### 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?"