Playing face or cross

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Three people decide to play to throw coins and see if they coincide in face or cross. Each one throws a coin and the one that does not match the other two loses. The loser must double the amount of money that each opponent has at that time. After three games, each player has lost once and has 240 cents.

How much did each one have at the beginning?

Solution

The first player started with 390 cents, the second with 210 and the third with 120.

This problem is easier to solve if we start from the final situation and go backwards, so that in the last play everyone ends up with 240 cents, the player who lost should have 480 and the other two 120 before the last game. If we go back we have that the development of the game was as follows:

In the first game loses the first player who is left with 60 cents, the second with 420 and the third with 240. In the next game, he loses the second player. The first one will have 120 cents, the second 120 and the third 480. In the last game, the third player loses and everyone gets 240 cents.