Here is a nice application/proof of Brouwer's Theorem in one
dimension from Mark Rubinstein's page
http://www.in-the-money.com/
which has some other nice material as well.
--
One morning, exactly at sunrise, a Buddhist monk began to climb a tall
mountain. The narrow path, no more than a foot or two wide, spiraled
around the mountain to a glittering temple at the summit. The monk
ascended the path at varying rates of speed, stopping many times along
the way to rest and to eat the dried fruit he carried with him. He
reached the temple shortly before sunset. After several days of fasting
and meditation he began his journey back along the same path, starting
at sunrise and again walking at variable speeds with many pauses along
the way. His average speed descending was, of course, greater than his
average climbing speed.
Prove that there is a spot along the path that the monk will occupy on
both trips at precisely the same time of day.
--
One can prove this by showing that the puzzle satisfies the
assumptions of Brouwer's Fixed Point Theorem but there is an intuitive
and satisfying answer also. The answer can be found in Rubinstein's
page but I'll also post it in a follow up-message - but no cheating!
Alex
--
Dr. Alexander Tabarrok
Vice President and Director of Research
The Independent Institute
100 Swan Way
Oakland, CA, 94621-1428
Tel. 510-632-1366, FAX: 510-568-6040
Email: [EMAIL PROTECTED]
