# Patrick and Rachel go to a tennis tournament with 7 other couples

Patrick and Rachel go to a tennis tournament with 7 other couples. Each round is a single's match (1 vs 1). Nobody plays against his/her partner and nobody plays twice against the same player. At the end of the tournament Patrick asked everyone how many people they played against, and found that each one answered with a different number. How many matches did Rachel play?

• This scenario does not seem possible... there are 16 players all playing a different number of matches, with no repeats. So that means each number from 0-15 is the number of matches per person. The person who played 15 matches played against every single person, including their partner, which is not allowed. May 22 at 14:52
• @Ankit Perhaps it may be possible, because it never said that Patrick played a different number of matches from everyone else, just the answers he got were unique, so it'd be 1-15 with 1 repeat May 22 at 15:17
• I found out this quiz during a test for a job, they used these words so I don't know how to re-word it. I wonder if "and found that each one answered with a different number" literally means every possible different number from 0 to 15, or just not the same number from everyone. May 22 at 15:34
• I've read this question (the essence, not the exact wording) in standard combinatorics problem book, so I'm quite sure it's correct (as also shown by hexomino's answer) May 23 at 10:21
• May 23 at 10:27

First notice the wording of the question: "Patrick asked everyone how many people they played against, and found that each one answered with a different number"

This means that

Patrick himself may have played the same number of games as one of the other players.

In particular, as noted in the comments

Nobody could have played 15 matches, hence Patrick heard each answer between 0 and 14 exactly once.

Now observe that

The person who played 14 matches plays everyone except their partner and so their partner must be the one who plays 0 matches (since every other player plays at least one match). Similarly, the person who played 13 matches plays everyone except the person who played 0 and their partner so must be coupled with the person who played 1 match.

Following this line of reasoning further, the person who played 12 matches is coupled with the person who played 2, the person who played 11 is coupled with the person who played 3... and the person who played 8 must be coupled with the person who played 6.
Finally, the person who played 7 matches must be coupled with the remaining person, Patrick, who must also have played 7.

Hence, we deduce that Rachel played

7 matches

• Sorry I didn't understand why "The person who played 14 matches plays everyone except their partner and so their partner must be the one who plays 0 matches". Why his partner could not have played some matches with some other player? May 22 at 21:06
• @MattiaVanzetto The reason being that every other player is guaranteed to have played at least one match (against the person who played 14) so the partner can be the only one who played zero. I'll edit the answer to clarify. May 22 at 21:13