### 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.

### Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

# Tying the Rope

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

There are six different possible ways of choosing two of the loose ends:

Imagine calling the ends of one rope A and B, and the ends of the other rope X and Y.

These are all the different possible pairings:
A & B
A & X
A & Y
B & X
B & Y
X & Y

Four of these pairings lead to the ropes being tied together:

A & X
A & Y
B & X
B & Y

Therefore the probability that Sam is left holding one piece of rope is $\frac{4}{6}=\frac{2}{3}$.

Alternative method:

When Pat picks up one end, there are three possible ends left that she can choose. One of these will be the other end of the same rope and will produce a loop. Either of the other two will produce the desired result.

Therefore the probability that Sam is left holding one piece of rope is $\frac{2}{3}$.

This problem is taken from the UKMT Mathematical Challenges.