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Absurdity Again

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

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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?

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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

Age 16 to 18 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.

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


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.