# Need some help with this.

How do we solve the following problem?

There are $n$ players participating in a sports tournament. Each player plays with every other player and each game ends in a win for one of the players (no draws). Prove that the players can be arranged in the following sequence: $P_1, P_2, P_3, \ldots , P_n$ where, player $P_i$ has beaten the player $P_{i+1} (i=1,2,3, \ldots , n-1)$ in at least one game.

Note by Deeparaj Bhat
4 years, 8 months ago

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I think, I have got a reasonable proof.I will be using induction.Consider the base case of two players.In two players, there will be a winner and a loser, so we can definitely form the sequence.Now,assume that we can form the given sequence for k players.So, we have the sequence P1,P2,P3,.........,Pk.We must now show that it is always possible to insert P (k+1) somewhere between the gaps of Pi (i=1,2,3.......,k).We know that P (k+1) will play exactly k games with each of P1,P2,......Pk.So,there will be k outcomes,where each of outcome is win (W) or lose (L).So,we consider the string of k letters where each letter is L or W.If at,say r(th) position,there is W,then it means that P (k+1) defeated Pr.Now,if first letter of the string is W,then P (k+1) will precede P1.Now,consider that the first letter of the string is L.If second letter is W,then P (k+1) comes between P1 and P2.Suppose second letter is L.Now,if third letter is W, then P (k+1) comes between P2 and P3.This logic can be extended,so as to get that,we can always find a consecutive L and W, if not, every letter of the string will be L.In that case P (k+1) will succeed Pk.So,it has been shown that it is always possible to form a sequence with (k+1) players.So,the question has been proved using induction.Does this solution seem fine?

- 4 years, 8 months ago

Hmm... This seems fine. It resembles the proof I'd written in the exam very much (but I missed one crucial point in my written proof 😣).

Btw, is there partial credit?

- 4 years, 8 months ago

Yes,definitely there is partial credit. Does my solution take into account,the crucial point you are talking about?

- 4 years, 8 months ago

Yeah, it does. Only after you gave your proof did I realise that I was pretty close to the actual proof.

Once again, thanks for the proof!

- 4 years, 8 months ago

Did you write the CMI exam?

- 4 years, 8 months ago