In 2005 Conway and Soifer published the famous shortest ever paper, asking whether an equilateral triangle of sidelength $n+\varepsilon$ can be covered by $n^2+1$ unit equilateral triangles and proving that it can be covered by $n^2+2$ unit equilateral triangles. The story is mentioned in several places but there is never an update on the current state of affairs for this problem. I found some generalizing results like here concerning higher dimensions or other shapes and but nothing on the problem itself. Does anyone have an up-to-date information?
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2$\begingroup$ You could try writing to Soifer directly. His email address can be found on the Geombinatorics website. But there's not much room left for partial results on this problem, so most likely the only "update" would be a full solution, which would probably be easy to Google if it existed. Incidentally, the claim to "shortest ever paper" can be disputed depending on your definitions. $\endgroup$– Timothy ChowFeb 22, 2023 at 2:45
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$\begingroup$ Writing to Soifer directly is a nice idea. And yes, the only real partial result i had in mind would be a lower bound on $n$ to make the triangles coverable or an elegant solution for the case of covering a hexagon with seven triangles (there seems to be a brute force kind of solution for this case mathoverflow.net/a/159902/127781). $\endgroup$– TakirionFeb 22, 2023 at 9:10
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$\begingroup$ @Tim yes, it's working now! $\endgroup$– David Roberts ♦Feb 22, 2023 at 20:30
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1 Answer
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Jineon Baek and Seewoo Lee recently posted to the arXiv a paper claiming to prove the conjecture in the case that every small triangle in the cover has edges parallel to the large triangle. (The small triangles can either point "up" or "down".) Note that the triangles in the $n^2+2$ triangle solution have this property.
I've read the proof; it looks good to me.
answered Nov 30, 2023 at 19:12