# Socks, shoes, gloves and boots

Socks, shoes, boots and gloves.

An old puzzle, the origin of which I was unable to find, is this:
1) My socks are in a dark room, where I can only feel them, not see them. I have 4 black and 4 blue socks. How many socks must I take out to be sure that I have 2 of the same color?

Solution: nr 2

There are numerous variations. A well known variation is where the socks are replaced by shoes. The colours are replaced with the left and right side of the shoes, and people are not allowed to fit them. Other variations are to replace the socks with boots or gloves. These variations are merely cosmetic, they do not change the combinatorial reasoning behind the puzzle.

3) The original puzzle has equal number of socks for both colours. But this is not necessary. It is perfectly valid to have different numbers of socks in the two colours:
You socks are in a dark room. You have 4 blue and 5 black socks. It is too dark to see what colour a sock is. How many socks do you need to take out to be certain you have 2 socks in the same colour?

Solution: nr 22

4) Raymond Smullyan, in his book “What is the name of this book?”, came up with two variations.
In the first one he asks: In the dark room are 4 blue and 4 black socks. How many socks should you take out of the room to be sure you have two socks in DIFFERENT colours?

Solution: nr 32

5) The second variation is: In a dark room are blue and black socks in equal numbers. Suppose the number of socks I must take out to be sure of 2 socks in the same colour happens to be equal to the number of socks I must take out to have 2 socks of different colours, how many colours are there?

Solution: nr 42

6) One variation I came up with, only to learn that others had thought of the same variation, is:
There are 3 socks in each of the colours blue, gray, brown and black in a dark room. How many socks must I take out to be certain that I have two socks in the same colour?

Solution: nr 52

Another variation:
7) Suppose there are an equal number of socks in several colours in a dark room. I tell you that you must take out 6 socks to be certain that you have 2 socks in the same colour. How many colours are there?

Solution: nr 62