A San Diego housewife with a high-school education read about an unsolved tiling problem in her son’s magazine — and, in her kitchen, using a notation she invented herself, found FOUR new kinds of pentagon that tile the plane, ones the professionals had missed. Marjorie Rice, amateur. Slide and watch her pentagons lock together with no gaps.
A convex pentagon’s interior angles always sum to 540°. It can TILE the plane only if its angles and edges can be arranged so that at every meeting point they sum to exactly 360° with no gaps or overlaps — a delicate combinatorial condition. Rice found new pentagon TYPES satisfying it. The instrument draws such a tiling. A fail-loud self-check throws unless the pentagon’s interior angles sum to 540° and the angles meeting at each shared vertex sum to 360° — the exact gap-free requirement.
The full classification of convex pentagon tilings (exactly 15 types) was only PROVEN complete in 2017 (Rao) — Rice found 4 of them by hand. The 540° interior sum and the 360°-per-vertex tiling condition are exact.