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Principle of pigeon coop in sixth grade mathematics.
1. The answer is at least 36.

The reason is that 6 colors *** 120 socks, each color can only have 20 socks. When carrying socks of size 36, there should be at least two colors and at most six colors. If the six colors are complete, there are three pairs of socks in each color. If the 36 pairs of socks are 2, 3, 4 or 5 colors, then there must be 3 pairs of socks of the same color.

2. Take out at least 10 checkers from the box.

Pigeon hole principle knows three different colors of checkers *** 18. There are only six pieces in each color. When you take 10 checkers, there are at least two colors. There are only three colors at most. 3*3=9 when three colors appear. There are now 10 teams. Obviously, at least four pieces are of the same color.