### 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?

# DOTS Division

##### Stage: 4 Challenge Level:

Take any pair of two digit numbers $ab$ and $cd$ where, without loss of generality, $ab> cd$. Form two 4 digit numbers $abcd$ and $cdab$ and calculate: $\frac{abcd^2-cdab^2}{ab^2-cd^2}$ Repeat this with other choices of $ab$ and $cd$. There is a common feature of all the answers. What is it? Why does this occur? Generalise this to $n$ digits for other values of $n$.