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?

Diophantine N-tuples

Stage: 4 Challenge Level:

Take any whole number $q$. Calculate $q^2 - 1$. Factorize $q^2 - 1$ to give two factors $a$ and $b$ (not necessarily $q+1$ and $q-1$). Put $c = a + b + 2q$ . Then you will find that $ab + 1$ , $bc + 1$ and $ca + 1$ are all perfect squares.

Prove that this method always gives three perfect squares.

The numbers $a_1, a_2, ... a_n$ are called a Diophantine n-tuple if $a_ra_s + 1$ is a perfect square whenever $r \neq s$ . The whole subject started with Diophantus of Alexandria who found that the rational numbers $${1 \over 16},\ {33\over 16},\ {68\over 16},\ {105\over 16}$$
have this property. (You should check this for yourself). Fermat was the first person to find a Diophantine 4-tuple with whole numbers, namely $1$, $3$, $8$ and $120$. Even now no Diophantine 5-tuple with whole numbers is known.