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?
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3$\begingroup$ 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. $\endgroup$– AnkitMay 22, 2022 at 14:52
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7$\begingroup$ @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 $\endgroup$– Ryan FuMay 22, 2022 at 15:17
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$\begingroup$ 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. $\endgroup$– DoctorVMay 22, 2022 at 15:34
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$\begingroup$ 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) $\endgroup$– justhalfMay 23, 2022 at 10:21
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$\begingroup$ Related: puzzling.stackexchange.com/questions/59507/… $\endgroup$– justhalfMay 23, 2022 at 10:27
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2 Answers
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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
answered May 22, 2022 at 20:03
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$\begingroup$ 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? $\endgroup$– DoctorVMay 22, 2022 at 21:06
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11$\begingroup$ @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. $\endgroup$– hexominoMay 22, 2022 at 21:13
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A less conventional solution:
If you subtract the number of played matches from 15 (the maximum), you'll still get the same inequalities but with the paired contestants playing with each other this time. The max number of the total sum of possible matches is $16*15 = 240$, and the new sum exceeds the old by $8*2=16$, so the old sum is $112$, making Patrick a 7-time player. If Rachel has played $r$ times, $15-r=r+1$, meaning $r=7$.
Test if the solution works:
14: Plays with the people from 1 to 6, then both Patrick and Rachel, and from 8 to 15. The only person they don't play with is 0, who's their partner.
13: Plays with the people from 2 to 6, then both Patrick and Rachel, and from 8 to 15. The only remaining person they don't play with is 1, who's their partner.
Same for 12-2 to 8-6. The Patrick - Rachel pair (7-7) also works.
PS. Assume there can be multiple possible solutions:
Let the possible sums be $s_1, s_2,...$ and $s_n$. Then $224-s_1, 224-s_2,...$ and $224-s_n$ are also possible. We can let $s_1+s_2=s_3+s_4=...=224$. So if Patrick can play $p$ matches, he can also play $14-p$, and $p_1+p_2=p_3+p_4=...=14$ (same for Rachel's $r$ matches and 14). If $p =/= r$ both can be anything due to symmetry, which is wrong because $p$ must be odd for $s$ to be even, so $p = r$. A smaller configuration can be achieved if a pair with different numbers of matches is ignored, which means the sum of their number of matches must be 14. $p=r=7$.
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2$\begingroup$ Clever. But this works only if you know a solution exists and is unique. And with that knowledge, using the symmetry between played and unplayed matches, you can note that Rachel must have played as many matches as she didn't. So, of the 14 possible matches she played exactly half. $\endgroup$ Nov 25, 2022 at 13:51
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1$\begingroup$ That's fair I guess, but if you assume a unique solution exists, you can test it out. If it works (which it did in my edit), great. If it doesn't, it's reduction to absurdity. Even if you assume multiple solutions can exist, there's nothing wrong with going for the single solution first, testing it out and stopping there if it works. $\endgroup$– NautilusNov 25, 2022 at 15:46
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$\begingroup$ Well, we can also prove that there's a single solution. $\endgroup$– NautilusNov 25, 2022 at 19:36