### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

### Plum Tree

Label this plum tree graph to make it totally magic!

# Pair Squares

##### Stage: 5 Challenge Level:

The numbers $2$, $34$ and $47$ are such that the sum of any pair of these numbers is a perfect square. Find a method for choosing three square numbers and from them finding a corresponding set of three integers with this property and give some examples.

The integers $-208$, $224$, $352$ and $737$ also have the property that the sum of any pair of these numbers is a perfect square. Find other sets of four integers with this property.