How many socks to guarantee two pairs?

Question

A drawer contains a large number of black socks and a large number of white socks, all jumbled. It is dark and you cannot tell them apart. The classic question asks how many socks you must pull out to guarantee one matching pair (the answer is three). How many must you pull out to guarantee two matching pairs? The pairs may be the same colour or different colours: two black pairs, two white pairs, or one of each all count. Then generalise to k pairs.

Accepted Answer

Five socks. Four is not enough. With four socks you might get three black and one white. That contains only one pair (two of the blacks) plus two odd socks. So four socks can leave you with a single pair. Five always works. Take five socks; say b are black and w are white with b + w = 5. The number of complete pairs is the whole-number part of b/2 plus the whole-number part of w/2. Check every split: (5, 0) gives 2 pairs; (4, 1) gives 2; (3, 2) gives 1 + 1 = 2; and the mirror-image splits are the same. Every case yields at least two pairs. The general rule. With two colours, the only socks th…

How many socks to guarantee two pairs?

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