 miles in the first 10 seconds, 4 miles in the next 10 seconds, 8 miles in the next 10 seconds, 16 miles in the next 10 seconds, on so on.

How much time does the exercise last?

The answer buttons represent the starting numbers for use in the algorithm.

As an example, start from 6 (step 1)
Remove all factors of 2 to give 3 (step 2)
x3 and add 1 to give 10 (step 3)
...

Which starting value takes the longest number of steps to END?

by Leslie Green

There are 40 ordinary coins on a table, with exactly 20 showing Heads, and exactly 20 showing tails.

You are blindfolded and wearing heavy gloves, so you can only feel the presence of a coin, but do not know which way up it is.
The coins showing Heads are randomly distributed across the table, and you do not know where they are.

A "step" consists of an operation such as separating the coins into four equal groups.
Turning over any number of coins in a single group would be considered as another single step. (Turning a coin over converts a Head into a Tail).

What is the minimum number of steps necessary to separate the coins into two groups, each having the same number of Heads showing?

Using the letters D, E, E, F, N, O, O, P, and S, we can form many 9-letter "words".

We accept all permutations, even if they don't exist in modern English.

If these "words" are arranged in alphabetical order, what position does the word "SPOONFEED" occupy?

In a distant galaxy there were three powerful gods, living on three nearby planets. Prota required his followers to give him 1 bag of gold dust every month for all time. Defteros required her followers to give her 1 bag of gold the first month, followed by at least ½ bag of gold the next month, then at least ⅓ bag, then at least ¼ bag, and so on. Tritos required its followers to give it 1 bag of gold the first month, then 2 bags the next month, then 3 bags the next month, and so on.

Quite why these petty gods required such gifts is a mystery, since they could create any arbitrary amount of anything they wanted at any time.

Each god was satisfied that they would be given an infinite amount of gold, since they could view backwards and forwards through time at will, unlike their pathetic linear-time followers.

The Tritonians had to develop space travel to get enough gold to settle the avaricious (having extreme greed for wealth) demands of their god. Each god "tuned in" to the activity of their followers, but when Tritonians landed on the planets of Prota and Defteros, all three gods suddenly disappeared from the galaxy.

by Leslie Green

Three students share a house. All have similar cars, with similar keys, and these keys are placed in a bowl in the hallway. It is nighttime, and the hall light has failed. They therefore take the keys from the bowl randomly.

What is the probability that exactly one of them takes their correct key?

Towards the end of 2020, what was the most dangerous social media information circulating?

Jane says the digits from 1 to 9, one by one, in a special sequence. Gerry writes the first digit down in the middle of the page. All subsequent digits are alternately placed to the left or the right of the already placed digits. Gerry has only one decision to make - Does the second digit go to the left or the right of the first digit.

Jane can see and react to the digits being written down. Gerry wants to make the final number as large as possible. Jane wants to make the final number as small as possible.

Both players are very intelligent, and expect the other player to make no mistakes. What is the final number?

Two drones patrol the perimeter of a super-secret establishment at regular intervals, but each drone patrols at a different speed for added security. This confuses would-be invaders.

The perimeter patrol times are: 30 mins and 49 mins respectively, with no breaks.

When they first start the patrol pattern, they set off from their perimeter base at 17 minute intervals, the fast drone leaving first.

Do they ever meet?

In a tag race, 4 stools are set out in a square, each stool having been systematically numbered as shown. Sitting on each stool is a child, and upon each child's head is a numbered hat, the hat numbers corresponding to the stool they are sitting on initially.

The teacher hits a gong to start the race between 3 teams: The All Jocks, The All Nerds, and the 50:50 Nerds and Jocks.

It is always the child in seat one who starts on the gong, or if all others are seated at anytime during the race.

Having started, the child runs to the stool found 3 positions clockwise and sits down. If that stool was occupied the seated child is tagged, and continues along following the same scheme. This tag race continues until all children are again seated in their original positions. Children cannot get up from their seats unless they are tagged (touched on the shoulder) or if they are starting in seat one.

The stools are a long way apart, so standing, sitting, tagging, and thinking times are all neglected.

Which team wins, given that jocks moves twice as fast as nerds?

In mathematics the formula

A + B x C = ?

is totally defined. There is no ambiguity at all.

Now we consider the clause shown (from European Directive 2014/30/EU).

It is a legal requirement, so you better get it right!

Which statement is correct?

A piece of string is 1 m long. It is cut into two unequal pieces, such that one piece is three times 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 unequal pieces, with one piece 3x as long as the other.

How many cuts are required to make all pieces smaller than one tenth of the original 1 m length?

As you get older your sphere of awareness tends to increase. You start off aware of your parents and your home, then it is the school and village, then the city, country, continent, and finally the whole world, and space. Too much awareness too soon can be a burden though.

Malala Yousafzai was blogging at the age of 12, and at the age of 15 was targeted by extremists who tried to kill her to shut her up. Greta Thunberg, at the age of 15, was promoting awareness of the climate change crisis. Fortunately, Big Petroleum hasn't yet taken out a 'hit' on her.

So here we see if your awareness has extended beyond first-person shooter games and cat videos.

Which group on this list most heavily facilitates crime?

Intelligence is all about problem solving. If you could memorise the entire contents of the 32-volume Encyclopedia Britannica that would not classify you as intelligent. The entire contents can be stored on a DVD, making you as intelligent as a piece of metallised plastic.

Non-numerical problems are arguably harder to solve than numerical problems. A mathematically difficult integral, for example, can be solved by using the free Wolfram Alpha website. What then is left for the future intellectual to consider?

