The moving sofa math problem: still unsolved 50 years later

Ever try to move a sofa down a hallway that has a corner? The underlying math behind it inspired a math problem that's been a puzzler since 1966. Gerver's Sofa above shows the parameters: a U-shaped sofa moving around a 90-degree corner in an even-width hallway. Gerver's got the record so far, and it is likely the optimal sofa.

Per the American Mathematical Society:

This question was first published by Leo Moser in 1966. The animation above, made by Claudio Rocchini, shows one attempt to solve this problem. It’s called the Hammersley sofa, since it was discovered by John Michael Hammersley in 1968... But it’s not the best known solution! In 1992, Joseph Gerver found a shape of area [...] that fits around the bend in the hallway. Basically Gerver rounded off some of the corners of Hammersley’s sofa. The resulting shape has a boundary consisting of 3 straight line segments and 15 curved segments.

AMS has a good detailed illustration of the Gerver Sofa, too.

Moving Sofa Problem (Wolfram)

Gif: Hammersley Sofa via Wikimedia Commons

Notable Replies

  1. This is not the optimal sofa. The optimal sofa is optimized for sitting on. It is comfy, yet durable. It is supposed to be difficult to move; that's how you know it's a sofa.

  2. The rest of us living in the real world know that the best way to move a sofa around a corner is to stand it on its end. For a real challenge, move a sofa up a flight of stairs in a hallway with a 90 degree corner.

    For extra credit, move a sleeper sofa up the same hallway stairs.

  3. Decades ago, I had the greatest couch a frequent mover could ask for...The arms bolted on.

    Flip the couch over.

    Peel back the velcro attached fabric.

    Remove the three bolts from each arm. (Note - Standard bolts that you can buy from any hardware store. Nothing exotic)

    Move the arms. Move the body. Re-assemble.

    Without the arms, it would go through a normal doorway vertically. No doorway, nor flight of stairs, nor wonky hallway would impede that couch from getting to its final spot.

    Sadly, its fabric was light colored and after a ill place glass of was never the same.

  4. Old story allegedly from (I think) the Royal Albert Hall in London: before committing to buying a new concert grand piano, they needed to be certain it would fit down a corridor with a bend in it. So they asked their in-house carpenter to make a full-size model for them to test. He did. He couldn't get it out of the workshop.

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