[shagie]

You might be interested in reading about Prince Rupert's cube. https://news.ycombinator.com/item?id=16399079

In particular from https://en.wikipedia.org/wiki/Prince_Rupert%27s_cube

> In geometry, Prince Rupert's cube (named after Prince Rupert of the Rhine) is the largest cube that can pass through a hole cut through a unit cube, i.e. through a cube whose sides have length 1, without splitting the cube into two pieces. Its side length is approximately 6% larger than that of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.

That 6% larger is the key.