Try this: How would you propose to deal with the COVID-19 pandemic, given information current at the time of writing (11 October 2020).

by Leslie Green

Fairly soon after the start of the COVID-19 epidemic of 2020, the US President Donald Trump appeared on TV, and his words have been transcribed as accurately as possible here.

What is your opinion of his statement?

Adult life is quite complicated, and the reasoning required to understand it is equally complicated.

Is it ever reasonable to discriminate for or against people on the basis of their color, gender, religion, or ethnicity?

To simplify the question we only consider employment opportunities.

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.
If Player 1 passes, Player 2 pays a fee of $0.50, but still gets $2 if he turns over a red card.
The above situation constitutes 1 game.

If both play optimally, which will win (on average) over a large number of games?

The 6 playing cards shown are jumbled up and placed face-down.

There are two players, neither of whom saw the cards being placed face down, and they use no trickery or deception, and they gets no clues from anyone.

Player 1 takes a card, and without looking at it himself shows it to Player 2.

If a red card is picked next, the player who picks it is rewarded.

Player 2 is given the choice. What is her optimum strategy?

Does a pure observation of a purely physical system change the system being observed?

We will use the Collins online dictionary definition: Observation is the action or process of carefully watching someone or something.

We use the term "pure observation" to distinguish between a measurement mechanism and the watching of the result of that measurement.

Be sure to have done the earlier question first.

In things like drugs trials, statistics and double-blind experiments are considered mandatory.

Such double-blind experiments are not done in more engineering related trials.

Why?

In a problem for young kids, the question asks:
"How far can a wolf run into a circular garden which has a 100 m diameter?"

The answer given was:
"50 m, since after more than that, the wolf would be running out of the garden".

Now we effectively ask the same question to a more sophisticated audience.

Which answer is the least acceptable?

Thomas Alva Edison was an American inventor and businessman who has been described as America's greatest inventor.

Which phrase belongs to Thomas Edison?

Five rabbits bought a scale which is imprecise for small and large weights. It is OK for weights between 9 and 19 kg. The rabbits weigh themselves in pairs in every possible combination, resulting in the following:

10, 11, 11, 11, 15, 15, 15, 16, 16 and 16 kg.

What is the weight of the lightest rabbit?

Lily and Billy share the same birthday, but Lily is 20% older.

How long must Lily wait until she is exactly 5% older than her younger brother on their birthday?

John tosses one fair coin, and Mary tosses three.

What is the probability that Mary gets an equal number of heads compared to John?

Gerry and Jane play a card game as follows: 8 cards are placed face-up in a row going left to right. All the cards are black apart from one, a red queen. Jane places the red queen wherever she likes - except at either end of the row.

At each step in the game one card is removed. The card is chosen by Gerry picking a number, and Jane counting in from a side of her choosing. For example if the row is BQBB the red queen is either 2 from the left of 3 from the right. Gerry tries to pick the red queen, and Jane acts to thwart Gerry's plans!

If the last remaining card is the red queen then Jane wins.

Given that both Jane and Gerry are clever, and make no mistakes, who wins?

What is 1234567890123456789012345678901234567891 mod 3 ?

Reminder: A mod 3 result subtracts the largest exact multiple of 3 and gives the remaining part so that, for example, 8 mod 3 is 2.



[HINT: You could try an easier problem first.]

by Leslie Green

You are shopping online for vitamin supplements. Your basket has £18 worth of items. If you spent £20 you get free shipping, which otherwise costs £2.99. Additionally, if you enter today's special promotional code, you can get 10% off from items costing £20 or more (not including shipping).

What should you do?

Whilst travelling in a strange and distant land, a stranger carries a purse of 5-mynt coins, and another purse of 7-mynt coins.

Assuming that he has an adequate number of coins, how many positive integer values of mynts from 20 to 100 cannot be formed from some combination of these two denominations?

Four children are randomly seated at a small circular table for a birthday party. Each will be assigned a unique integer identifier.

A child is only allowed to swap places with a child sitting right next to them.

As a cruel twist, these children (of mathematician parents) must correctly evaluate the mean number of swaps required to seat themselves in numerical order (clockwise or anti-clockwise orientations are both allowed) before being allowed to eat the delicious cake sitting in the middle of the table.

What is the mean number of swaps required for a random initial seating arrangement?

High on a Tibetan mountain, a mystical mathematical cult spends its time in quiet contemplation. At their annual Xmas festivities, 8 of them sit at the circular high table for their feast of gruel. Each member, having renounced their earlier mundane lives, wears a badge showing their number. All integers from 1 to 8 are present, but by some calamitous catastrophe, they have arranged themselves in numerical order counter-clockwise as seen from the viewpoint of the All Seeing Eye in the ceiling.

The High Priest (numbered 0) of the cult is displeased that such a sacrilegious event should occur during his tenure in office. Without any words he strides over to the man-sized golden gong in the corner of the room. Each time he strikes the gong, any number of pairs of cult members can swap places with a person sitting next to them.

To be clear, for each pair, they swap places with each other, and so they only move one space around the table for each strike of the gong.

How many times does the gong need to sound in order for the members to be seated in a suitable clockwise arrangement?

by Leslie Green

A wandering magician and his rather homely assistant are required to entertain the Sultan in his Palace. The magician faces the Sultan, and on either side of the magician is a highly ornamental cubical box. The assistant reveals an excessively large key, as you might find in a clockwork mechanism, except this key is two feet wide. She inserts the key into the box on the left and winds away merrily. Discordant and unmelodious noises are emitted by the box. The magician reaches into the box and pulls out what looks like a cannon ball. He places the ball on the hard flat marble palace floor and it does nothing. The ball is returned to its box with a grunt of disgust from the magician.

