### 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?

# Loopy

##### Stage: 4 Challenge Level:

Sunil sent us his work on this problem:

When I tried this out, no matter what numbers I started with, I always seemed to find that the $6$th term was the same as the $1$st term, and the $7$th the same as the $2$nd, so it kept repeating. I used a spreadsheet to help me do the calculations! Then I used algebra to try to explain why it worked.

The first term is $a_1$.

The second term is $a_2$.

The third term is $\frac{1+a_2}{a_1}$.

The fourth term is $\frac{1+a_1+a_2}{a_1 a_2}$.

he fifth term is $\left(\frac{1+a_1+a_2+a_1 a_2}{a_1 a_2}\right)\times\left(\frac {a_1}{1+a_2}\right)=\left(\frac{(1+a_1)(1+a_2)}{a_1 a_2}\right)\times\left( \frac{a_1}{1+a_2}\right)=\frac{1+a_1}{a_2}$.

The sixth term is $\left(\frac{1+a_1+a_2}{a_2}\right)\times\left(\frac{a_1 a_2} {1+a_1+a_2}\right)=a_1$.

The seventh term is $\left(a_1+1\right)\times\left(\frac{a_2}{1+a_1}\right)=a_2$.

This explains why the pattern always works, no matter what you start with! (As long as the bottom is never $0$, so we can't have $a_1=0$, or $a_2=0$, or $a_1=-1$, or $a_2=-1$, or $a_1+a_2=-1$.)