Antipodal Temperature

Answer: At any given time, there exist two antipodal points with the same temperature.

Let’s pick any great circle around the Earth, for example, the equator. Choose a starting point A on this circle, and let B be the point directly opposite to A (the antipodal point).

Case 1: If point A and B have the same temperature, we're done! But more likely than not, they'll have different temperatures, in which case we consider case 2.

Case 2: Point A and B have different temperatures.

Let's start from point A and walk around the equator. At the same time, we'll have a friend start from point B and walk around the equator in the same direction, always keeping on the opposite side of us. As we do so, each of us carry a thermometer to measure the temperature.

Eventually, we'll reach point B, and our friend would've reached point A.

Case 2A: Point A's temperature is lower than point B's temperature. We started with a lower temperature than our friend, but now we have a higher temperature than our friend. Then at some point, our temperatures would have to “cross” each other.

We can even cross multiple times:

Or even cross outside our original temperature range:

Notice that since we're always on the opposite side of each other at all times, the crossing point indicates a pair antipodal points with the same temperature.

Case 2B: Point A's temperature is higher than point B's temperature. We can use the same argument as case 2B to prove that there's a pair of antipodal points with the same temperature.

Across case 1, 2A, and 2B, we can always find a pair of antipodal points that have the same temperature. And that's our solution!