The Moving Sofa Problem: What's the Largest Sofa That Can Turn a Corner?

The Genesis of the Moving Sofa Problem

• 💡 The "moving sofa problem" originated in 1966 with mathematician Leo Moser, initially posing it as a generic shape in 2D, not specifically a sofa.
• 🎯 The core mathematical interest lies in finding the shape with the maximum possible area that can navigate a 90-degree corner.

Early Attempts and Shapes

• 🚀 John Hammersley proposed a shape resembling a retro telephone handset, achieving an area of approximately 2.2074 square units.
• 📈 Hammersley also hypothesized an upper bound for the solution at about 2.8284 square units, though without a corresponding shape.
• 🛋️ By the early 1970s, the problem was commonly framed in terms of sofas, with Hammersley's shape serving as the benchmark.

Gerver's Sofa and Baek's Proof

• 🧩 Joseph Gerver developed a more complex sofa shape in 1992, composed of 18 lines, arcs, and curves, with an area of about 2.2195 square units.
• ⏳ For over three decades, mathematicians believed Gerver's sofa was the solution but lacked a mathematical proof.
• ✍️ In 2024, Jineon Baek published a paper on arXiv providing a proof, using a novel function (Q) to establish an upper limit for any sofa's area.
• 🧠 Baek's proof, developed with pencil and paper, confirmed that Gerver's shape indeed maximizes the area for a 2D corner.

Beyond the 2D Corner: Ambi-Turning Sofas

• 🔄 The problem extends to navigating two corners, leading to the concept of an "ambi-turning sofa."
• 🚗 Shapes like John Conway's "car" have been explored for this more complex scenario, with ongoing research to find a proven solution.
• 🧐 While Zhipeng Deng posted a paper in 2024 attempting to solve this, it has not met mathematical standards for proof.