Puzzles

There are one hundred people standing in a circle. They count off beginning at one and ending at one hundred. Since they are in a circle, ONE is next to TWO and ONE HUNDRED.

ONE has a sword and kills TWO. He passes the sword to THREE, who kills FOUR, and so forth. NINETY-NINE kills ONE HUNDRED and passes the sword to ONE. Then ONE kills THREE and passes the sword to FIVE. This goes on until only one person is left standing.

Which number is he?

You’re creating a new coin system for your country. You must use only four coin values, and you must be able to make every value from 1 to 10 using at least one coin and at most two coins.

Which 4 coins do you choose? Can you find a second set of 4 coins that also works?

In certain parts of rural Russia, a would-be bride would gather six long pieces of straw or grass and grasp them in her hand. She would then randomly tie pairs of ends together at the top and at the bottom. Since there are six blades sticking out above and below her hand, she ties three knots at the top and three knots at the bottom.

The story goes that if she formed one big ring, she would get married soon.

What is the probability that she will get married?

Consider a list of all the ways you could take four different digits from 1 to 9 and arrange them to make the sum of two 2-digit numbers. Some answers will appear several times: 134, for example, is 93 + 41, and also 91 + 43.

What are the mean, median and mode of all the numbers on this list?

(The mean is the total of the numbers divided by how many there are, the median is the middle value when they are in order, and the mode is the number that appears most often.)

Every day you make a trip from A to B, 2 miles away. But your trip is unpredictable.

On 50% of days, you can walk between A and B without any obstruction.

The other 50% of the time, a troll appears at the halfway point, blocking your trip. The troll also switches on an invisible barrier that stretches 1 mile at right angles to your path in both directions, forcing you to walk around it.

You cannot see the troll or the barrier until you are halfway from A to B, so you can’t plan in advance whether you need to walk around it. You do know the troll appears at random 50% of the time.

If you can walk in straight lines to any points between A and B, what is your best strategy to walk the least distance on average? What is that distance?

Ali Baba found a cave full of gold and diamonds. A bag full of gold weighs 200 kilograms, and a bag full of diamonds weighs 40 kilograms. Ali Baba can carry only 100 kilograms at a time. A kilogram of gold is worth $20, and a kilogram of diamonds is worth $60.

What is the most money Ali Baba can make from the gold and diamonds he carries out in one trip, in one bag?

The game starts with 6 empty glasses in a row, numbered 1 to 6. You roll an ordinary dice. If the glass with that number is empty, it gets filled up. If it is full, you drink it, so it becomes empty.

There is a special rule when 5 glasses are full. If you roll the number of the one empty glass, it gets filled and you have to drink all 6 glasses. Then the game ends.

From the start of the game, what is the average number of rolls until the game ends?

A rajah left his daughters some pearls in his will, with instructions to divide them as follows. The oldest daughter was to receive 1 pearl plus 1/7 of what was left. The next oldest was to receive 2 pearls plus 1/7 of what was then left. The next was to receive 3 pearls plus 1/7 of what was left, and so on. The youngest daughter received whatever was left after all the others.

At first, the youngest daughter thought this was unfair, but after some careful calculation she realised that all the daughters would receive the same number of pearls.

How many pearls and how many daughters were there?

Suppose we play the following game. I toss a coin repeatedly until it comes up heads. If heads appears on the first throw, I pay you $2. If it appears on the second throw, I pay you $4. On the third, I pay $8, and so on, doubling each time.

How much would you be willing to pay me to play this game?

Albert and Bernard have just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:

May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17

Cheryl then tells Albert the month of her birthday, and tells Bernard the day.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know either.
Bernard: At first I didn’t know when Cheryl’s birthday was, but I know now.
Albert: Then I also know when Cheryl’s birthday is.

When is Cheryl’s birthday?

Two mathematicians meet in the street, and one says to the other, “I bet you can’t guess the ages of my 3 children.” (All the ages are whole numbers.)

The second mathematician says, “I’ll take your bet, but only if you give me enough information to solve it.”

The first agrees, and says, “The product of my 3 children’s ages is 36.”
“That isn’t enough information.”
“Right you are!” He looks around, spots the number on a building and says, “The sum of my children’s ages is the same as the number on that building.”
After thinking a little: “That still isn’t enough information.”
“Right you are again! My eldest son has red hair.”

After some thinking, the second mathematician won the bet. What are the ages?

Imagine there is a steel band around the equator of the Earth, nice and snug (about 25,000 miles long).

If you add one foot of steel to this band, how far is it raised off the ground all the way round? High enough to slip a hair under it? Your hand? High enough to walk under it?

A school has 1,000 students and 1,000 lockers, all closed. On the first day of term, the headteacher asks the first student to go along and open every locker. She asks the second student to go to every second locker and close it, the third to go to every third locker and close it if it is open or open it if it is closed, the fourth to do the same with every fourth locker, and so on. The process ends with the thousandth student.

How many lockers are open at the end?

You die, and the devil says he’ll let you go to heaven if you beat him in a game. He sits you down at a round table and divides a large pile of quarters (coins) into two equal piles, one each. He says, “We’ll take turns putting quarters down on the table. No overlapping is allowed, and the quarters must lie flat on the table. The first one who can’t put a quarter down loses. You can’t shift or squeeze any quarter into a space if it moves another quarter.” The devil says he wants to go first.

You realise that if the devil goes first, he probably has a winning strategy. You ask for some time to think, and he gives you 15 minutes. By the end of that time you know how to beat him, but you also know that you must go first for your strategy to work.

You convince him to let you go first. What is your winning strategy?

A woman came to market to sell some eggs. The first buyer took half of her eggs plus half an egg. The same happened with the eggs that were left: the second buyer took half of them plus half an egg. A third buyer bought what was left over: 1 egg. No eggs were broken.

