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We have a square 2n*2n, where n belongs to N. The main problem is to find how many different equal area connected figures could be produced by cuttings without self-crossings. The orientability of the figures is not in the question. For n=1 it is evidently 1, for n=2 it is 19 (I have written a computer program of brute force cutting), for n=3 the number is more than 6643 (this number I have got from another program based on random cutting). It's very interesting to find an exact mathematical formula for this number. Another question is the length of the cut. For n=2 all cuts from 4 to 10 except 5 is allowed, for n=3 -- all cuts from 6 to 26 except 7 are also allowed. For n>=4 it is possible to use closed cuts, that form a boundary between inner and outer areas of the square. I want to know about maximal lengths of the cuts and banned lengths of the cuts. Another problem is based on the idea of covering the figures by rectangles (with intersections and without intersections), question is about the max number of the rectangles.

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  • $\begingroup$ I take it the picture you have in mind is of the big square being demarcated as a grid of $1\times1$ squares, and you're only allowed to cut along the "dotted lines". $\endgroup$ Apr 3, 2021 at 0:39
  • $\begingroup$ Would this problem have originated from a textbook or a research article? As @Gerry implicitly suggests, a picture could be helpful here! $\endgroup$ Apr 3, 2021 at 7:01

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