### 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?

### Nine or Ten?

Is a score of 9 more likely than a score of 10 when you roll three dice?

### Racing Odds

In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?

# A Dicey Paradox

##### Age 11 to 14 Challenge Level:

Four fair dice are marked on their six faces, using the mathematical constants $e$, $\pi$ and $\phi$ as follows:

 A: 4 4 4 4 0 0 B: $\pi \pi \pi \pi \pi \pi$ where $\pi$ is approximately 3.142 C: e e e e 7 7 where e is approximately 2.718 D: 5 5 5 $\phi \phi \phi$ where $\phi$ is approximately 1.618

The game is that we each have one die, we throw the dice once and the highest number wins. I invite you to choose first ANY one of the dice. Then I can always choose another one so that I will have a better chance of winning than you. You may think this is unfair and decide you want to play with the die I chose. In that case I can always chose another one so that I still have a better chance of winning than you. Investigate the probabilities and explain the choices I make in all possible cases.

Does it make any difference if the dice are marked with 3 instead of $\pi$, 2 instead of $e$ and 1 instead of $\phi$?