# Mathematical Puzzles

Click on a puzzle for its solution.

## Mixing

Start with a half cup of tea and a half cup of coffee. Take one tablespoon of the tea and mix it in with the coffee. Take one tablespoon of this mixture and mix it back in with the tea. Which of the two cups contains more of its original contents?

## Sequencing

Extend the sequences:

## Mutilated Chess Board

A domino is exactly the same size as two squares of a standard 8x8 chessboard. 32 dominoes can be arranged to tile the chessboard with no gaps or overlaps. If two diagonally opposite corner squares are removed, is it possible to tile the remaining 62 squares with 31 dominoes?

## Black-Balled

You are in prison with 1000 black balls, 1000 white balls and two large baskets. You must put all the balls in the baskets before next morning, when you will be brought grovelling before the king and the baskets placed next to him. He will first choose a basket at random, and then put in his hand and choose a ball at random. If it's white then you walk free, and if it's black then... gulp. How should you distribute the balls between the two baskets? (Oh, and if the king chooses an empty basket: it's the chop.)

## Subset Sums

Given the set X = {8462, 75693, 51089, 19293, 5664, 9826, 20781, 22195, 119392, 10352, 19987, 26532, 1662, 1894, 19556, 8325, 99810, 20029, 11828, 87206}, you must show that there exist at least two disjoint subsets of X having the same sum of elements. The solution does not involve exhibiting the sets. (Note that "disjoint" means that the two sets do not have any elements in common.)

## Mr Sum and Mr Product

Mr. S. and Mr. P. are both perfect logicians, being able to correctly deduce any truth from any set of axioms. Someone thinks of two numbers between 2 and 500 inclusive. He then adds them up and whispers the sum to Mr. Sum. He also multiplies them together and whispers the product to Mr. Product. The following conversation then ensues.

• Mr Product: I don't know what the two original numbers were.
• Mr Sum: I already knew that you didn't know.
• Mr Product: Well now I know.
• Mr Sum: Aha! So do I.

What were the original two numbers?

## Truth Tellers

There is a road which forks. One path leads to heaven and the other to hell. At the junction there are two oracles who will answer any yes/no questions they are asked. One of them always tells the truth, and the other always lies. You do not know which is which. To discover which path leads to heaven, you are allowed to ask one yes/no question to one oracle.

At another fork there are three oracles, one of them always tells the truth, another of them always lies and the third tells the truth or lies purely at random. To discover the path to heaven you can now ask two yes/no questions (not necessarily to the same oracle).

## Gameshow

You're on a gameshow, and there are three doors, only one of which contains a prize. As happens every week, you initially choose one door, then the gameshow host opens a different door to show that the prize was not behind that one. Finally you have the option of keeping your door or changing to the other closed door. Should you change?

## Missing Square

A selection of mathematical puzzles chosen by Joe Leslie-Hurd.