**From New Scientist #2444, 24th April 2004** [link]

Some players entered a “round robin” tennis tournament, where each player plays each of the others once, with each match resulting in a win for one of the players. At the end of the tournament I noted how many matches each of the players had won. The men were pretty pathetic, winning only one match each. Even so, Alan beat Barbara in their match. On the other hand, Christine beat David: had that result been reversed all the women would have won the same number of matches.

How many men and how many women entered the tournament?

**Note:** I am still waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment. The latest estimate is that I’ll have a connection by the end of October 2014.

[enigma1286]

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This Python 3 program constructively produces possible tables that satisfy the constraints of the problem. It runs in 51ms.

I’ve assumed that Alan (A) and David (D) are men and Barbara (B) and Christine (C) are women.

Solution:There were 2 men and 4 women in the tournament.Here’s a diagram of the possible matches played, a square with a 1 in it indicates that the player in that row beat the player in that column:

Analytically we can see that

mmen playT(m – 1)matches amongst themselves, and each of them must be won by a man.We are told that there are at least 2 men, and one of them wins a match against a woman. So there can be at most

m – 1matches amongst the men.So, we are looking for

mwherem ≥ 2andT(m – 1) ≤ m – 1.The only possible value is

m = 2.So the two men are

AandD, andDbeatsAwhen they play. They each lose the rest of their matches.The only remaining variable is

w, the number of women(w ≥ 2).There are

T(w + 1)matches in total, 2 of them won by men and the remainingT(w + 1) – 2must be 1 more than a multiple ofw.So, in order for

kto be a whole number it follows thatwmust be a divisor of 4(w ≥ 2). The only possible values are 2 and 4.Hence there are 2 men and 4 women (6 players in total). The men win 1 game each, the women win 3 games each, except for Christine who wins 4.

It is now easy to construct a set of possible matches, as follows:

If the men are

A,D, and the women areB,C,X,Y. Then:AbeatB, and loses the rest of his matches (D,C,X,Y). (1 win).DbeatA, and loses the rest of his matches (B,C,X,Y). (1 win).Xalready has 2 wins (againstAandD), let’s suppose they also beatY, and lose againstBandCto give the required 3 wins.Yhas 2 wins (againstAandD), and lost againstX. So needs one more win againstBorC. Let’s sayYbeatsBfor 3 wins.Bhas wins againstDandX, and has lost toAandY, so needs to win againstCto get 3 wins.All matches are now specified and

Chas wins againstA,D,X,Yand lost againstBto give 4 wins, as required.