# 1975 AHSME Problems/Problem 25

## Problem

A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player?

$\textbf{(A)}\ \text{the woman} \qquad \textbf{(B)}\ \text{her son} \qquad \textbf{(C)}\ \text{her brother} \qquad \textbf{(D)}\ \text{her daughter}\\ \qquad \textbf{(E)}\ \text{No solution is consistent with the given information}$

## Solution

We know that the worst player's twin and the best player are of opposite sex, and the worst and best players are the same age.

Case 1: Suppose the daughter was the worst player. The son would have to be her twin because nobody else can be her twin. Then, that means the son and the best player are of opposite sex. But we initially made the daughter the worst player, so she cannot also be the best player. That only leaves the woman to be the best player. However, the daughter and the woman are not the same age. Therefore, the worst player cannot be the daughter.

Case 2: Suppose the woman was the worst player. Suppose again that her and her brother are twins. Her brother and the best player are of opposite sex. Using the same logic as in Case 1, the best player would be the daughter. The woman and the daughter are not the same age, so the worst player cannot be the woman.

Case 3: Suppose the brother was the worst player and that him and the woman are twins. The woman and the best player are of opposite sex, so the best player would be the son. If the brother is the woman's twin (our initial assumption), then the son is not the same age as the brother, which means that the son is not the best player, and the brother is not the worst player.

However, the woman and the brother do not necessarily have to be twins. For this case, we had to assume they were twins or else there was no way to continue the problem, since the brother would have no twin.

Case 4: Suppose the son was the worst player. The daughter and the best player are of opposite sex, so the best player is the brother. The woman and her brother do not have to be twins. So it is, in fact, possible that the brother and the son are the same age.

From this, we conclude our answer as $\boxed{\textbf{(B)}\ \text{her son}}$. ~jiang147369