Colour and size of balls

Two boxes contain between them 65 balls of several different sizes. Each ball is white, black red or yellow. If you take any five balls of the same colour, at least two of them will be of the same size(radius). Prove that there are at least three balls which lie in the same box have the same colour and the same size(radius).

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Originally Posted by malaygoel

Two boxes contain between them 65 balls of several different sizes. Each ball is white, black red or yellow. If you take any five balls of the same colour, at least two of them will be of the same size(radius). Prove that there are at least three balls which lie in the same box have the same colour and the same size(radius).

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I like this problem.

Becuase there are two boxes we know one of them contains at least 33 balls. 33 divided by 4 is 8 with a remainder of 1, so we know there are at least 9 of one color balls in one box. There are at most 4 different sizes of balls, so 9 divided by 4 is 2 with a remainder of 1. Therefore, there are at least 3 balls which lie in the same box, have the same colour, and the same size. :D