The process is repeated with the box on the right, but this time the noises are less unpleasant. The ball is placed in the same spot as before, but this time it seems to accelerate away from where it was placed, rolling across the floor. The ball was clearly held only on the sides, so the Sultan can clearly see that the ball was not pushed. He is highly amused and entertained.

How was this trick done?

by Leslie Green

A magician has a standard card deck with 52 cards. Half of them are red and another half is black. You can choose 2 cards or take a pair from the top. If they are both black, then you get the pair. If they are both red, then the magician takes the pair. If they are different colors, then the pair is discarded. You continue taking pairs until there are no cards in the deck. The game cost you $1 to play. The magician pays you $1 for every extra pair you have.

What do you expect to win at the end of the game?

Which situation is the most correct interpretation of the Standard Mathematics of Infinity?

A) You throw small pebbles into a bucket for 8 hours a day for your entire life. Every third pebble goes to the bucket on the right, whereas normally pebbles go into the bucket on the left. When you are too old to continue, the task falls to your relatives, and so on through the ages. At some indeterminate time in the future the number of pebbles in each bucket will be the same.

B) You have a hotel with an infinite number of rooms, each of which has a guest in it. A new guest arrives, so you get everyone to move to a room with a number one-greater than their current room number, and the new guest goes in room number 1.

C) The number of counting numbers that are evenly divisible by 1E98765 is equal to the number of counting numbers.

by Leslie Green

The pirate captain has to share a collection of gold coins with the quartermaster. The quartermaster is both illiterate and innumerate, whereas the pirate captain is well educated, especially in mathematics. Each will alternately take at least one, but not more than three gold coins from the collection, the number taken being entirely their own decision for each separate choice.

 The person who takes the last coin (or coins) from the collection then gets to take half of the coins from the other participant. The captain starts piling the coins up into stacks of 8, but the quartermaster is impatient so only three stacks are formed. There are at least 50 coins in the heap, in addition to the three stacks, but the exact number is unknown. Coins are taken from the three stacks only when the heap of coins is depleted.

Who gets the most coins?

(Hint: HINT: make sure you have done the easier problem first.)

by Leslie Green

In Ancient times, ceramic jars (called amphorae) were used to store and transport many different products, both liquid and dry.

A rich father has decided to equally split his wealth between his three sons. The total to be split consists of: 15 amphorae filled with gold dust, 15 amphorae half-filled with gold dust, and 15 empty amphorae (which are also valuable). The requirement is that each son gets the same number of amphorae, and the same total amount of gold dust, but this is to be achieved without transferring gold dust between the amphorae.

How many distinct arrangements of amphorae are possible if we do not consider the brothers as being distinct?

(HINT: do the easier one first)

by Leslie Green

In a tight-knit group of 4 children, everyone has 3 friends. To make life interesting, every week each of them randomly chooses a best friend from within this group.

What is the probability of exactly 3 of these children being best friends with somebody who is also their best friend?

The pirate captain has to share a heap of gold coins with the bosun. The bosun is both illiterate and innumerate, whereas the pirate captain is well educated, especially in mathematics. Each will alternately take either one or two gold coins from the heap, the number taken being entirely their own decision for each separate choice.

The person who takes the last one or two coins from the pile then gets to take half of the coins from the other participant. The captain starts stacking the coins up into heaps of 6, but the bosun is impatient so only three heaps are formed. There are at least 20 coins in the heap, in addition to the three stacks, but the exact number is unknown. Coins are taken from the three stacks only when the heap of coins is depleted.

Who gets the most coins?

(HINT: make sure you have done the easier problem first.)

by Leslie Green

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) totaling between $1 and $9 (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.

What is the expected amount of the money for you?

A clock chimes once on the quarter hours, and chimes the hours according to the hour.

For example, at 1 pm there is 1 chime, and at both noon and midnight there are 12 chimes.

There are in 220 chimes in 23 hours and 59 minutes.

When did the time interval start?

An engineering company builds a road network connecting 6 cities that are at the same distance on a circle. The same number of people travel from each city to 5 other cities every day.

For which design of the roads the average distance that a person has to ride is the smallest?

Balls numbered 1, 2, ... are put into a box as follows. At 1 minute to noon the balls numbered 1 to 10 inclusive are put in, then ball 1 is taken out. At 1/2 a minute to noon balls 11 to 20 are put in, then ball 2 is taken out. At 1/3 minute to noon balls 21 to 30 are put in, and ball 3 is taken out, and so on.

How many balls are in the box at noon?

adapted from A Mathematicians Miscellany by J.E. Littlewood (1953) by Leslie Green

John and Mary, rival treasure hunters, enter the treasure room. In the room are 5 identical closed cardboard boxes, placed conveniently in a circle. The treasure hunters cannot get close to or touch the boxes at first.

In one of the boxes there is a large jewelled crown, all the other boxes being empty. One of the treasure hunters will get first choice of a box, but will not get to open it immediately. When the first box has been chosen, three large guards (who know where the crown actually is) will walk directly to 3 of the empty boxes and jump on them, crushing the boxes completely to show that they are empty. The guards will never go near the chosen box under any circumstances. The remaining box will be given to the treasure hunter who did not choose.

The treasure hunters, having now been fully briefed on the procedure, are free to act. The first person to touch a box makes the choice.

John pushes Mary aside, and she falls to the ground clutching her ankle. John sprints off and touches a box.

