### Plum Tree

Label this plum tree graph to make it totally magic!

### Magic W

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

### 2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

# Triangles Within Triangles

##### Age 14 to 16 Challenge Level:

This diagram shows how the first triangular number can be added to 3 copies of the second triangular number to make the fourth triangular number:

That is: $$T_1 + 3 \times T_2 = T_4$$ Here is a diagram showing how the second and third triangular numbers can be combined to make the sixth triangular number:

$$T_2 + 3 \times T_3 = T_6$$ Can you generalise this rule?

Can you find a rule in terms of $T_n$ and $T_{n+1}$?