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Square Dissections
By Anonymous on Monday, March 19, 2001 - 04:09 am:

I am really troubled with this question. If one can answer it and explain how you approached the answer it would be greatly appreciated. Thanks in advance

A number N is called "nice" if a square can be dissected into N non-overlapping squares. What numbers are "nice"?

By James Lingard (Jchl2) on Monday, March 19, 2001 - 09:04 am:

Think about some obvious ways of putting together small squares to make a large square.

You could use n2 squares of the same size, tiled in the obvious way. So all square numbers are nice.

Another way would be to use one big n x n square, and put 2n + 1 small (1 x 1) squares along two of the edges of the big square. This gives you a dissection of a square into 2n + 2 squares. Which numbers can you get in this way?

Now you can get a long way from thinking about this second method and a couple of modifications thereof. Can you see where to go from here? (Hint: think about adding more small (1 x 1) squares to make an even bigger square.)

James.