Why is Mary now sitting on the floor laughing?

by Leslie Green

Statistics (ethically applied) can be a powerful way of improving overall performance. Unfortunately, when financial or other rewards are applied to such success, unethical application seems to automatically be applied.

In the UK (2019) schools were rewarded for the number and grade of exam passes per student.

What was the result?

You are given two sets of the numbers from 1 to 12 inclusive. You are required to pick a set of 12 numbers from the 24 numbers given, then arrange them in the circle shown. It is required that the difference between any two adjacent numbers is not more than 1.

How many different valid arrangements of the numbers can you make?

(Do not count rotationally or mirror symmetric variants.)

To be clear, each different arrangement starts by you intelligently choosing 12 numbers from the original 24.

For the years and companies provided, which can legitimately claim to be the fastest growing energy supplier in terms of customers.

by Leslie Green

Data Analysis:

Which group represents the best liars?

by Leslie Green

Everyone who has "freedom of speech" knows what that is. Or do they?

What is freedom of speech?

by Leslie Green

You have a two-pan balance scale and 12 identically looking coins. One of the coins is counterfeit and it can be either lighter or heavier than the others.

How many weighings do you need to determine the counterfeit coin?

Physics teachers like to use the word "velocity" rather than "speed".
For example, they might say: "A car is moving with a velocity of 10m/s."

Why?

by Leslie Green

There are three university students working part-time at a supermarket on the checkout tills. All three are currently studying dimensional analysis on their courses, and have realised that you can't add terms with incompatible units.

The physicist is always late, claiming that her temporal uncertainty is higher than the others because of her knowledge of quantum mechanics. The mathematician passes through people who only have fruit. He can add apples and oranges because they are both types of fruit. The engineer can only pass through people who have tinned goods because all tins are dimensionally equivalent. The high-school dropout always has a long queue (line) at her checkout.

Who is the best employee?

by Leslie Green

A Maharaja in Ancient India is training his dancers to perform in a spectacular musical. At various stages in the act they all form well-aligned columns of 3, then 5, then 7, and then 11 widths. When arranged in 7 or 11 columns there are no dancers out of formation. However, when arranged in 3 or 5 columns there are only two dancers in the front row.

 What is the minimum number of dancers in the Maharaja's troupe?

by Leslie Green

If we accept as fact that 80% of statistics are made up on the spot, what is the chance that at least one statistic out of the two unrelated statistics we have been given from independent sources is false (made-up)?

by Leslie Green

Leslie Green asks:

"When you read in a technical journal that a particular problem is 'trivial' what should you understand by that?"

An integer on a 64-bit computer can handle values up to around 2•10¹⁹.
The number of electrons on the planet is crudely estimated to be around 2•10⁵¹.

What possible use is there for whole numbers in excess of 1•10²⁰⁰?

by Leslie Green

Boolean Logic

There are three balls X, Y and Z. They are colored red, white and blue, but not necessarily in this order. One, but only one, of the following statements is true:

X is red

Y is not red

Z is not blue

How many solutions does the puzzle have?

The original question can be found in the Plus Magazine.

The picture shows George Boole (1815-1864), the author of the Boolean algebra. Boolean logic is credited with laying the foundations for the information age (computers and cell phones).

What is the least useful answer to the question: what is zero divided by zero?

by Leslie Green

Dihydrogen monoxide, also known as hydric acid, is difficult to detect since it is both colorless and odourless and yet it has horrific consequences when mishandled. There are estimated to be over 300,000 deaths per year (world wide) caused by the unintentional inhalation of this substance, and yet efforts to ban it are dismissed out of hand.

In the US state of California addiction to this substance is so bad that people literally die if it is withheld from them for more than a few days. In Wisconsin, every household has at least one cold-water tap which has significant levels of dihydrogen monoxide present.

Dihydrogen monoxide has three distinct forms: solid, liquid, and gaseous. Prolonged exposure to the solid form can cause the skin to blacken; amputation of fingers and toes can then be necessary. Even brief exposure to the gaseous form can lead to an extremely painful burning sensation.

How should this substance best be handled?

by Leslie Green

The image shows a sliding square puzzle. Each square tile can move left/right/up/down if and only if the tile moves into the empty space.

How many moves are needed to go from the setup on the left to the setup on the right?

by Leslie Green

Which is more important, a good question or a good answer?

( HINT: consider several different situations.)

by Leslie Green

Adults often worry about things like the cost of living and house prices. John has just bought his first home, by which we mean that he has saved up at least 5% of the price and has borrowed the rest as a mortgage. He hopes that he will be able to trade up to a larger house in the same area in a few years time. He hopes and expects that the larger house will always cost 50% more than the house he has just bought.

Assuming that house prices rise faster than the general level of inflation, is it easier or harder to trade up in the future?

by Leslie Green

A typical problem for kids goes something like this:

"1 man digs 1 trench in 1 day. How many days would it take for 9 men to dig 2 trenches?"

This is a stupid question which an adult would not even bother with.

Why not?

(Pick the best answer)

by Leslie Green

Jane has 33 cards: white, yellow and blue.
There is at least one white card in any randomly chosen 21 cards.
There is at least one blue card in any randomly chosen 22 cards.

What is the maximum number of yellow cards that Jane could have?

A printer sometimes moves certain characters up. It can print

₂³ ₅³

instead of

₂₃₅₃

The first can be interpreted as the product of
2³ = 8 and 5³ = 125 that is 2000.

These two numbers are different.

Is there a 4-digit number that does not change its value when two of its digits move up?

