The question now is: Can we be certain that no matter how he climbs there will necessarily exist some instant between 6 a.m. and noon on the two days when the climber is at exactly the same elevation? (Answer below)
The Poincaré Conjecture
So, with these appetizers under our belts, we can proceed to Dunwoody's possible accomplishment.
Poincaré's conjecture says that a certain property of a sphere in space holds for higher-dimensional analogues of a sphere. To understand the property, imagine stretching a rubber band around an orange. We can contract this rubber band slowly, making sure it neither breaks nor loses contact with the surface of the orange. In this way we can shrink the rubber band to a point. We can't do this with a rubber band stretched around a doughnut (either around the hole or around the body).
The orange, but not the doughnut, is said to be "simply connected." Henri Poincaré, who, incidentally, almost discovered relativity theory before Einstein, was aware of the fact that a ball like an orange could be characterized by this property of simple connectivity. He wondered if this held true for balls in higher-dimensional space.
If Dunwoody's proof survives the vetting of other topologists, Poincaré can stop wondering and rest in peace. And with his million dollars, Dunwoody can buy doughnuts and oranges for all the world's topologists.
Answer to question: The answer is Yes, and the proof is vivid and convincing. Imagine the ascent and descent, exact in every detail, being made simultaneously by two climbers. One climber starts at the base and the other at the summit and they both begin their journeys at 6 a.m. of the same day mimicking what the original climber did on Monday and Tuesday, respectively. It's clear that these two climbers will pass each other going in opposite directions and at that instant their elevations will be the same. Since they're only re-enacting the original climber's ascent and descent, we can be certain that the original climber was at the same elevation at the same time on the two successive days.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.