Five problems that were open for decades and fell between 2024 and 2026, each one a shape you can pick up and turn over — 14 interactive demonstrations. A point set with more unit distances than Erdős thought possible, found by an AI model. A 90-vertex solid that cannot pass through a copy of itself. The largest sofa that turns a corner. A needle that turns around in no area at all, and the 2025 proof that in three dimensions the set it sweeps is never thin. And a chair-shaped tile that claims to fill space without ever repeating. Each lesson is honest about what is proved, what is computer-checked, and what is still a preprint.
See also: Aperiodic Tilings for the hat, the spectre, and the two-dimensional story the chair continues, Sphere Packing for kissing numbers and another place machines are now finding records, Measure Theory for the measure-zero sets behind Kakeya, and Number Theory for the sums of two squares that power the unit distance grid.
How many pairs of n points can be exactly 1 apart? Erdős bet on the square grid for eighty years, and in 2026 an AI model found something better
A cube can pass through a hole bored in an identical cube. In 2025 two mathematicians built a 90-vertex solid that cannot
The largest shape that turns a corner in a unit-wide hallway: Gerver guessed it in 1992, and Jineon Baek proved him right in 2024
A needle can turn around in as little area as you like, yet in 3D the set it sweeps can never be thin — proved by Hong Wang and Joshua Zahl in 2025
Seven cubes in the shape of a chair, a pyramid on every face, and a 2026 claim of the first single tile that forces space-filling without repetition