The solution was suggested by Henry Ernest Dudenay in the book Amusements In Mathematics, 1917

MENSA question.

What number can be added to the set below to complete it?

1, 2, 6, ?

For every day in January, Gerry sent three letters, one to each of his three girlfriends.
The postman put the letters into three different boxes without looking at the name of the recipient.
For ten of the days, nobody received their letters correctly.
For twelve of the days, only one girl received her letter correctly.

For how many days did all three girls receive their letter correctly?

When my coconuts were counted in twos, there was one extra.
When counted in threes, there were two extra.
When counted in fours, there were three extra.
When counted in fives, there were four extra.
When counted in sixes, there were five extra.
When the coconuts were counted in sevens, no extra coconut was left.

What is the minimum number of coconuts I could have?

One hundred and one coins have the property that when one is removed, the others can be divided into two groups having equal value.

If three coins have 3-piastra value, what value do the other coins have?

Inspired by: The USSR Problem Book by Shklarsky, Chentov, and Yaglom, Freeman, 1962.

Leslie Green asks:

In many homes, old and new, you find radiators directly underneath the windows.

Why?

29A + 30B + 31C = 366

What is the sum of the three positive whole numbers A, B, and C?

Leslie Green asks:

Here is a number sequence with a definite mathematical rule to move from one number to the next.

3,   7,   11,   17,   27,   43

The tricky thing is that one number (not necessarily at the end) has been deleted from the sequence, but which number?

(Hint: This is just mathematics. It works in any language, not just English.)

Gerry arranged the digits 2 through 9 in a particular order as follows:

8   5   4   9   7   6   3   2

Where should the digit 1 be placed in this order?

Leslie Green asks:

Two hundred and thirty two boys and two hundred and twenty nine girls came out of school at different speeds.
After a while, 38 adults rushed out of the same building even faster than the kids.

The question: 'How many students and teachers went home after school?' has already been asked of 10 year olds, to which the correct answer was 499.

Now we ask a deeper question.

How many reasonable assumptions had to be made to get the correct answer?

Logic Puzzle.

Many years ago a king of a North European country decided to establish which is more dangerous for health: tea or coffee.
He ordered that one prisoner should only be given coffee, and that another prisoner should only be given tea.

What do you think? Who died first?

Four teams are in group A of a soccer (football) tournament for young girls.
Each team plays one match against each of the other teams, with three points for a win, one point for a draw and none for a defeat.

If the three teams that pass to the next stage get 5, 4 and 4 points, how many points does the losing team get?

Number SEVEN is mystic and frequently used in human culture.

What is the most reasonable cause of that fact?

A jeweler has 2016 pieces of chain, each with two links. He wants to make one big closed chain of them. To do this he has to open some links and close them afterwards.

What is the smallest number of links he has to open?

The average score for dance of boys and girls in class A are 16 and 21, respectively.
The average score of boys and girls in class B are 15 and 20, respectively.  

Twenty percent of class A students are girls.
Forty percent of class B students are girls.  

Which class has a higher average score?

The picture shows a project plan. Arrows indicate where an activity must be finished before the next activity can start. The number shows the duration of the activity in days. An activity is critical if a delay by 1 day delays the whole project.

How many critical activities are there?

The teacher walked into her classroom to find a scene of devastation. There was red paint on the walls, her lunch had been half eaten, and books were thrown around the room. There were only three children in the room: Alex, Betty, and Clive.
All three said that Betty ate the lunch. Betty said Alex painted the wall.
Clive said that Alex threw the books. Alex said that Clive painted the walls.

The Headmaster was called in to resolve the crisis. On his way to the scene he found Wesley hiding in the corridor. Whilst Wesley would not directly implicate anyone, he did admit (under duress) that each of the three had done one of the crimes, and that every statement they made had been untrue.

Given that Wesley is telling the truth, who threw the books?

Author: Leslie Green

These are two identical six‐sided dice. The opposite faces of a die add up to 7.

What is the sum of the number of dots on the two faces that touch each other?

What is the main difference between the pictures on the left page and on the right page?

The problem is similar to 100 Bongard Problems on visual pattern recognition.

You are pitching your new idea to a panel of Venture Capitalists (VCs) to secure increased funding. Using your advanced mathematical skills, you have dumbed-down the probability of success to something even VCs can understand. You tell them that if they were to throw 10 normal dice and sum the dots on top, the probability of your success is the same as the sum being less than 50.

One of the VCs seems very antagonistic, but you must still give the best possible answer, quickly – and using only mental arithmetic.

His question is "Can you guarantee that the sum of dots would be less than 50?"

Author: Leslie Green

Forty-eight liters of water are poured into an aquarium of dimensions 50cm length, 30cm breadth, and 40cm height.

How high (in cm) will the water rise?

[1 liter = 1,000cm³]

This is a typical SAT question.

Leslie Green asks:

Peter changes his old fashioned single 100W incandescent light bulb to a super efficient 4 LED bulb system which gives the same light output, but only consumes 20W in total.

Estimate the saving per year if the bulbs are switched on for 8 hours every day of the year and you pay $0.1 per kWh.

(1 kWh is 1000 Watts for 1 hour)

The chocolate biscuit factory you are now in charge of has a problem. There are three 8 hour shifts and 11 shift supervisors who each have their own treasured setup of the machines to give an optimum biscuit pass rate. Every time the controls are adjusted the pass rate drops for several hours until the process settles down again. Each supervisor adjusts the controls to their “optimum” settings when their shift starts! Having analysed what they are doing, you have summarized the settings into 11 controls with two positions each. You need to devise a series of experiments to establish the optimum settings in a convincing manner to improve the productivity of your plant.

