B0.40 Puzzles

Some puzzles that have amused us over the last couple of years. Some easy, some hard.
Don't be put off by the first few which are harder than most.
Sequencing I is an easy place to start.
(If you hate doing puzzles and want to laugh at people who do, then see this.)

The Triangle Problem

What is the angle a?

[Slightly easier version available here.]

Wreck Me

Some rectangles, each with the property that at least one of its sides has integer length, are placed together to form a big rectangle with no holes.
An example is shown below.
Prove that the big rectangle has the same property.

Extension: does this generalise to fitting together n-cuboids with the property that m<n of their sides have integer length?

Note added later: This question was answered in the affirmative by Postlethwaite in 2005, using advanced slicing and hand-waving techniques. It has now come to be known as Postlethwaite's Integer Length Cuboid Hypothesis for Any Real Dimension (PILCHARD).

Clear Out

Setup: a semi-infinite chess board with counters in the three bottom left squares, as shown below.
How to move: iff the squares above and to the right are free, a counter can be removed and replaced by two counters, one in the square above and one in the square to the right.

Challenge: prove that it is not possible to leave the three bottom left squares empty.

Sequencing I

What is the next number in the sequence?
Or as James would say: Which number, the answer being an integer, best fits as the next number in the sequence which begins on the next line, continues for a n lines, where n is an integer, and then terminates with the number, indeed the integer, 312211?
1
11
21
1211
111221
312211
?

Sequencing II

What is the next number in the sequence?
1210
2020
21200
3211000
?

Basic Maths

Add one straight line to the following to make it true.
5+5+5=550
[Hint: a line through the equals sign, making it a not-equals sign, is not the required solution.]

[The stupidist (incorrect) answer so far: add a horizontal line through the center of the 0 and define \theta to be 3/11.]

Secondary Maths

How many seconds are there in six weeks?
[Hint: the answer only uses three symbols.]

Boil me an egg-and-soon

You have an egg and want to boil it for exactly 9 minutes. To do this you are provided with two egg timers - one which lasts for 4 minutes and one which lasts for 7 minutes. How should you use the egg timers in order to boil the egg in the shortest possible total time?

Wait for 12

You have 12 coins which are all identical except for their mass - 11 have the same mass and the other has a different mass (either more or less). You are given a set of scales and are allowed three weighings to determine which coin has a different mass. How do you do it?

Plain Colours

Consider the infinite 2D plane with a unit length defined on it. Can you colour the entire plane with three colours such that no two points unit distance apart are the same colour?

Bouncing Ants

100 ants are distributed randomly on a 1m long pole. They each constantly walk at 1cm/s. When they meet another ant they turn round and walk in the other direction. If an ant reaches the end of the pole it falls off. Consider one of the ants in the middle - i.e. number 49 or 50. Where will it be after 100s?

Gossiping Women People

Each of n people has one bit of information. They can phone each other and exchange information. How many phone calls does it take for all the people to know all the information?

Bag the Gold

You enter a room with contains 10 bags each full of coins. One of the bags is full of real gold coins weighing 2g each, whilst the others are all full of fake coins, weighing 1g each. The fake and real gold coins cannot be told apart by looking. There is a set of scales and you are allowed to make one measurement. How do you identify the bag full of real gold coins?

Cube Job

Prove that a cube cannot be dissected into smaller cubes that are all of different sizes.
[Note: however, this can be done for a square, e.g. see here.]

Coin a Baize

Consider the following two-player game: players take it in turn to place identical coins on a circular table such that the coins do not overlap. The first player not to be able to place a coin is the loser. With perfect play, who wins and what is their winning strategy?

Basic Maths 2

What mathematical symbol can be put between 5 and 9 to get a number bigger than 5 and smaller than 9?

Basic Maths 3

Move one digit in the following to make it true.
101-102=1
[The required solution does not involve a not-equals-to sign.]

Bridge Over Troubled Water

Four people need to cross a dark bridge. They take 1, 2, 7 and 10 minutes to cross respectively. They must cross with the use of a torch and, since there is only one torch, they must cross either alone or in pairs. What is the shortest total time in which they can all cross?

Prime Time

The following number is a sum of two primes. Which? 879976242195951958890801816612768566943805170226410617823301865416003514546684111640331356490455690766475 339038983303063831818394885482954417406863802340357540397021808027209610884076158915519334125353771492981

Tuesday's Child

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

Splitting 'ell

Cut the following shape into four identical pieces.

