The problem
Question: does there exist a single connected prototile that tiles the plane but admits NO periodic tiling (no translation symmetry whatsoever)? Such a shape is called an einstein (German: 'one stone').
The Einstein Problem (Aperiodic Monotiles)
historicReferences
History & significance
Hao Wang's 1961 domino problem birthed aperiodicity: Berger (1966) produced the first aperiodic SET — over 20,000 tiles — proving Wang's decidability guess wrong; Penrose famously reduced to two tiles (1974); Socolar–Taylor achieved one tile in 2010 only by disconnecting it and adding matching rules. Whether a single CONNECTED, RULE-FREE shape sufficed stood open.
The resolution (computer-assisted proof)
Solved March 2023: David Smith, a retired print technician and lifelong tiling hobbyist, found a 13-sided 'hat' polykite; with Joseph Samuel Myers, Craig Kaplan and Chaim Goodman-Strauss they proved it tiles aperiodically (two proofs: combinatorial substitution + computer-assisted case analysis for the hard direction). Weeks later the team eliminated the small caveat of reflected copies with the 'Spectre', a strictly chiral einstein. Hobbyist insight, careful human proof, machine verification — the modern triad, delivering one of recreational geometry's holy grails.