Socks

Date: 8/31/96 at 17:34:15
From: Anonymous
Subject: Socks

Jack's sock drawer contains 10 blue socks and 12 gray socks.  
The room is dark and he cannot turn on the light.  What is 
the least number of socks he must take out of the drawer to 
be certain of each of the following conditions?

a. He has a pair of the same color.
b. He has a pair of the blue socks
c. He has a pair of gray socks.
d. He has one pair of each color.

I've just looked at it a lot and I can not figure it out. 
Thank you.

Date: 9/1/96 at 3:36:53
From: Doctor Mike
Subject: Re: Socks

>a. He has a pair of the same color.

Least number = 3, because if the first two do not make a pair, 
then the third will match one of the first two.

>b. He has a pair of the blue socks

Least number = 14, because the worst case is he will pick all 
the grey ones on the first 12 tries, and then he is guaranteed 
that the next two will be blue.

>c. He has a pair of gray socks.

Least number = 12, because the worst case is he will pick all 
the blue ones of the first 10 tries, and then he is guaranteed 
that the next two will be grey.

>d. He has one pair of each color.

Least number = 14, following essentially the same logic as (b).  
He could go for 12 tries and get only grey, and then the next 
two would be blue.  If you select 14 of them to take care of 
this worst case scenario, then all the other situations would 
be taken care of also.
     
Another answer would be to pick your socks out the previous evening 
when you could turn the lights on, in which case the first three 
answers would be 2 and the answer to d would be 4 (unless you are
color-blind).
  
I hope this helps. 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/