➡ The Problem
Pick three points at random anywhere on the globe. What is the probability that all three of the points will be in the same hemisphere?
➡ The Solution
There are two answers to this question, depending on whether you think about the word “hemisphere” with a capital H.
If you split the Earth into the Northern and Southern Hemispheres, then we’re talking about a problem with simple probabilities. There is a 1-in-2 chance that a randomly chosen location will be in one of the two hemispheres. Given three random points, that’s a simple 1-in-8 chance, or 12.5%. Same as when you compare East and West.
But what if one of those points is right on the Equator? Mathematically, the Equator is simply one of infinitely many Grand Circles, which we get anytime we slice a sphere perfectly in half. The center of a Grand Circle is the same point as the center of its sphere. In theory, if we sliced the Earth in half at the Equator, then both of the halves would contain the Equator! So, we might as well say that points on the Equator count as being in both hemispheres at once. This follows for any possible Grand Circle.
That idea is the crux of the second answer to this question, which is 100%.
Imagine that two of the points you chose were on the extreme opposite sides of the Earth. Create a grand circle that connects those two points and split the Earth perfectly in half to create two hemispheres. Let’s call them Left and Right. The two points you chose count as being on both hemispheres. The third point is on either the Left or Right, which means one of those two contains all three points.
What if the third point also happens to be on that grand circle? Then suddenly all three points are on both hemispheres! Still 100%.
To generalize, pick any three points on the globe. Choose the two points that are farthest apart and draw a grand circle around the globe that passes through those two points. This is always possible. The third point will naturally be on one of the two created hemispheres.
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Richard Malena is a puzzle and game designer, creating codebreaking challenges for the launches of the Tomb Raider movie and Call of Duty: WWII, as well as for board games, film festivals, immersive theater, and all kinds of strange and esoteric endeavors. When he isn’t building puzzles, Rich is likely rewatching National Treasure 2: Book of Secrets and searching for hidden mysteries in the world.