What is the minimum number of experiments necessary to find the optimum settings for each control?

Author: Leslie Green

Leslie Green asks

A phrase you will hear on the news or from people speaking is "the vast majority of".

As a silly example you might hear something like
"The vast majority of people with big noses also have big ears."

What is the mathematical definition of the phrase "the vast majority of"?

Leslie Green asks:

"The more debt you get into, the more credit card companies profit. If you buy $100 of goods at a shop you owe the credit card company $100, but typically the shop only gets $98. The shop has to inflate its prices to pay the credit card company. Typically the shop is contractually not allowed to give a discount for cash.

The CostCrashers supermarket chain deals only in cash. The CardPayers supermarkets deals both with both cash and credit card transactions, although 75% of people pay by credit card.

All else being equal, how much cheaper could the CostCrashers prices be compared to the CardPayers supermarkets, given the figures stated earlier?"

Leslie Green tells a story and asks :

It has been several years since the Apocalypse, but the Zombies still seem to be everywhere. I have been caught out in the open on my own and am now surrounded by 3 hungry Zombies, intent on eating my brains. Fortunately I have 6 rounds (“bullets”) in my gun. Unfortunately the ammo is old and degraded so it only works 80% of time. Also, although my aim is excellent on a shooting range, my shots are inaccurate when I am nervous, for example when I am surrounded by Zombies! It turns out that the closer they get, the more nervous I get, so the chance of my getting a shot to their head is only 63%. (Everybody knows that only a shot to the head will kill a Zombie). I worked it out, there is roughly 50% chance that any particular round will end up killing a Zombie.

There is just enough time to fire off all 6 rounds.

What is my chance of living to fight another day?

Leslie Green asks:

I am allergic to washing powder, but I don’t want my shirt to smell. The dermatologist has told me to reduce the total amount of washing powder residue on my shirt to less than 1 pico-gram. (I think he just made that number up on the spot!)

My dry shirt measures 280g, but after washing and spin-drying it weighs 350g. The washing and rinsing uses 21 L of fresh water for each operation. I wash my shirt with one 30g tablet of washing powder.

How many times do I have to rinse my shirt to reach the required non-allergenic state?

(remember that the density of water is 1g / mL, 1000 mL = 1 L, 1 pico-gram = 10^-12g).

Divide the analog watch face with five straight parallel lines so that the sums of the numbers in each part are equal.

What is the sum?

There are two gods named Orbis and Fidelis, one on your left, the other on your right, but you do not know which is which. Fidelis always answers correctly. Orbis only answers alternate questions correctly; you do not know if his last answer was correct.

You must determine which god is which using the minimum number of YES/NO questions. How many questions do you need to be certain?

To clarify the matter, if Orbis is answering incorrectly your entire question is evaluated correctly and then the answer is reversed. A single question to both gods counts as two questions.

Author : Leslie Green

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

I can cut off 4 circles from a circular sheet of dough.
I use the left-overs from 3 dough sheets to produce a new one.

How many circles can I make from 20 dough sheets?

Three parrots yell at Gerry every 5, 7 and 9 seconds, respectively.

If they start yelling at Gerry at 6:00:00, how much time will elapse until they yell at Gerry at the same time again?

Crazy Logic. Choose a number.

Six students travel in a car.

How many different ways can they take seats if only 3 of them may drive and there are no empty seats?

Gerry won 30% of the first 40% of the annual Discus Throw competitions.

What percent of his remaining competitions must he win to finish the year with 50% wins?

On average there are 100,000 strands of hair on a person's head.
Hair grows at a rate of about 15cm a year and each hair lasts up to 6 years before it falls out.

If two sisters have 222,222 strands of hair together, and one has 20% more hair than the other, what is correct?

A grandmother prepared bowls of fruit for her family. The only fruits available to her were cherries, apricots and red currants. Of course no bowl was empty and each bowl contained one kind of fruit.

All but five bowls contain some cherries,
all but four contain some apricots, and
all but three contain some red currants.

How many bowls were at the dinner?

I choose a 4-digit number in which the first digit is one-sixth the last, and the second and third digits are the last digit multiplied by 6.

What is the sum of all digits?

Guess!

After an hour, Evguenia catches and frees 10 fish.

If there are 100 fish in a lake how many fish remain uncaught after 5 hours?

At the end of each week, Gerry forgot 50% of what he knew the week before.

At the end of which week Gerry knew less than 50% of all he studied?

If Gerry gives Jane one apple, they will have the same number of apples.

If Jane gives Gerry one apple, he will have twice as many apples as she has.

How many apples do they have?

Gerry is a gentleman and so always gives Jane apples.

Two successive increases of 25% and 25% are followed by two successive discounts of 25% and 25%.

What percentage is the result?

If each coconut is priced at $9, then the shopkeeper loses $11.
If each coconut is priced at $11, then the shopkeeper gains a profit of $9.

How many coconuts are there?

Two trains are 1 kilometer apart on a single track railway line. They set off towards one another at 1 m/second.
A bee sitting on the front of one of the trains sets off and starts to fly along the railway line at 3 m/second.
When the bee meets the other train it immediately turns around and flies towards the first train and so on . . .
How many times does the bee turn before the trains bump into each other?

"This is a capital letter of the alphabet that's been folded just once.

Which letter is it if I unfold it? "

The puzzle was created by puzzlemaster Scott Kim

Sixteen teams enter a football tournament.
Each team plays one match against each of the other teams, with three points for a win, one point for a draw and none for a defeat.
How many games does our team lose if it scores 20 points and wins 3 games?

