Worked solution: Smith–Myers–Kaplan–Goodman-Strauss's aperiodic monotiles: the hat and the Spectre (2023)
Putting the pieces together: one purely geometric argument (coupling the hat to two incompatible periodic relatives) rules out periodicity with no computer needed, while a second, independent computer-assisted argument (metatiles substituting into an infinite hierarchy of supertiles) both proves a tiling exists at all and independently reconfirms that it must be non-periodic. Two months later, a small modification of the same shape family closed the last loophole by removing the need for mirror images entirely.
A problem open since the 1960s — whether a single tile shape could force non-periodicity all on its own — was answered completely within a single year, and the same toolbox (continuum of shapes, metatile substitution, chirality tricks) immediately became the starting point for a wave of follow-up research into other aperiodic monotiles.
The 'einstein problem' — whether a single tile could be an aperiodic monotile, tiling the plane but only non-periodically — dates back at least to Hao Wang's 1961 work on Wang tiles and the successive shrinking of aperiodic tile sets from Berger's tiles (1966) down to Penrose's celebrated -tile set (1974) and various other -tile sets found later. Smith, Myers, Kaplan and Goodman-Strauss (2023) closed the question for ordinary (reflection-allowed) tilings by proving the hat is aperiodic via two logically independent routes: a classical geometric contradiction (coupling to chevron and comet tilings, no computer needed) and a computer-assisted hierarchical substitution argument (metatiles into supertiles) that also constructs an actual tiling.
Two months later, in the companion paper (Smith, Myers, Kaplan & Goodman-Strauss 2024, arXiv:2305.17743), the same team produced the Spectre family, resolving the sharper 'chiral einstein problem' — a single shape tiling aperiodically using only rotations and translations, with no reflections permitted even in principle — by geometrically deforming the edges of the equilateral member of the same continuum.
Together these results are widely regarded as one of the most significant developments in tiling theory since Penrose's work in the 1970s, and as a striking recent example (alongside the four colour theorem and the sphere-packing proofs of Hales and Viazovska) of a major geometric theorem whose proof genuinely requires computer assistance for at least one of its essential steps.