### 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?

# Kangaroo Subtraction

##### Stage: 3 and 4 Short Challenge Level:

The difference between '$KAN$' and '$GAR$' is less than $100$ and since $K\neq G$ we must have $K=G+1$.

Next we must have $N < R$ else the difference between '$KAN$' and '$GAR$' would be at least $100$.

Let $R=N+x$ where $1< x< 9$. Then $OO=100-x$ and hence $O=9$ and $R=N+1$. Also we must have $K\leq 8$

We want the largest value for $KAN$ so we try $K=8$. This forces $G=7$, hence must have $A\leq 6$. Set $A=6$, this forces $R\leq 5$ and hence $N\leq 4$ since $R=N+1$. So $864$ is the largest possible value for $KAN$, and we have $864-765=99$.
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This problem is taken from the UKMT Mathematical Challenges.
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