Yes

The area of the square would be exactly 1/2 the area of the outer square. Lots of "square roots of 2" involved in my proof of that. You probably have a much simpler proof.

Note: there's a relationship between the area of the two circles based on the polygon one draws in the middle. It happens that when a square is drawn the outer circle is exactly twice the inner. A triangle would form a different ratio. The more sides to the polygon, the closer the two circles approach each other in size. A polygon with an infinite number of sides is a circle, assuming the polygon maintains symmetry.

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I think the question of "understanding" starts to touch on rather abstract things like "intuition," and "sense" and "gut feel." The chess example is a little difficult because chess can be reduced to a mathematical solution set. But what if, for example, a chess grandmaster was able to discern the emotional state of his opponent, and altered his style accordingly? For instance, if the grandmaster sensed a certain bravado in his opponent, are there things that could be done to draw the opponent into making a small tactical error that later turns into a strategic blunder? The question that follows is: would the discernment of that bravado be based on the opponent's chess moves, or a reading of something else? If merely chess moves, then a computer could be programmed to "intuit" the opponent's mood.

I am a believer in an undefinable "sense" about things. People seem to have it to greater and lesser degrees. I've seen it described as "Emotional Intelligence," or "EQ" rather than "IQ." What I can't say is whether the person possessing this higher EQ is really just factoring more perceptions into a more complex equation. If we learned what perceptions and the algorithm they used, could we program a computer to do the same? I'm skeptical, but not convinced it's out of the question.

Still, there's something beyond even that. I can't begin to explain it.

Ultimately, I think "understanding" is a concept that is not definite. We acquire greater degrees of understanding about something, but never fully understand it. I think that can be said for even the simplest of things. Can you think of anything in our realm of existence for which it can be said that we -- mankind -- understands it completely?