P.G. Zhenoarove

In a box, there are 111 balls: red, blue, green, and white. It is known that if you pull out 100 balls from the box without looking, there will definitely be four balls of different colors among them. What is the smallest number of balls that need to be pulled out without looking to ensure that there are three balls of different colors among them?

#
There are no less than 12 balls of each color (otherwise, they could all end up among the 11 remaining in the box). Therefore, the number of balls of two colors is no more than 

$111-24=87$. Hence, among any 88 balls, there will be balls of at least three colors.

87 balls are insufficient, for example, for the color distribution 75, 12, 12, 12.

## Answer

88 balls.