The Width of a Circle

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?

Yes for sure. A prime example happened in the Morelia-Linares chess tournament yesterday. Ex-world champion Veselin Topalov was playing a 16 year old Norwegian kid (Magnus Carlsen) at the first leg of the tournament in Mexico. Somewhere in the middle of the game Magnus offered an exchange of queens which, due to the state of the board, would have almost surely led to a drawn game. But Magnus is smart, Topalov is one of the most arrogant players ever and there was no way he was going to accept a mere draw from this kid. The only move to avoid the draw slightly weakened Topalov's position - but he thought he could recover it, but the kid was too strong and Topalov lost :)

Also certain computer chess engines now have an "anti-human" mode, so if they detect that their opponent is trying to blockade all the pawns to obtain a draw (which these days is the only thing GMs can do to get half a point), they just adjust their evaluation of the position to give say 0.5 pawn bonus to an open file, and that's usually enough to blast open the position and send the human home in a body bag.

Can you think of anything in our realm of existence for which it can be said that we -- mankind -- understands it completely?

Our senses are not unlimited - so I suspect that therefore by definition we cannot understand anything completely. We can only understand things to the limits of our senses (or brains I guess).

+++

Yes I got the area of the inner square as half that of the outer square also. The way I did was like this.

Imagine a big square, a circle inside it, and the little square inside that. Three objects.

Big square has a side of length x.

Circle has a radius of lenght x/2.

(Now this bit is the key) The distance from the centre of the little square to it's corner is the same as the radius of the circle, so x/2.

Anyone who knows Pythagoras' Theorem can do the rest.

+++

Two questions:

1. (This is a "how is your calculus or series?" type question) Can you find the sum of all the areas of the little circles (or squares) as they get smaller and smaller and smaller? If you could add them all up, what would their areas add up to as a percentage of the big (topmost) circle (or square)?

2. Why is it that the same rule applies to the square as it does the circle? (ie. object n-1 has half of the area of object n). After all, a square has a rational area, but a circle has an irrational area (based on pi). I have no idea!

+++

PS. The append title is a David Bowie song from "The Man Who Sold The World"