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Mind School·Wonder·Honor-system

The Moving Sofa Problem

What is the largest shape that can navigate an L-shaped hallway? Still unproven.

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Characterization

The Moving Sofa Problem asks: what is the largest area of a rigid two-dimensional shape that can be moved around an L-shaped corridor of unit width? The problem was posed by Leo Moser in 1966, and it is disarmingly physical — one can cut shapes from cardboard and test them against an L-shaped template. The simplest solution is a unit square (area 1), but a semicircle of radius 1 achieves π/2 ≈ 1.571. In 1968, John Hammersley proposed a shape resembling a telephone handset with area π/2 + 2/π ≈ 2.2074. In 1992, Joseph Gerver refined this with a more complex shape composed of eighteen curves, achieving an area of approximately 2.2195. Dan Romik's 2018 work introduced the ambidextrous moving sofa problem — navigating two opposite L-turns — and found the optimal shape for that variant. But for the original single-turn problem, no one has proved that Gerver's sofa is optimal. The best known upper bound, due to Kallus and Romik (2018), is approximately 2.37. The gap between 2.2195 and 2.37 remains open. The Academy hosts the Moving Sofa in the Mind School because it is the rare mathematical wonder that is simultaneously deep and physical: a problem a child can explore with scissors and paper, whose resolution has eluded professional mathematicians for sixty years.

Lineage

Leo Moser, "Moving Furniture Through a Hallway" (problem statement), SIAM Review 8(3), 1966. John M. Hammersley, "On the Enfeeblement of Mathematical Skills," Bulletin of the IMA 4, 1968. Joseph L. Gerver, "On Moving a Sofa Around a Corner," Geometriae Dedicata 42(3), 1992. Dan Romik, "Differential Equations and Exact Solutions in the Moving Sofa Problem," Experimental Mathematics 27(3), 2018. Yoav Kallus and Dan Romik, "Improved Upper Bounds in the Moving Sofa Problem," Advances in Mathematics 340, 2018. Philip Gibbs proposed a numerical approach via computational optimisation (2014).

Quests

Three quests — one for each archetype. Choose the one that fits your way of taking up the discipline.

  • Design and draw (or compute) a connected planar shape that can navigate a unit-width L-shaped hallway, and calculate or estimate its area. Compare your shape's area to Hammersley's constant (π/2 + 2/π ≈ 2.2074) and to Gerver's shape (≈ 2.2195). You may use any construction technique — geometric, parametric, or computational optimisation. Record your shape, its area, and what the construction process taught you about the constraints of the problem.

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  • Build a physical scale model of the L-shaped hallway (cardboard, paper, or any available material) and experiment with rigid cut-out shapes to navigate the corner. Begin with a semicircle; then try Hammersley's telephone-handset shape; then improvise. Record the largest shape (by estimated area) that you successfully manoeuvred around the corner, and note what the physical manipulation revealed about the problem that a diagram alone could not.

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  • Explain the moving sofa problem to a reader who knows basic geometry. State the problem; describe the progression from the semicircle through Hammersley's shape to Gerver's 1992 construction. Cite Moser (1966), Hammersley (1968), and Gerver (1992). Mention Romik's 2018 ambidextrous variant. Explain why Gerver's shape is believed to be optimal but the proof remains elusive.

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