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There are 20 gloves in a drawer: 5 pairs of black gloves, 3 pairs of brown, 2 pairs of grey. You select the gloves in the dark and check them only after you have made the selection. What is the smallest number of gloves you need to select to guarantee getting the following?

a) at least one matching pair

b) at least one matching pair of each color

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For b), $19$ is enough, anything less and we may be in trouble. – André Nicolas Dec 23 '12 at 19:08

2 Answers

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HINTS:

(a) If you select $10$ gloves, you might get every left glove in the drawer; what if you select more than $10$?

(b) You could select $16$ gloves and still have only two colors; how? How many more than that are needed to be sure that you actually have a pair of each color?

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Hi @brian for part a) I was thinking that I would select all of the gloves for the wrong hand. So 5 black gloves, 2 grey gloves, and 3 brown gloves. A total of 10 gloves. And then the next glove I pick would need to be a matching pair? – Justin Dec 23 '12 at 19:03
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@Justin: You’re right: I was thinking of something without handedness, like socks, rather than gloves. I’ll change the answer. For (b) just use the same kind of thinking. – Brian M. Scott Dec 23 '12 at 19:07
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In general, to do this sort of problem, you want to find the "worst-case scenario" – in this case, the greatest number of gloves you can select that don't fit the conditions you've set out. Then, selecting any more will satisfy the conditions.

For example, selecting either white or grey socks out of a drawer, the most you can pull out without having a matching pair is two - one white and one grey. The third one must be either one of the two colours, meaning that there is at least one matching pair.

In this case, the gloves can either be left or right gloves, which puts another condition on the problem.

a) The most gloves you can have without a single matching pair is all the gloves of a single hand. So if you have more than 10 gloves (at least 11), at least one of them will be a matching pair.

b) The most gloves you can have without a matching pair of each colour is all the gloves except the gloves of one hand and one colour (so that you never have a matching pair of that colour). For example, if you have none of the left black gloves, you'll never have a black pair, and you have 15 gloves. However, it's possible to do better than this. Which colour will give you the worst-case scenario?

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