1970 Canadian MO Problems/Problem 3

Problem

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either $1$ pound or $2$ pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.

Solution

Let $R1, R2, B1, B2$ each be the 4 types of balls such that the number is the weight and the letter be the color of the ball.

We can prove by contradiction.

Assume $R1$ exists, then, $B2$ can't exist. We know there has to be at least 1 of each color, so $B1$ has to exist. Since $B2$ exist, $R2$ can't exist. The question states that there has to be at least 1 of each weight, but there isn't any ball that weighs 2 pounds. There is a contradiction.

Thus, there has to be a ball with different color and weight.