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?

It seems to me there is no maximum.

Consider a drawer with 9 pairs of color A, 10 pairs of color B and an unlimited supply of color C.

If she finds one left glove she knows it will take 20 right gloves to guarantee a matching pair.

Finding more than one left glove would be worse, because these might match each other so she would still need 20 rights.  (Even if she had 20 lefts, they could all be color C.)

[I know you could argue that if there is an infinity of pairs of color C you would be guaranteed never to pick A or B, but my argument still shows there is no upper bound.]

-Jer

Posted by Jer on 2004-06-03 09:21:01