### Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

### Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

# Triangles Within Triangles

##### Age 14 to 16Challenge Level

Well done Tom, from Finham Park School, for clear use of notation :

Whenever you add 3 triangles ( as in $T_2$ ) together with a triangle one size smaller ( as in $T_1$ ), a new triangle is formed ( $T_4$ ) , twice the height of the triangle which was used three times ( $T_2$ ) .

The smaller triangle can be called $T_n$ , while the 3 triangles one size up can be called $T_{n+1}$.

One of the $T_{n+1 }$ joins with the $T_n$ to form a square of side length $n+1$ .

The two remaining $T_{n+1}$ fit to that square producing a large triangle that has a height twice that of $T_{n+1}$ ,

So the sum of all four triangles is the triangle $T_{2(n+1)}$

So $T_n$ + $3T_{n+1} =T_{2(n+1)}$ or, if you prefer, $T_{2n+2}$