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During their holidays Martina, Karl and Jim played badminton every day.

The winner of each match continues to play in the next match against the one who just had a break, while the loser of this match pauses.

At the end of their holidays, it turned out that Martina played 18 matches, and Karl played 25 matches.

What is the largest number of matches that Jim could have played?

(A) 29 (B) 31 (C) 33 (D) 35 (E) 39

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Answer

(B) 31

Reasoning

Given that each player will play in, at least, every second match and that Martina plays 18 matches, there could have been, at most, 37 matches. This means that there are, at most, 74 player slots and since we have already counted those slots for Martina and Karl (18 + 25 = 43), there are, at most, 31 player slots left for Jim. It remains to show that this is achievable. It could have been done as follows.

Karl beats Jim,
Karl beats Martina,
Karl beats Jim,
Karl beats Martina,
Karl beats Jim,
Karl beats Martina,
Karl beats Jim,
Karl beats Martina,
Karl beats Jim,
Karl beats Martina,
Karl beats Jim,
Karl beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,
Jim beats Martina,
Jim beats Karl,

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The solution is

B

Reasoning:

To get the solution you have to consider how often Jim has won. To get the largest number of Tobias wins:

J - Jim,
M - Martina,
K - Karl,
- : against

First match: Starting with K - J because K has more plays than M, J wins to plays also the next match. Second match: M - J, J wins J has 2 plays, M and K have 1 play

...

35th match: K - J
36th match: M - J
Now M has 18 plays. So M can't play more games. Karl has also 18 plays but need 7 more. Jim has 36 plays. 37th match: K - J, K has 19 plays. He needs 6 more. Jim has 37 plays. So Jim can't win every time. He has to lose 6 times, because then Karl has 6 plays more. 37 plays - 6 plays = 31 plays.

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