Size of the largest square that will fit in the unit cube?

I think that this puzzle is interesting to "ponder about"

What is the size of the largest square that will fit in the unit cube?

Moved to "Odds & Ends".

I have seen this before. I know the answer. Now to come up with an original solution.

Gives me a headache just thinking about it!

I'm only gonna "ponder":

The square will have sides larger than 1 and smaller than sqrt(2).

It is inserted in the cube with a side along one of the bottom diagonals;
the diagonal exceeds the side by an equal length at each end;
the square is initially perpendicular to the bottom of the cube.

Now, the square is "pushed" from the top (bottom stays where it is!),
gently, until the top 2 corners simultaneously touch 2 adjacent sides
of the cube, and right at the top of the cube.

Now go figure...