## Square Laminas

#### 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,

^{2}2

^{2}= 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 `x`^{2} `y`^{2} = 240.

Writing `x`^{2} `y`^{2} = (`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` 2`t`). 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`) = 2`x`, that is the sum of the two factors is 2`x`. 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?