How many eggs were there at the start?

Add just two kinds of maths symbol to each line to make it true. You can use symbols more than once, and factorials (!) and square roots count as symbols. Brackets are free.

0   0   0   =   61   1   1   =   62   2   2   =   63   3   3   =   64   4   4   =   65   5   5   =   66   6   6   =   67   7   7   =   68   8   8   =   69   9   9   =   6

Three people each lie 1/3 of the time. They all watch a coin being flipped, and they all say it landed Heads.

What’s the probability that it really is Heads?

The warden meets 23 new prisoners when they arrive. He tells them, “You may meet today and plan a strategy. But after today, you will be in separate cells and will have no communication with one another.

“In the prison is a switch room, which contains two light switches labelled 1 and 2. Each can be either up or down. I am not telling you how they are set now. The switches are not connected to anything.

“After today, whenever I feel like it, I will choose a prisoner at random and take him to the switch room. He must choose one of the two switches and flip it. He must flip exactly one switch. Then he’ll be led back to his cell.

“Given enough time, everyone will visit the switch room as many times as everyone else. At any time, anyone may say to me, ‘We have all visited the switch room.’ If it is true, you will all be set free. If it is false, you will all be punished horribly.”

What strategy should they use to guarantee their freedom?

Warm-up: You have a box of matches and a piece of rope. The rope takes exactly an hour to burn, but it may not burn evenly. For example, it might be that the first tenth of the rope takes 50 minutes to burn and the other nine tenths take only 10 minutes. How can you measure exactly 30 minutes?

The real problem: You have a box of matches and two of these ropes, which are not necessarily the same. How can you measure exactly 15 minutes?

On Bagshot Island, there is an airport. It is home to an unlimited number of identical aeroplanes. Each plane holds enough fuel to fly exactly halfway around the world, along a great circle. The planes can refuel each other in flight without losing speed or spilling fuel. There is unlimited fuel, but the island is the only place to get it.

What is the smallest number of aircraft needed to get one plane all the way around the world, if every aircraft must get back safely to the airport?

Notes: every plane must take off from and land at the same airport. Ignore the time and fuel used for refuelling.

A wholesale merchant came to me one day with this problem. Every day he has to weigh amounts from 1 pound to 121 pounds, to the nearest pound, using a set of balance scales.

What is the smallest number of weights he needs, and how heavy should each weight be?

A box holds lots of balls with different numbers on them. You pick two balls at random.

Someone offers you a bet on whether the two numbers are relatively prime (they have no common factor other than 1):

If you pick, say, 6 and 13 (relatively prime), you lose $1.
If you pick, say, 5 and 25 (not relatively prime), you win $2.

Will you accept the bet?

Four people need to cross a rickety bridge at night. They have only one torch, and the bridge is too dangerous to cross without it. The bridge can only hold two people at a time, and a pair has to walk at the slower person’s speed.

The four people take 1 minute, 2 minutes, 7 minutes and 10 minutes to cross.

What is the shortest time needed for all four of them to cross the bridge?

Mr John has 25 horses, and he wants to find the fastest 3. He has only 5 tracks, so only 5 horses can race at a time, and he doesn’t have a stopwatch.

What is the smallest number of races he needs to find the 3 fastest horses?

There are twenty coins on a table: ten heads and ten tails. You are blindfolded and wearing gloves. You can feel where the coins are, but you can’t see or feel whether they are heads or tails.

You must make two groups of coins, with the same number of heads in each group. You can move coins and flip them over, but you can’t find out which way up any of them is.

How do you do it?

People are queueing to board a 100-seat aeroplane. Steve is first in the queue. He gets on the plane but can’t remember his seat number, so he picks a seat at random. After that, each person who boards sits in their own seat if it’s free; otherwise they pick a free seat at random.

The flight is full, and you are last in the queue.

What is the probability that you get to sit in your own seat?

In a country where everyone wants a boy, each family keeps having babies until they have a boy. After some time, what is the proportion of boys to girls in the country?

(Assume a boy and a girl are equally likely each time.)

An island has 10 inhabitants. One day a monster arrives and says he intends to eat every one of them, but will give them a chance to survive.

In the morning, the monster will line everyone up in single file, so the last person sees the other 9, the next person sees the 8 in front, and so on, down to the person at the front, who sees nobody. The monster will then put a black or white hat on each person’s head at random (they could be all white, all black or any mixture).

Starting with the last person (who sees everyone else’s hat), the monster will ask each person to guess the colour of their own hat. They may answer with only one word: “white” or “black”. The monster will eat all ten as soon as a second person answers wrongly. Everyone can hear each guess, but not whether it was right.

Devise a strategy that guarantees everyone survives.

A building has 100 floors. You are given 2 identical eggs. Every egg has a “threshold” floor, N: the egg will not break if dropped from any floor below N, but it will definitely break if dropped from floor N or any floor above.

These are very strong eggs, so they can be dropped again and again without breaking, as long as they are dropped from below floor N.

What strategy should you use to find floor N with as few drops as possible? What is the smallest number of drops that guarantees you find it, however unlucky you are?

Five pirates have found 100 gold coins and must divide up the loot. The pirates are all extremely clever, treacherous and selfish (especially the captain).

The captain proposes how to share out the coins, and all the pirates vote. If half the crew or more vote “Aye”, the coins are shared as proposed, as no pirate would take on the captain without a majority behind him.

If the captain doesn’t get the support of at least half his crew (including himself), there is a mutiny: the pirates turn against him and make him walk the plank. They then start again, with the next most senior pirate as captain.

What is the largest number of coins the captain can keep without risking his life?