## 2008年6月19日 星期四

### A "Football Theorem"

With the European Football Championship being a hot topic these days (this is especially true for me as I am now in a country that is crazy about the sport!), let me share with you a theorem about football. It goes like this:

"Place the ball in the center of the court, and start the game. The next time when the ball is being placed at the center of the court again, there exists one point on the ball which is at exactly the same position as it was in the first time."

The proof only involves a simple concept in elementary Linear Algebra, so do think before looking at the hint below!

Highlight the space in the right for the hint: Use the fact that every rotation has an eigenvalue 1.

#### 2 則留言:

Tastic 提到...

Must a continous map from S^2 to S^2 has a fixed point? (It must be so if the map is anti-podal preserving by Borsuk-Ulam Theorem.)

Another interesting Theorem:
If the football is pressed so that all the gas inside it is relased, then there must be a pair of antipodal points overlapping.
(by Borsuk-Ulam Theorem again)

Alex 提到...

By Lefschetz's fixed point theorem, a continuous map from S^2 to S^2 must have a fixed point, since the Euler characteristic of S^2 is nonzero.