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?

AMGM

Age 14 to 16 Challenge Level:

Choose any two numbers. Call them $a$ and $b$ ($b < a$). Work out the arithmetic mean $(a + b)/2$ and the geometric mean $\sqrt{ab}$. Which is bigger? Repeat for other pairs of numbers. What do you notice?

In the diagram PQRS is a rectangle measuring $a$ units by $b$ units. The green rectangle measures $(a + b)/2$ by $b$ and the orange and blue rectangles both measure $(a - b)/2$ by $b$. By considering the areas of the rectangles explain why this diagram shows that $$ab < ({{a + b}\over 2})^2.$$ What does this tell us about the arithmetic mean and the geometric mean of two numbers?