Suppose that a new version of Ping-Pong is invented and that the rules of the ga

Question

Suppose that a new version of Ping-Pong is invented and that the rules of the game are different. The game is still played between two people and the first player to get 21 points wins the game. However, each time the score reaches a ratio that can be reduced, it is immediately reduced to lowest terms. For example, if the score reaches 4-2, it is reduced to 2-1. If it reaches 12-9, it is reduced to 4-3. If it reaches 14-12, it is reduced to 7-6 and so on. Even a score of 21-7 will be reduced to 3-1. What are the possible final scores for this game?

Explanation / Answer

If the first player's score reaches to 21 points he will win the game.In this time there should not be any common divisor to the scores of first and second players.so the greatest common divisor(GCD) for the two scores must be 1.so there are some possible combinations to win the game for the player.

(21,1),(21,2),(21,4),(21,5),(21,8),(21,10),(21,11),(21,13),(21,15),(21,16),(21,17),(21,19),(21,20)

But if we clearly observe the scores there is a problem in the above scores.suppose take the second combination (21,2) here first player wins the game.But before he wins the game second player had already got 2.First player get 21 points after getting 20 points.In this case score will be (20,2) and it is reduced to (10,1) so final score (21,2) never possible.Similarly for the remaining combinatoins also first player score should cross the second player score so remainning scores also not possible.So only two final scores are possible those are is (21,1),(21,0)

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