### 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?

### Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

# Bang's Theorem

##### Stage: 4 Challenge Level:

Thank you Hyeyoun Chung, age 14, St Paul's Girls School, London, for your solution. In a tetrahedron each face, a triangle, shares a different side with every other triangle. Therefore every one of its sides has to be equal in length to one side of each of the other triangles and we can label the side lengths $a, b, c, p, q$ and $r$ as in the diagram. All the faces have the same perimeter (say $k$) so
\begin{eqnarray} a + b + c &= k \quad (1) \\ a + r + q &= k \quad (2) \\ b + p + r &= k \quad (3) \\ c + p + q &= k \quad (4). \end{eqnarray}
Adding the last three of these equations gives $$a + b + c + 2(p + q + r) = 3k$$ so $$p + q + r = k \quad (5).$$ From equations (2) and (5) we get $a = p$, from equations (3) and (5) we get $b = q$ and from equations (4) and (5) we get $c = r$. As we can see from the diagram all the faces of the tetrahedron have sides of lengths $a, b$ and $c$ so all four faces are congruent.