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### Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

### Medallions

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized one has radius 9 cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

# Our Ages

##### Stage: 4 Challenge Level:

Many thanks to Robert Simons for this question:

"I am exactly $n$ times my daughter's age. In $m$ years I shall be exactly $(n-1)$ times her age. In $m^2$ years I shall be exactly $(n-2)$ times her age. After that I shall never again be an exact multiple of her age. Ages, $n$ and $m$ are all whole numbers. How old am I?

Now suppose there is some wishful thinking in the above assertion and I have to admit to being older, and indeed that I will be an exact multiple of her age in $m^3$ years. How old does this make me?"

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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