Answer to the Insane King Puzzle | ||

Alas, the king is trapped:
This picture is one way the demoness could confine our lost-marbled king. The circles represent bars on the king's forthcoming cage.
Specifically, here is the demoness's strategy:
We must be absolutely certain that the demoness can satisfy
this condition. As the king moves, most of his windows will
only gain a single circle at a time. Hence those windows
are easy to deal with. However, there are 4 places where
a move of the king will increase the number of circles in
his window by two at once. This happens whenever the king
moves from a colored square toward the same-colored circles.
But the demoness can deal with this by coloring in one of
those same-colored circles in advance, i.e., as soon as the
king moves From here it is easy to verify that we have covered all possible cases of window-moves on the king's part, and that the king is in fact, royally hosed. It turns out that there are much smaller cages than this one. This one just allows for a sort of elegant (-ish) proof.
So the question becomes: what is the |