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