As far as my investigation has gone, I can see that people have worked on the optimal way to pack incrementally larger squares into rectangles (page 2, .pdf), as well as the optimal ways to pack equal squares into a square. I have not, however, found a reasonable algorithm or method for packing incrementally larger (or smaller, depending on your point of view) squares into a larger square area. It feels similar to the first link, but with additional constraints on the rectangle container.

As an example of what I am asking about, I manually created a packing of a group of squares (1x1, 2x2, 3x3,...8x8) in a "square" container. Is this something that has been researched at all, or is it largely just a meaningless corner case (ha!) of packing squares into a rectangle?

Since I have just joined this stack, I cannot embed my image, but it can be found here: http://i.stack.imgur.com/sD7Jw.png



You can buy it at Kadon Enterprises

I posted various solutions on 2 March 2003. The most recent imporovement I know about was Shigeyoshi Kamakura's packing of 23 squares into 66x66, in 2004.

Shigeyoshi Kamakura's pack of 23 squares

More solutions are posted here. Many of the solutions were found by Robert Reid. There is also the Oct 99 Math Magic.

  • $\begingroup$ Oh, this is amazing! Thank you! $\endgroup$ – Fritz Sep 25 '15 at 18:57

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