Sharon has a number of pairs of gloves of identical design, but of several (at least three) different colors. She has at least three pairs of each color. In the dark she can distinguish the handedness of a glove, but not its color. Unfortunately, she keeps the gloves jumbled up in a drawer in an unlit cellar.

Sharon knows that if she takes out 21 gloves, in the dark, she can be sure of getting at least one pair.

What is the maximum number of pairs of gloves that she could have?

(In reply to re: Different solution? by np_rt)

Let's take one example of what Jer said, to concretize it:

She has 9 pairs of color A, 10 pairs of color B and 10,000 pairs of color C.

She takes one left-hand glove. As luck would have it, it's color C. If she chose up to 19 right-hand gloves, the worst-case scenario would still allow the possibility she had no color-C right-hand glove.  But if she took one more right-hand glove, she'd be assured of having a pair.

When the number of pairs of color C is said to be infinite, that is only the "limit".  That is, there is no limit. No matter how many of color C there were, it would require 21 gloves to assure a match.

Posted by Charlie on 2004-06-03 15:22:53