### 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:

We have received a very clearly explained solution to this, but unfortunately whoever sent it did not include their name. If it was you, let us know!

Let $ab=x$ and $cd=y$ (where $ab$ means $a$ as the tens digit, and $b$ as the ones digit, not $a$ times $b$).

$\frac{abcd^2-cdab^2}{ab^2-cd^2} = \frac{(100x+y)^2 - (100y+x)^2}{x^2-y^2}$ $= \frac{(10000x^2+200xy+y^2) - (10000y^2+200xy+x^2)}{x^2-y^2}$ $= \frac{9999x^2 - 9999y^2}{x^2-y^2}$ $= \frac{9999(x^2-y^2)}{x^2-y^2}$ $= 9999$

(since $x> y$ we are not dividing by zero)