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# Factoring a Million

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### Coin Tossing Games

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Age 14 to 16

Challenge Level

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

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 a head (you win). What is the probability that you win?

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?

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