Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

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?

Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Diophantine N-tuples

Age 14 to 16Challenge 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.