# Puzzles

## The blue-eyed sisters

If you happen to meet two of the Jones sister (two sisters chosen at random from all the Jones sisters), it is exactly an even-money bet that both will be blue-eyed. What is your best guess of the total number of Jones sisters?

#### Show answer & extension

#### Hide answer & extension

If there are \(n\) sisters and \(k\) of these sisters have blue eyes, then the probability of the two sisters having blue eyes is:

$$\frac{\left(\begin{array}{c}k\\2\end{array}\right)}{\left(\begin{array}{c}n\\2\end{array}\right)}
\\
=\frac{k(k-1)}{n(n-1)}=\frac{1}{2}
$$

This means that:

$$2k(k-1)=n(n-1)$$

The smallest integer solution of this is when there are 4 sisters, 3 of whom have blue eyes.

The next smallest integer solution is when there are 21 sisters, 15 of whom have blue eyes. There are unlikely to be as many as 21 sisters, so 4 sisters are the most likely.

#### Extension

What is the next integer solution after 21 sisters?