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A generalized Gerver sofa for angled corridors

A mix of human-written and AI-generated textHuman understanding: some partsmath.MG — Metric Geometrymath.HO — History and Overview

Contributed by Henrik Schou Guttesen

Version 2 / Oct 01, 2026 / CC BY 4.0

Abstract

The moving sofa problem asks for the maximal area of a two-dimensional object capable of being moved through a right-angled corridor of unit width. J. L. Gerver constructed a sofa of area 2.2195…2.2195\dots, which was only recently proven to be optimal by J. Baek. Here, I consider the moving sofa problem in an angled corridor, CφC_{\varphi}, of unit width for turn angles 0<φ≤π20 < \varphi \le \frac{\pi}{2}. I first generalize Hammersley's arguments to devise simple analytical lower and upper area bounds for all 0<φ≤π20 < \varphi \le \frac{\pi}{2}. ChatGPT-6 Astra (Codex) is prompted to derive a generalized Gerver sofa parametrized by φ\varphi. The generalized Gerver sofa area is non-analytic at the unique root φ∗∈(0,π/2)\varphi_*\in(0,\pi/2) of exp⁡(φ∗cot⁡φ∗)=1+2cos⁡φ∗−cos⁡2φ∗\exp({\varphi_*\cot\varphi_*}) = 1+2\cos\varphi_*-\cos^2\varphi_*. For turn angles smaller than φ∗\varphi_* a closed-form expression of the sofa area is obtained. The generalized Gerver sofa is conjectured to be the solution to the moving sofa problem for turn angles 0<φ≤π20 < \varphi \le \frac{\pi}{2}.

Provenance statement

Section III concerning the Generalized Gerver sofa was obtained by prompting ChatGPT-6 Astra Medium in Codex based on an initial note with manually derived Hammersley bounds. The section (III) has been broadly checked and read, but details of the method have not been fully understood nor verified.

Tools used

OpenAI
CodexVersion Astra Medium

References

  1. Baek, Jineon. Optimality of Gerver's Sofa. arXiv e-prints, pp. arXiv:2411.19826. 2024DOI
  2. He, Xingyi. A Gas-Driven Algorithm for Variants of the Moving Sofa Problem. arXiv e-prints, pp. arXiv:2608.11206. 2026DOI
  3. Dan Romik. Differential Equations and Exact Solutions in the Moving Sofa Problem. Experimental Mathematics, vol. 27, no. 3, pp. 316–330. 2018DOI
  4. Yoav Kallus; Dan Romik. Improved upper bounds in the moving sofa problem. Advances in Mathematics, vol. 340, pp. 960-982. 2018DOI
  5. Hanson Bai; John Bai. Modeling the moving sofas in circular hallways using geometric methods. Journal of Emerging Investigators. 2025DOI
  6. Gerver, Joseph L. On moving a sofa around a corner. Geometriae Dedicata, vol. 42, pp. 267–283. 1992DOI
  7. Moser, Leo. Moving Furniture Through a Hallway. SIAM Review, vol. 8, no. 3, pp. 381. 1966DOI
  8. Hammersley, J. M. On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities. Bulletin of the Institute of Mathematics and Its Applications, vol. 4, pp. 66–85. 1968
  9. Wikipedia contributors. Moving sofa problem. 2026. [Online; accessed 9-January-2026]link

Version history

  1. v1Submitted by Henrik Schou GuttesenInitial depositSupersededOct 01, 2026
  2. v2Submitted by Henrik Schou GuttesenSubmitted as a new versionCurrentOct 04, 2026