Absurdity Again

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

Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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 $1 = 1^2 + 0^2$