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

Stage: 5 Challenge Level:

This is a problem by Lewis Carroll and it revolves around some of the many interesting properties of sums of squares of integers. Is it always true that if you double the sum of two squares you get the sum of two squares? If so can you prove it? Here are some examples.

$2(5^2 + 3^2) = 2(25 + 9) = 68 = 64 + 4 = 8^2 + 2^2$
$2(7^2 + 4^2) = 2(49 + 16) = 130 = 121 + 9 = 11^2 + 3^2$

NOTES AND BACKGROUND

In his book Pillow-Problems Lewis Carroll extends this idea with a further problem. Prove that 3 times the sum of three squares is also the sum of 4 squares.

For further problems like this see Lewis Carroll's Games and Puzzles compiled by Edward Wakeling published by Dover Books ISBN 0-486-26922-1.