A 44-month long project is divided into three consecutive activities.
The time between the midpoints of the first and last activities is 33 months.

What is duration of the middle activity?

How many uninteresting numbers exist?

The question is based on the interesting number paradox, which is a semi-humorous paradox, which arises from the attempt to classify natural numbers as "interesting" or "dull."

Find the logic:

12345678 → 4
  234567   → 2
   3456      → 2
     45        → X

Find X.

Mary can complete an activity in 40 days.
John can complete the activity in 50 days.

Estimate for how many days they work on two activities if they change them at the best moment.

Students at a school are on average 165 cm tall.
The average male student height is 172 cm and the average female student height is 160 cm.

What is the ratio of boys to girls in the school?

There are 123 part-time employees and 321 full-time employees in a company.

If the part-time employees switch from 60% occupation to 40%, how many of them can be hired for a full-time position?

The rear tires of a car wear out after 40,000 miles, while the front tires wear out after 50,000 miles.

Estimate how many miles the car can drive if I change the tires (front - rear) at the best moment.

Working alone, Mr. Smith earns the family's monthly budget in 40 calendar days.
Working alone, his beloved wife earns the budget in 2 months.

How long will it take the family to earn an additional monthly budget?

On an island, three-quarters of the men are married to four-fifths of the women.

What is the minimum possible number of people on the island?

Mr. Rapid always starts driving on a road at exactly 7:00 a.m.
If he drives on cruise control at 42 miles per hour, he arrives at work late.
If he sets the cruise control to 48 miles per hour, he arrives at work early.
The amount of time he is early is also the amount of time he is late.

How fast should he go to get to his office exactly on time?

Which symbol goes next?

Which number goes next?

1 3 6 10 ?

Thinking outside the box.

Solve this:

Thinking outside the box:

Which statement is correct?

A committee consists of 7 people. The committee keeps an important object in a safe.

How many mechanical locks must the safe have so that it can be opened precisely when at least 4 members of the committee are present?

You may suggest your design of the lock(s).

Vladimir Arnold (1937-2010), one of the greatest 20th century Russian mathematicians told the following story:

“Our schoolteacher, I. V. Morozkin, gave us the following problem:

Two old women started at sunrise and each walked at a constant (different) velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m.

At what time was the sunrise on this day?"

There are 12 people in a village.
They consume 12 coconuts altogether.
Each man eats two coconuts, each woman eats a half, and each child eats a quarter.

How many men are there?

"Three sailors come across a pile of coconuts. The first sailor takes half of them plus half a coconut. The second sailor takes half of what is left, plus half a coconut. The third sailor also takes half of what remains, plus half a coconut. Left over is exactly one coconut, which they toss to a monkey.

How many coconuts were in the original pile?"

Martin Gardner

Bob earns $48,000 per year and has two weeks of paid vacation.
He works five days a week and eight hours a day.

What is his hourly wage?

Anna, Beatrice, and Cindy run a 2000-meter race.
All of them run at a constant speed.
Anna beats Beatrice by 100 meters.
Beatrice beats Cindy by 100 meters.
By how many meters does Anna beat Cindy?

I drive to my country house at a constant speed of 100 miles/hour.
I immediately turn and drive back at another constant speed.
The entire journey takes one hour and the distance to my country house is 40 miles.

What was my speed on the back way?

Divide the circles into two disjoint groups so that the sums of the numbers in each group are equal.

What is the sum?

After a three-hour drive, I stopped my car in front of a wall with three doors: a silver door on the left, a gold door in the middle, and an iron door on the right. Which door would I open first?

There is only one correct answer to the question below.

Which answer is correct?

Kevin cleans rooms in a student hostel; he earns twice as much for cleaning a large room as he does for cleaning a small room. Yesterday he cleaned five large rooms and three small ones. If, instead, he had cleaned three large rooms and five small rooms, he would have earned $10 less.

How much does he earn for cleaning a small room?

Alex greeted Bill.
Bill greeted Cindy.
Alex did not greet Cindy.

If the first two statements are true, the last one is:

In any 24-hour period, how many times are the digits of a 24-hour digital clock in an increasing gapless counting sequence, eg 3:45?

You have a part of a black-and-white map of a reef coast.
You are at point A on the coast, while a treasure is at point B.
Is the treasure under water?

I arrange five marbles randomly in a ring.
There are two green and three blue marbles.

What is the probability that all blue marbles are adjacent?

A donkey must transport 900 carrots to the market, which is 300 miles away.
The donkey carries a maximum of 300 carrots, and eats 1 carrot every mile.

What is the largest number of carrots that can be delivered at the market?

One hundred soldiers form a 10 x 10 square.

From every column, the tallest soldier is chosen, and from these 10 soldiers, the shortest is chosen. His height is X.

At the same time, the shortest soldier is chosen from every row, and from these 10 soldiers, the tallest is chosen. His height is Y.

Compare the heights.

Logic puzzle: a secret door.

A man approached the door of a secret laboratory and the door robot said "SIX." The man replied, "THREE" and was let in.

Another day, the robot said, "TWELVE."
The man replied, "SIX" and was let in.

Today, the robot said, "FOURTEEN."

What should the man say?

I want to measure exactly six minutes from the moment I touch an hourglass, using only a five-minute hourglass and a four-minute hourglass.

How many times do I flip over an hourglass?

Count the initial flip(s) and find the minimum possible number.

Which number comes next?

2, 12, 1112, 3112, . . . .

