mathschallenge.net

Problem

A square lamina can be made from 1, 4, 9, 16, 25, ... unit square tiles. But by placing a square hole in the centre of the lamina it is possible to make a "hollow" square lamina from a non-square number such as 32,

   6² 2² = 36 4 = 32

How many different square laminas can be made from 240 unit square tiles?

Solution

If we let x be the dimensions of the lamina and y be the dimensions of the hollow, we are trying to solve x² y² = 240.

    (where t is the thickness)

Writing x² y² = (x + y)(x y) = 240.

We are now looking for two integers that multiply to make 240, but some further analysis is required...

If x is odd and the hollow is centrally placed, y must also be odd (equal thickness either side, that is, y = x 2t). Similarly if x is even then y will be even.

In both cases (x + y) and (x y) will be even, so we are looking for pairs of even factors of 240: They are (2,120), (4,60), (6,40), (8,30), (10,24) and (12,20).

Now (x + y) + (x y) = 2x, that is the sum of the two factors is 2x. Then y can be found by knowing that (x y) is the smaller of the two factors.

	2, 120	x = 61, y = 59
	4, 60	x = 32, y = 28
	6, 40	x = 23, y = 17
	8, 30	x = 19, y = 11
	10, 24	x = 17, y = 7
	12, 20	x = 16, y = 4

Can a square lamina be made from any number of starting tiles?

Is there a connection between the number of tiles and the number of solutions?