Realizing that this sort of thing is possible is half the
challenge. A friend, the daughter of a physicist, thought
it was non-obvious. She asked "Is that a physics thing?"
I claim yes, there is physics in this. Moderately
sophisticated physics, after a fashion. Physics cares a lot
about symmetry. You can win a Nobel prize for discovering a
new symmetry ... or breaking an old one.
This is an example of what we call a /broken symmetry/. By
that I mean the answer has a lower symmetry than the
There is a proverb that says "Between two stools one sits
in the ashes" ... but this is the opposite of that. Cutting
along the meridian (a line of symmetry) doesn't work, and
cutting along the diagonal (another line of symmetry)
doesn't work, but between the two symmetries there is a