The problem
Classify all convex pentagons that admit an edge-to-edge-free tiling of the Euclidean plane. (Triangles and quadrilaterals all tile; hexagon families were classified by Reinhardt; the pentagon case resisted.)
Classification of Convex Pentagon Tilings
historicReferences
History & significance
Reinhardt found five families in 1918; Kershner added three (1968); Richard James IV one (1975); Marjorie Rice, an amateur following a Scientific American column, discovered two more in 1976-77 via her own notation; Stein found a fourteenth (1985); Mann, McLoud and Von Derau a fifteenth (2015) using computer search. Repeatedly declared complete, repeatedly reopened.
The resolution (computer-assisted proof)
Closed August 2017 by Michaël Rao (CNRS, Lyon): an exhaustive computer-assisted analysis reduced all possibilities to 371 families of angle/equation systems and eliminated each, proving the fifteen families complete. The result settled a century of cat-and-mouse and, en passant, completed the answer to the tiling component of Hilbert's 18th problem's neighbourhood. Human pattern-hunting supplied the families; the computer supplied the certainty.