A drawer has 10 black socks and 10 white socks mixed together, in a dark room

Why

Maansi's number (11) would be right for a different question — 'how many socks guarantee you a MAJORITY of one color', which really would need more than half of one color's total count. But the question only asks for ANY matching pair, and that needs far fewer socks, because there are only 2 colors in play. Think about the worst possible luck: pull 1 sock — no pair yet, obviously. Pull a 2nd sock — in the worst case it is the OTHER color, so still no pair (1 black, 1 white). Now pull a 3rd sock — it has to be either black or white, and both colors already have one sock pulled out, so this 3rd sock MUST match one of the first two, no matter which color it happens to be. Test it: this reasoning does not depend at all on how many total socks of each color are in the drawer — it would work exactly the same with 2 socks of each color or 1000 of each, because the argument only ever used the fact that there are 2 colors, never the total count. Her brother is right: just 3 socks guarantee a matching pair — with only 2 possible colors, the 3rd sock can never be a brand-new, unmatched color.