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Square Dissections
By Anonymous on Monday, March 26, 2001 - 08:33 am:

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

I have discovered that all numbers other than 2, 3, and 5 are nice. For instance 1,4,6,7,8,9. 10 are nice, however I cannot seem to draw the configuration showing that 8 is a nice number, any suggestions. Anyway, the question I have is how do I show a formula or rule illustrating the patterns of discovering which numbers are nice. For example, How can I be convinced that 1587 is nice? Please help me out! Thanks !!!

By Kerwin Hui (Kwkh2) on Monday, March 26, 2001 - 08:06 pm:

Anonymous,

I shall not spoil the question by answering fully, but here is some hints.

To show 8 is 'nice', consider a 4x4 square, with a 3x3 square block and 7 1x1 squares.

To show any even number >2 is nice, consider writing the number in the form 2n, and use a similar construction to the one above.


To show any odd number >5 is nice, consider quadrisecting a particular square from the even case.


Kerwin