The Moving Sofa Problem

What’s going on

A corridor is 1 unit wide and turns a right angle. You want to carry a sofa round the corner. You may slide it and turn it on the floor, but not tip it up or bend it. What is the largest area such a sofa can have? Here a “sofa” is any flat shape at all.

Pick a sofa in the scene above and drag across the picture (or press Play) to carry it round.

• The square. A 1 × 1 square slides into the corner, then slides up the other arm. It never turns. Area: 1.
• The semicircle. A half-disc of radius 1, flat side along the inner wall, can spin about the red inner corner like a door on a hinge. Area: π/2 ≈ 1.571. A bigger half-disc cannot even enter the corridor: from its flat side to the top of its arc it measures its radius, more than 1.
• Hammersley’s sofa. In 1968 John Hammersley cut the half-disc into two quarter-discs, pushed them apart and put a block between them. A plain block would jam on the corner, so he bit a half-disc of radius r out of it. As the sofa turns, the corner stays on the edge of the bite, and the centres of the two quarter-discs slide along the inner walls. Area: π/2 + 2r − πr²/2.
• Gerver’s sofa. In 1992 Joseph Gerver bent and smoothed Hammersley’s outline. His sofa slides and turns at the same time, and its boundary is made of eighteen different curves. Area ≈ 2.2195.

The best bite

Choose Hammersley and move the r slider. The two quarter-discs always give π/2. The block adds 2r, because it is 2r long and 1 high. The bite takes away half a disc of radius r, that is πr²/2. A bigger bite makes room for a longer block, but eats more. The area is largest when r = 2/π ≈ 0.64, and then it is π/2 + 2/π ≈ 2.2074. The little graph shows the trade-off: the curve is a parabola, highest at 2/π.

Half a percent

For 24 years many people believed Hammersley’s sofa was the best. Gerver’s sofa is only about half a percent bigger: 2.2195 instead of 2.2074. Switch to Gerver in the scene: the dashed outline behind it is Hammersley’s sofa, making the same trip. The difference is small, but it shows that the “obvious” shape was not the best one.

Is Gerver’s sofa the champion? For a long time nobody knew. In late 2024 Jineon Baek posted a proof, more than a hundred pages long, that no sofa can be larger. So the answer to a puzzle about furniture is a shape whose area is about 2.2195, with an outline stitched together from eighteen pieces of curves.

What the scene computes

For each motion, the scene computes the largest shape that stays inside the corridor all the way round; that is the sofa you see. For the square, the semicircle and Hammersley’s sofa this gives exactly the known shapes. Gerver’s exact formulas are complicated, so ours is a numerical reconstruction: the best motion we found of a fairly simple kind. Its area is 2.2193, within 0.01% of Gerver’s. It matches his sofa to the width of a line, but it is an approximation, not his exact curves.

Try it yourself

1. The longest pole

Instead of a sofa, carry a very thin pole round the corner of the same corridor, keeping it flat on the floor. How long can the pole be?

Answer

2√2 ≈ 2.83. The hardest moment is when the pole is at 45° and touches the inner corner. Then each half of it crosses one arm of the corridor diagonally, from the inner corner to an outer wall: a diagonal of a unit square, √2 long. Two halves give 2√2. Any longer pole gets stuck at that moment.

2. Check Hammersley's formula

Take r = 1/2. What is the area of Hammersley’s sofa? Is it bigger or smaller than with the best r = 2/π?

Answer

π/2 + 2 · ½ − π · ¼ / 2 = π/2 + 1 − π/8 ≈ 1.571 + 1 − 0.393 = 2.178. That is smaller than 2.2074. Set r to 0.50 in the scene to see the same number.

3. A sofa with no bite

What happens to Hammersley’s sofa when r = 0, and when r = 1? Which of the two is bigger?

Answer

With r = 0 there is no block and no bite: the two quarter-discs close up into the semicircle, area π/2 ≈ 1.571. With r = 1 the block is 2 long and the bite is a whole half-disc of radius 1, so the area is π/2 + 2 − π/2 = 2. Both are worse than r = 2/π, which sits in between.

Go deeper

• The problem was posed by Leo Moser in 1966. Hammersley also proved that no sofa can have an area larger than 2√2 ≈ 2.83; in 2018 Yoav Kallus and Dan Romik lowered this upper limit to about 2.37.
• The same question for a corridor in which the sofa must be able to turn both left and right has a different champion, found by Romik in 2018.