Where the Heat Goes
Take a flat metal plate whose edges are insulated, so no heat leaks out. Warm it unevenly and wait. The temperatures even out, but one spot stays a little hotter than the rest for longest. In 1974 Jeffrey Rauch asked: does that last hot spot always end up on the edge?
1 · Heat a plate and watch
Pick a shape. Paint heat by dragging across the plate (or press Scatter heat), then press Run. The marker follows the hottest point as the heat spreads.
A simulation on a grid of squares: each step, every square moves toward the average of its neighbours on the plate, and no heat crosses the edge. The colours are rescaled as the plate evens out, so the last contrast stays visible. On every shape here the hot spot settles on the edge. The shapes where it does not are delicate constructions, too fine to draw on this grid — see section 2.
2 · Where it holds, and where it fails
| Shape | Does the hot spot end on the edge? |
|---|---|
| Some symmetric convex shapes | Yes — proved in the 1990s and 2000s |
| Every triangle | Yes — Judge and Mondal, 2020 |
| A plate with two holes | Not always — Burdzy and Werner, 1999 |
| A plate with one hole | Not always — Burdzy, 2005 |
| Some star-shaped solids in three dimensions | Not always — Atar and Burdzy, October 14, 2026 (first draft) |
| Convex shapes in very high dimensions | Not always — de Dios Pont, 2024 |
| Every smooth plate with no holes | Claimed 2026, machine-checked; not reviewed by specialists |
| Every plate with no holes, smooth or not | Claimed two days later, with ChatGPT, from the smooth result (weaker form) |
Holes matter: the counterexamples all use them, or extra dimensions. The question that stayed open was the plain one — a flat plate with no holes — and it was posed as a conjecture by Krzysztof Burdzy in 2005.
3 · The reply, two days later
The 2026 paper assumes the plate’s edge is smooth. On October 8, Krzysztof Burdzy — who had stated the open case in 2005 — posted a short paper removing that assumption, and said how it was made:
“This paper was generated by ChatGPT in response to multiple prompts from the human author. KB has not checked every detail, because the theorem is, to a human, an obvious corollary of the results in [3].”
“Independently and at the same time, Rodrigo Bañuelos used ChatGPT to generate an almost identical theorem and proof.”
He also wrote that the OpenAI model “missed the theorem presented in this note,” and that it was “highly surprising” the release did not cite de Dios Pont’s 2024 high-dimensional counterexample. The release does cite a related 2025 paper by de Dios Pont, Hsu and Taylor.
Fifty-two years, and one week
- 1974Jeffrey Rauch poses the question in a lecture at Tulane University. He never publishes it; it first appears in print in 1985, in a book by Bernhard Kawohl.
- 1999Burdzy and Werner find a plate with two holes where the hot spot stays inside.
- 2005Burdzy does it with one hole, and states the no-hole case as a conjecture.
- 2020Judge and Mondal prove it for every triangle.
- 2024de Dios Pont shows it fails for some convex shapes in high dimensions.
- Oct 6, 2026OpenAI releases a machine-checked proof for every smooth plate with no holes.
- Oct 8, 2026Burdzy and Bañuelos, separately and with ChatGPT, extend it to plates with rough edges.
- Oct 14, 2026Atar and Burdzy post a three-dimensional counterexample, citing the 2026 proof as the planar state of play.