Score

[radosr]

Two competitors play badminton. They play two games, each winning one of them.
Then they play a third game to determine the overall winner of the match.
The winner of a game of badminton is the first player to score at least 21 points with a lead of at least 2 points over the other player.
In this particular match, it is observed that the scores of each player listed in order of the games form
an arithmetic progression with a nonzero common difference.

What are the scores of the two players in the third game?

 

[euroazn]

In this particular match, it is observed that the scores of each player listed in order of the games form
an arithmetic progression with a nonzero common difference.

Can you be more clear on the order of the player scores?

 

[Tsotne]

And one more thing, I'm not sure how the points are counted.

 

[pruse]

1) Both progressions are increasing. Then clearly the loser of the first game - say player I - wins the series because he overtakes the other player by larger leaps. Assume player I's scores are (a<a+d<a+2d), where the last two are winning scores. If (a+d>21), player II's scores in games 2 and 3 must be (a+d-2,a+2d-2), respectively. But then his score in game 1 must be (a-2), which is impossible. So we must have (a+d=21). This means that player II's game 2 score is (\leq 19). However, his game 1 score must be (\geq 21) because he won that game. This means that his scores are in decreasing order - a contradiction. So we can't have both progressions be increasing.

2) Both progressions are decreasing. Then the loser of the first game - say player I - wins the series because his decreases are smaller. Assume player I's scores are (a>a-d>a-2d), where the last two are winning scores. This means (a,a-d>21), from which we get that player II's scores in the first two games are (a+2,a-d-2), which means that his score in the third game is (a-2d-6). But this is 6 less than player I's score in that game, which means that player I's score in that game must be exactly 21. This makes 21-15 the final score of game 3.

3) One increases, the other decreases. Clearly the increasing one wins the series. Assume player I's scores are (a<a+d<a+2d). If he wins games 2 and 3 , then (a+d\geq 21). If (a+d>21) then as in case (1) we get a contradiction. So we must have (a+d=21). His scores are then (21-d < 21 < 21+d), and player II's scores are, say, (a > b > 19+d). But clearly (b\leq 19) since player II lost the second game, making (b>19+d) a contradiction.

So the only possibility is 21-15.

 

[euroazn]

Wow, well done!