Prove that at any time there are two opposite points along the Equator, which have exactly the same temperature. Assume the temperature function varies continuously as you move along the Equator.
Counterargument: This is patently impossible. If there are such points on the Equator, there must also be similar points on any circle around the Earth, such as a meridian. But in that case, we'd have one point in the north hemisphere, in winter, and the other in the south, in summer; that doesn't make sense!
What's wrong with this reasoning?
Pick any point on the equator and its opposite point. Call the point where the temperature is higher A and the other one B. (If they're the same temperature, we're done.) Now picture both points moving eastward, maintaining their opposite positions. At some point they'll reach each other's position and B will be warmer than A. So, as A and B move their temperatures will vary but sooner or later a time will come when A is no longer warmer. Because the temperature change is continuous, the two points cannot switch hotter-colder positions without both being the same temperature for an instant. The two points' position at that instant defines at least one pair of two opposite points with the same temperature. Obviously, there may be more.
Edited on March 28, 2005, 1:07 am