Problem 1593

1. Let $k$ be the number of men and $m$ be the number of women, with $n = k + m$ being the total number of players.
2. Each player scores the same total points against women as against men. This implies that the total points scored by men against women equals the total points scored by women against men.
3. The sum of the scores of man-against-man matches is $\binom{k}{2}$. This is because each pair of men plays one match, and there are $\binom{k}{2}$ such pairs.
4. Similarly, the sum of the scores of woman-against-woman matches is $\binom{m}{2}$.
5. The total points scored by men against women must equal the total points scored by women against men. Since there are $km$ matches between men and women, we have:
$$\binom{k}{2} + \binom{m}{2} = km$$
6. Expanding the binomial coefficients, we get:
$$\frac{k(k-1)}{2} + \frac{m(m-1)}{2} = km$$
7. Multiplying through by 2 to clear the fractions:
$$k(k-1) + m(m-1) = 2km$$
8. Simplifying, we get:
$$k^2 - k + m^2 - m = 2km$$
9. Rearranging terms, we obtain:
$$k^2 - 2km + m^2 = k + m$$
10. This can be rewritten as:
$$(k - m)^2 = k + m$$
11. Let $n = k + m$. Then:
$$(k - m)^2 = n$$
12. This implies that $n$ must be a perfect square, since $(k - m)^2$ is a perfect square.

The final answer is $\boxed{ n }$ must be a perfect square.