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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?
A Dicey Paradox
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.
Squaring the Circle
Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.
Stage: 3 Challenge Level: 
Both 9 and 10 can be made up in 2 different ways by adding pairs of the numbers 1, 2, ... 6, that is
9 = 3 + 6 = 4 + 5
10 = 4 + 6 = 5 + 5
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?
NRICH is part of the family of activities in the Millennium Mathematics Project.