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(4): huh? by nikki)

I'd like to support nikki's solution in the comment this is replying to.

28 pairs may be the solution if you limit the glove selection to a single right glove and 20 left gloves.  If there are 3 pairs blue, 3 pairs of green and 10,000 pairs of red, it only takes 7 left gloves and 7 right gloves to guarantee a red pair.

I'm going to argue some pedantics now.  The puzzle says, "...if she takes out 21 gloves, in the dark, she can be sure of getting at least one pair."  One might say, "But it only takes 14 gloves!" but if she takes out more than 14, it does not make her any less sure that she has a pair. 

The puzzle also seems to imply that she can choose between left and right gloves in the dark.  If not, then I suppose the maximum pairs is only 20 (or getting 21 left gloves is a possibility).

Posted by Tristan on 2004-06-03 20:18:22