Greg EganIn 1907, Henry Dudeney showed that you could dissect a square into 4 pieces that can be rearranged to form an equilateral triangle.
[Edit: In 1902, in fact. “1907” seems to be a typo in the Demaine et al paper.]
In 2024, Erik D. Demaine, Tonan Kamata and Ryuhei Uehara finally proved that this cannot be done with *less than* 4 pieces!
https://arxiv.org/abs/2412.03865
(H/T @robinhouston )
Dudeney's Dissection is Optimal
In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney demonstrated a beautiful four-piece solution, which today remains perhaps the most famous example of dissection. In this paper (over a century later), we finally solve Dudeney's puzzle, by proving that the equilateral triangle and square have no common dissection with three or fewer polygonal pieces. We reduce the problem to the analysis of discrete graph structures representing the correspondence between the edges and the vertices of the pieces forming each polygon.
Gerard Westendorp
gormster
Matthew Dockrey
Steve Root
thomas yager-madden
robinhouston
Simon Zerafa
Nixie