Sock Drawer Puzzle

A visual proof using the Pigeonhole Principle.

Your drawer holds:
  • 2 red socks
  • 20 yellow socks
  • 31 blue socks

Question: What is the minimum number of socks you must pull out, without looking, to guarantee a matching pair in the same color?

Red

2 socks

Yellow

20 socks

Blue

31 socks

Worst-case draw simulation

To delay getting a matching pair as long as possible, you would try to draw one sock of each color first.

No socks drawn yet.
There are 3 possible colors. So without making a pair, you can draw at most 3 socks: one red, one yellow, and one blue.

Pigeonholes = colors

Think of the colors as the pigeonholes:

Red
0
Yellow
0
Blue
0
3 + 1 = 4

With 4 socks placed into only 3 color categories, at least one category must contain 2 socks.

Minimum number required
4 socks

The first 3 socks could be one red, one yellow, and one blue, so no pair is guaranteed yet. But the 4th sock must be red, yellow, or blue again, so it must match one of the socks already drawn.

Therefore, the answer is 4.