A game that demands a logical approach using systematic working to deduce a winning strategy

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by. . . .

Two brothers belong to a club with 10 members. Four are selected for a match. Find the probability that both brothers are selected.

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

Which of these games would you play to give yourself the best possible chance of winning a prize?

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

From the atomic masses recorded in a mass spectrometry analysis can you deduce the possible form of these compounds?

When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.