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Is it possible to tile an equilateral integer-sided triangle with smaller equilateral integer-sided triangles, with no congruent triangles touching? This has been answered in the negative for the case that no like sizes repeat, I would like to know if it can be done with repeats but like sizes not touching, even at a corner.

Some more info: I've been running a computer search for such a tiling and just finished the side-47 triangle which took 109 hours. But just because a 'reachable' tiling is hard to find, one should not expect no tiling at all. In related "squaring the square" problems tilings have been known to crop up of squares well over 100 on a side.

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  • $\begingroup$ It would help to give a reference for the negative answer in the special case. $\endgroup$ – Matt F. Dec 30 '19 at 7:24
  • $\begingroup$ Karl Scherer's paper proves this: eudml.org/doc/141300 "The impossibility of a tesselation of the plane into equilateral triangles whose sidelengths are mutually different, one of them being minimal." $\endgroup$ – theonetruepath Dec 30 '19 at 8:31
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It is not possible. Tutte shows in "The dissection of equilateral triangles into equilateral triangles" that any triangulation (with equilateral tiles) of a big equilateral triangle will contain two adjacent triangles that share a side, and therefore have equal size.

In order to get past this obstruction he defines triangles of size $\pm k$ where the side length is $k$ and the sign is $+$ if their sides are parallel to the big triangle and $-$ otherwise. With this convention it is possible to give a triangulation of an equilateral triangle with all triangle tiles of different sizes, and the proof is quite similar to that of squaring the square.

Since you are interested in computing the smallest cases of such triangulations you can also look at "An enumeration of equilateral triangle dissections", where the authors list all dissections of triangles of side $\le 20$ satisfying the property from the prevous paragraph. The smallest triangle admitting such a dissection has side $15$.

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  • $\begingroup$ Thanks, that sounds like I can rest a bunch of CPUs or at least put them to more rewarding work. Pity I can't read the paper since it is hidden behind a pay wall. $\endgroup$ – theonetruepath Jan 1 '20 at 11:57
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    $\begingroup$ @theonetruepath It is in Elsevier but I can download it without any problem. The second paper that of Tutte has a pay wall, yes. $\endgroup$ – juan Jan 3 '20 at 18:56

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