An S-tileset is a collection of n oriented tiles, where no two tiles have the same size, each tile is one unit thick, and its non-zero-integer length and width add up to n+1. (So, an S-tileset has n tiles, of sizes 1×n, 2×n−1, ...n×1.).

Conjecture: No two disjoint subsets of any S-tileset can be tiled to form two areas of identical size and shape.

Can you either prove the conjecture or find a counter-example.

  • $\begingroup$ Another conjecture: A subset (of an S-tileset) having more than one tile in it cannot tile a rectangle $\endgroup$ Aug 17, 2014 at 23:18
  • 2
    $\begingroup$ I know this is a math puzzle, but the way it's written feels like it should go on Math.SE under the "puzzle" tag rather than actually here. $\endgroup$
    – user88
    Aug 17, 2014 at 23:49
  • $\begingroup$ Do you allow rotation? If so, the answer is trivial - simply use the 1xn tile normally, and the nx1 tile rotated 90 degrees. They are disjoint, yet form two areas of identical size and shape. $\endgroup$
    – Trenin
    Oct 3, 2014 at 18:41
  • $\begingroup$ @Trenin No, the tiles are oriented, so rotation is specifically disallowed. $\endgroup$
    – Penguino
    Oct 5, 2014 at 20:30
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    $\begingroup$ @jwpat7 - a counterexample to your conjecture: imgur.com/k11J8zm. This is actually parametric in the n=14 set, there's a number of rectangles with the same layout. I suspect this might help find a counterexample to the posted conjecture. $\endgroup$ Oct 8, 2014 at 1:19

1 Answer 1


Here's a counter-example to the conjecture, for N=860

(you may need to open the image in a new tab to read the sizes in the image) A counter-example

The set of rectangles used:

Shape 1

  • 602x259
  • 210x651
  • 28x833
  • 133x728
  • 469x392
  • 679x182
  • 707x154
  • 840x21

Shape 2

  • 196x665
  • 70x791
  • 161x700
  • 413x448
  • 609x252
  • 770x91
  • $\begingroup$ That is an excellent result. Did you solve it using Diophantine equations? $\endgroup$
    – Penguino
    Oct 20, 2014 at 22:41
  • $\begingroup$ Sort of. I set up systems of linear equations, and used sympy to solve or simplify them as much as possible. This ruled out a lot of candidates on its own (two sides would have the same algebraic length, for example). The system describing these two shapes ends up having only 1 free variable, so it's just a matter of picking the first one that gives integers for all sides. $\endgroup$ Oct 21, 2014 at 0:29

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