# Counting Socks

Can you work out the smallest number of socks Rachel needs to take out of the box to be guaranteed a pair?

## Problem

A box contains two white socks, three blue socks and four grey socks.

Rachel knows that three of the socks have holes in them, but does not know what colour these socks are.

She takes one sock at a time from the box without looking.

How many socks must she take for her to be certain she has a pair of socks of the same colour without holes?

This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

Imagine the worst luck that Rachel could have.

She could take out all 3 socks with holes in them before she finds a pair.

If all of the white socks or all of the blue socks have holes in them, then having removed the socks with holes in them, Rachel will only have socks of two colours left, which will speed the process up.

So if Rachel is unlucky, there will still be socks of all 3 colours left after discounting the socks with the holes in them, and she will use up 3 choices taking a sock of each colour. Then, the 7th sock will have to make a pair with one of them.

So the most choices that Rachel could need is 7.