The moving sofa problem

The challenge is to find the sofa with the largest possible area that can be moved around a right-angled corner in a one-meter-wide corridor.

The shape shown below consists of two quarter-disks of radius 1 and a rectangle of length 4/π with a semicircular cutout along its entire length. It can pass through the corridor and has an area of $\frac{\pi }{2}+\frac{2}{\pi }\backslash frac\{\backslash pi\}\{2\}+\backslash frac\{2\}\{\backslash pi\}$, or approximately 2.207 m². It was proposed in 1968 by the British mathematician John Michael Hammersley (1920–2004).

The result has since been improved several times. In 2017, the American mathematicians Yoav Kallus and Dan Romik also established an upper bound for the maximum area, namely 2.37 m². To this day, the problem remains unsolved!