### Absurdity Again

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

### 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?

# Sums of Squares

### Why do this problem?

It provides an opportunity for learners to experiment with numerical examples, to observe a pattern and make a conjecture, to try to explain their own conjecture and to formulate a proof of the conjecture which involves only simple algebra.

### Possible approach

Suggest that the learners make up their own examples similar to the ones given and look for a pattern.

### Key questions

Can you see any relationship between the numbers in the examples?

Is there any pattern in the relationship between the numbers?

If you spot a pattern can you prove it always occurs?

### Possible extensions

The problen page suggests an extension to three times the sum of three squares.

The pattern can be generalised to four times the sum of four squares and so on (see the solution).

### Possible support

Suggest that learners find out which of the numbers from 1 to 10 can be written as the sum of 2 squares.

eg $1 = 1^2 + 0^2$