moving sofa problem

About 60 years old · traced to 1966

Let

H=\{(x,y)\in\mathbb{R}^2:\ x\le 1,\ 0\le y\le 1\},\qquad V=\{(x,y)\in\mathbb{R}^2:\ 0\le x\le 1,\ y\le 1\},

and let $L=H\cup V$ be the L-shaped corridor formed by these two legs of unit width meeting at a right angle.

Call a compact connected set $S\subset\mathbb{R}^2$ a moving sofa if there is a continuous family $(g_t)_{t\in[0,1]}$ of orientation-preserving isometries of $\mathbb{R}^2$ (that is, $t\mapsto g_t$ is continuous in the natural topology on the group of rotations and translations) such that

g_t(S)\subset L \quad\text{for all } t\in[0,1],\qquad g_0(S)\subset H,\qquad g_1(S)\subset V .

Thus $S$ is rigidly maneuvered from the horizontal leg to the vertical leg without ever leaving $L$ .

Writing $|\cdot|$ for two-dimensional Lebesgue measure, define the sofa constant

\mu=\sup\{\,|S|\ :\ S\subset\mathbb{R}^2 \text{ is a moving sofa}\,\}.

Determine the exact value of $\mu$ , and determine every moving sofa $S$ with $|S|=\mu$ , up to isometry.

References

Primary source

Wikipedia

Additional references

Wikipedia, Moving sofa problem, the article this problem comes from.

geometry unsolved-problem

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A proof claim, later reported as a 2026 solution, identifies Gerver’s shape as the largest, but it has not been independently verified and does not classify all largest shapes.

Leo Moser posed the problem in 1966: determine the largest rigid shape that can negotiate a unit-width right-angle corridor. The leading candidate is Gerver’s sofa, whose area is approximately $2.219531668871$ .

Known results

Hammersley, 1968: construction of area $\frac{\pi}{2}+\frac{2}{\pi}\approx 2.2074$ .
Gerver, 1992: an $18$ -section construction of area approximately $2.2195$ .
Kallus–Romik, 2018: computer-assisted upper bound $\mu\le 2.37$ .

January 2026 reports of a solution

Baek’s November 2024, $119$ -page preprint claims that Gerver’s sofa attains the maximum; January 2026 news reports describe this as a solution. The supplied sources provide no independent verification, and none states a uniqueness theorem for all extremizers.

Current status (as of September 2026): Gerver’s area is claimed, but not independently verified, to equal $\mu$ ; classification of every equality-case moving sofa up to isometry remains unreported.

Sources

No solutions have been posted yet.