TITLE: The Pigeonhole Principle
QUESTION [5 upvotes]: Vera has $10$ white socks, $10$ black socks, $10$ brown socks, $10$ blue socks, and $10$ red socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least two matching pairs?
  (The two pairs cannot share a sock -- i.e., three socks of the same color don't count as forming multiple pairs.)

I thought, since we need $6$ socks to make at least one pair, we'd need $7$ to make at least two pairs, but that is incorrect.

REPLY [3 votes]: Another argument why $8$ socks is sufficient: have Vera throw away one sock in each color of which she has an odd number of socks. Since she cannot have an odd number of socks in all colors (because then the total number of socks would be odd), she throws away at most $4$ socks and is left with at least $4$ socks. Furthermore, she now has an even number of socks in each color, so these at least $4$ socks must form at least $2$ pairs.