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?

Relative Powers

Stage: 4 Short Challenge Level:

Let the number be $x$. Then $x^3 = 9x$, so either $x=0$ or $x^2 = 9$, giving $x = 3$ or $x = -3$. Of these three values of $x$, only $-3$ is also a solution of the equation $x^2 = x + 12$.

This problem is taken from the UKMT Mathematical Challenges.