How many different ways are there to dissect a 6x6 square into 1x3 rectangles?

Two dissections of a square are equal if there is an isometric mapping of the square into itself, which maps the first dissection into the second one, but in this case no dissected squares can be equal.

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    $\begingroup$ One could write a computer program to enumerate these, e.g. using dancing links if naive approaches were too slow. What do you mean by “no dissected squares” can be equal? If you dissect into a $6×2$ grid of horizontal rectangles, that would be equal to a version rotated by $180°$, so they can be equal. Do you mean to say that we must make sure to only count these once? $\endgroup$
    – MvG
    Feb 28, 2016 at 19:17
  • $\begingroup$ Yes, we must make it sure, as there cannot be any isometric mapping that maps a given dissection into another one. $\endgroup$
    – user290325
    Feb 28, 2016 at 21:47
  • $\begingroup$ @Laszlo Maybe we can add a bonus section to this question: Produce a proof that no tiling can exist which has no 3x3 sub-square? Seventh of my diagram having only one. $\endgroup$ Jan 9, 2018 at 3:24

1 Answer 1


Sorry about the delay, I just spotted this looking for something else. About two minutes to set up my program which finds all 11 tilings in 0.00 seconds (ie too small to measure without changing my method of displaying the time taken). It's a really small job though, only 405 'piece placements' total. The trickiest part is in rejecting rotations and reflections of which there were 53. I use an algorithm that makes all rotations and reflections of each solution as it is found, then throws away any that aren't 'the smallest' by some arbitrary measure.

All tilings of 6x6 with 1x3


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