A project manager has to determine whether to purchase or rent four cars for a construction project. The car is available at a rental fee of $900 per day. Purchasing the cars costs $140,000 for the investment and $100 for the daily cost.

After how many days would the cost of renting and purchasing the cars be equal?

One hundred and thirty shareholders decide to buy 30,000 shares of company A.
They decide that five gentlemen together will buy a thousand shares, and that four ladies together will buy a thousand shares.

How many gentlemen are there?

On a 5 × 5 grid, I place a 1x2 card so that it exactly covers two squares.
I continue until there is no place for a card.
There is no card overlapping and all cards are inside the grid.

What is largest number of empty squares I can leave at the end of the exercise?

How many three-digit numbers containing only even digits are divisible by 9?

A bee crawls along the sides of the honeycomb hexagons.
It has moved from point A to point B along the shortest possible path.
It can move in three different directions and made an even number of steps.

What statement is correct?

Use each digit, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, exactly once to form two 5-digit numbers that when multiplied produce the largest quantity.
Which is the larger of the two numbers?

Source: Mathematics Teacher, NCTM Journal

How many zeros are there at the end of 100! ?

You have four piles: three piles with real coins and one pile with fake coins.
All the real coins weigh 20 grams each, and the fake coins weigh 21 grams.

How many times do you need to use a digital kitchen scale to find the pile with the fake coins?

Multiply all positive two-digit numbers.

How many zeros are there at the end of the result?

Each of these seven cells contains one number from 1 to 7, using all seven numbers.
The sum of the four horizontal cells is 16.
The sum of four vertical cells is 17.

What is the number in the shared, corner cell?

Which of these diagrams could be drawn without taking the pen off the page and without drawing along a line twice?

John has just had a 1% net pay rise.
He used to take home $5,000 a month.
He told his wife that the rise was a third of a percent.

How much more money does he keep for personal spending this year?

There are a number of apples, all of different weights.
The 10 lightest apples weigh 40% of the total weight.
The 5 heaviest apples weigh 25% of the total weight.

How many apples are there?

Anna bought 1 slice of mushroom pizza and 2 slices of cheese pizza for a total of $3.
Bill bought 4 slices of mushroom pizza and 5 slices of cheese pizza for a total of $9.

What is the cost of one slice of mushroom pizza?

Four painters can complete a painting job in 20 days.
12 more painters join the team 4 days after starting work on the job.

How many days does the painting job take from start to finish?

On January 1st, Anna and Bill have $100 each.
Each month Anna saves $10 more than she spends while Bill spends $5 more than he saves.

At the beginning of what month is Anna 10 times richer than Bill?

Alex has bills of different dollar values.
There are five times as many ones as there are fives.
There are ten times as many ones as there are tens.
There are twice as many tens as there are twenties.

How much money does he have?

Find the minimum possible value.

If 12 workers take 7 hours to build a brick wall 21 m long and 4 m high, how long will it take 11 workers to build a brick wall 1 m longer and 1 m higher?

I put $990 into 10 envelopes.
I try to compose all possible whole amounts from $1 to $990 by giving you a certain number of these envelopes.

What is the minimum amount that cannot be composed by a set of these envelopes?

The addition shown here has 21 terms and the last element consists of 21 9's.

Find the three last digits of the sum.

A lady, attempting to avoid revealing her real age to her husband, says:
I'm twenty-two years old if you do not count weekends and one summer month of every year.

Guess her real age.

The product of two natural numbers is 1000.

Neither of the numbers contains a zero.

Find the sum of these two numbers.

Sixteen European teams enter a football tournament.
Each team plays one match against each of the other teams, with three points for a win, one point for a draw and none for a defeat.
The probability of a draw is 0.5.

What is the most likely score of a team?

What is the largest number you can write with just two different digits?

An analog clock loses 15 minutes each hour.
If the clock is set correctly at noon, what time is shown when it first reads the correct time again?

In-Out Machine transforms 111121 into 100.
The same machine transforms 121244 into 100.
The same machine transforms 131369 into 100.

Find the result of the transformation, if the input is 141486.

A hunter met two shepherds, one of whom had three loaves of bread and the other, five loaves. All the loaves were the same size. The three men agreed to share the eight loaves equally between them. After they had eaten, the hunter gave the shepherds eight bronze coins as payment for his meal.

How should the two shepherds fairly divide this money?

Paul Sloane, Lateral Thinking Puzzles, 1991

You take half of the syrup and mix it with the water, and then take the same quantity from the water jar and mix it with the syrup.

Does the syrup now contain more water than the water does syrup, or the other way round?

1. At least 1 of these 6 statements is false
2. At least 2 of these 6 statements are false
3. At least 3 of these 6 statements are false
4. At least 4 of these 6 statements are false
5. At least 5 of these 6 statements are false
6. At least 6 of these 6 statements are false

How many sentences are true?

Two gears, one with 11 teeth and the other one with 18 teeth, have teeth marked as indicated.

After how many rotations of the small gear will the marked teeth be in the same position again for the first time?

How many straight lines are needed to separate each star from all the others?

Statement 1: There are 3 animals in a room.
Statement 2: All of the animals in the room are kittens.

Question: How many legs are there in the room?

What is sufficient to answer the question?

Ten years ago, Bob was three times as old as Anna.
Today, he is twice as old as Anna.

How old is Bob today?

How many two-digit integers are there in which the sum of the digits is equal to 10?

The twins are the same height.
They place four identical blocks as shown in the figure.

How tall are the boys?

Mr. Smith can read 1 page in 2 minutes. His wife can read 2 pages in 1 minute.

Reading together two different books, how many minutes will it take them to read 55 pages?