You need to pack for a midnight flight and the power is out. Your closet contains six pairs of shoes, six black socks, six gray socks, six pairs of brown gloves, and six pairs of tan gloves, but it’s too dark to distinguish colors or to match shoes. How many of each item must you take to be sure of having a matched pair of shoes, two socks of the same color, and a pair of matching gloves?
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Imagine the very worst case: You might pack six non-matching shoes, two non-matching socks, six left-hand gloves of one color, and six right-hand gloves of the other. You can make each of these failures impossible by adding one more item. So you’ll need to pack seven shoes, three socks, and 13 gloves.
From Peter Winkler’s excellent collection Mathematical Puzzles, 2021.
04/20/2021 UPDATE: I made a mistake here — in Winkler’s original, you’re not able to distinguish right-handed gloves from left in the dark. So instead of “six left-hand gloves of one color, and six right-hand gloves of the other,” I should have said, “six brown all left or all right, and six tan, also all left or all right,” as in Winkler’s original. In that case 13 gloves are needed.
(Thanks, Chris and Adam.)