### The Lady or the Lions

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path to follow. Which room should she wait in to give her lover the greatest chance of finding her?

Four fair dice are marked differently on their six faces. Choose first ANY one of them. I can always choose another that will give me a better chance of winning. Investigate.

### Flippin' Discs

Identical discs are flipped in the air. You win if all of the faces show the same colour. Can you calculate the probability of winning with n discs?

# Monkey Puzzle

##### Stage: 3 Short Challenge Level:

Let the hats of B, H and L be b, h and l respectively. Draw a table showing the ways in which the monkeys can select hats.

In only two of the six ways do none of the monkeys have the same hat as when they arrived, hence the required probability is

$\frac{2}{6}=\frac{1}{3}$.

Alternatively, there are $3\times2\times1=6$ possible ways to choose the three hats.
There are two hats that B could choose.
If B chose h, then L would have to choose b and H would have to choose l.
If B chose l, then H would have to choose b and L would have to choose h.

So once B has chosen his hat the other two are fixed. So there are just the two possible alternatives out of six ways.
So the probability is $\frac{2}{6}=\frac{1}{3}$.

This problem is taken from the UKMT Mathematical Challenges.