High Stakes

The problem is simple. You are given two wooden stakes, and a single loop of rope. The stakes are hammered into the ground vertically, a certain distance apart.

It is possible to loop the rope around the stakes such that it cannot be removed by pulling it horizontally. One example of how this can be done viewed from above is:

The problem is to satisfy the above requirement in such a way that when either of the two stakes is removed, the rope can then be pulled away horizontally. (The configuration drawn above is not good enough, as the rope will still be looped around one stake even when the other has been removed.)

The second part of the problem is to achieve the same for any number of stakes - when any single stake is removed, it should be possible to remove the rope by pulling horizontally.

Half-a-Dozen of the Other

Aim: to go from the arrangement of six coins shown on the left to the arrangement shown on the right in three moves.
How to move: pick a coin and slide it, without disturbing any other coins, so that it ends up touching two other coins.

People are not conserved

How many people are in this picture?

Joined-up Writing

Join all four pairs of letters with lines which do not leave the rectangle, do not go through other letters and do not cross.

M-theory

Make 9 separate triangles by passing 3 straight lines through an M.

Black and White?

You are one of a group of one hundred men held in a high-security jail. Unfortunately, your sadistic guards have devised a game to test the intelligence of their prisoners.

Each inmate will be wearing either a white cap or a black cap, but will not know which colour it is. You will all then stand in a long line, each man facing the back of the man in front, so that you can see the caps of the men in front of you (but not your own, nor any behind you).

The guards then ask each man what the colour of his own hat is, beginning with the prisoner at the back of the queue. A correct answer means prolonged life for that prisoner; a mistake will lead to a swift (and painful) death. The guards will continue from the back to the front of the queue, asking each inmate in turn.

You are to discuss strategy with the other prisoners beforehand. What is the best tactic to adopt in order to save as many men as possible, and consequently how many can you save?

Mutiny on the High Seas

Five pirates have brutally murdered a wealthy gentleman and taken possession of his one hundred indivisible gold coins. The five decide to go their separate ways, each pirating a different high sea. But first the question of how to divide their spoils must be resolved. The plan is as follows:

Fifthbeard, the lowest-ranked pirate, must make a suggestion as to how to divide the money. If he can gain the support of a majority of the five (including himself) then the coins will be allocated as he suggests. If not, he will be cutlassed into tiny pieces.

If he is killed, then Fourthbeard, the pirate of the next lowest rank must make a suggestion, and if this gains the support of a majority, he will live. Otherwise he will be killed, and Thirdbeard must make a suggestion. This goes on until the (remaining) pirates reach agreement.

Each pirate values life above anything else, followed by wanting to make as much money as possible. All other things being equal for him, a pirate will choose to kill another pirate rather than allow him to live. Assuming that the pirates always think logically, what should Fifthbeard suggest in order to achieve his best possible outcome?

Confused?

You are a sergeant in the British Army, in rather a tricky situation. Your CO has left you and your troops sitting under a bridge, with instructions to explode it in order to block enemy supply routes. You have been given enough explosives to destroy a small country and three fuses.

Each fuse burns for exactly four minutes, but they do not burn at an even rate. One fuse is attached to the explosives and lit, the explosion occurring four minutes later. The problem is that your men must leave one minute before the explosion - any later and you risk being caught in the explosion, any earlier and the enemy will have time to defuse the explosives. However, you have not watch or clock with which to measure time, just the three fuses and the explosives.

How will you light the fuses to ensure your safety?

A Switch in Time Saves Twenty-Three

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

"In the prison is a switch room, which contains two light switches labeled 1 and 2, each of which can be in either up or the down position. I am not telling you their present positions. The switches are not connected to anything.

"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell.

"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back.

"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.' and be 100% sure.

"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators. You will be carefully monitored, and any attempt to break any of these rules will result in instant death to all of you"

What is the strategy they come up with so that they can be free?

Numbers Two

A teacher thinks of two integers greater than one and asks two of her students to determine what they are. The first student knows their product and the second knows their sum.
First: I do not know the sum.
Second: I knew that. The sum is less than 14.
First: I knew that. However, now I know the numbers.
Second: And so do I.
What were the numbers?

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