![]() If you see A, your comparison will show it to be smaller than G. The decision that the comparison procedure forces you to make is given in bold.Ĭase 1. In the following illustrations, cover the numbers A and B in turn to simulate choosing the number in each of the host’s hands. ![]() Note that only the relative positions of the numbers is relevant, not their actual values. I’ve shown them below on the number line with numbers increasing as usual from left to right. How does this solution work? There are three cases to consider. If it is larger, stay with the number shown. If it is smaller than G, switch to the other hand. Depending on which hand you selected, you will see either A or B. Following Erica, let’s call this the guide number G, the original numbers being A and B, with B being the larger. Now comes the key step: You generate an arbitrary or random number entirely from your imagination. The solution is simple: First choose one of the host’s hands at random. The puzzle was first solved in our column’s comments by Jason, and his solution was well explained in detail by Erica Klarreich, g g, Jeremy Magland, Greg Egan, Steven Alexander and Jamie H., among others. Let’s consider the abstract or “pure math” version of this game: It is played just once and is non-adversarial (i.e., the game host is not intentionally trying to get the guesser to fail). Incidentally, Cover mentored my colleague Joseph Chang, who brought this puzzle to my attention. Cover, “Pick the Largest Number,” in Open Problems in Communication and Computation, ed. Cover of Stanford University in 1987 (Thomas M. Is there a strategy that will give you a greater than 50 percent chance of choosing the larger number, no matter which two numbers I write down?Ī solution to this problem was published by Thomas M. Then you can decide to either select the number you have seen or switch to the number you have not seen, held in the other hand, as your final choice. Which is larger? You can point to one of my hands, and I will show you the number in it. You have absolutely no idea how I generated these two numbers. I write down two different numbers that are completely unknown to you, and hold one in my left hand and one in my right. I have tried to keep the actual math to a minimum so readers who are not professional mathematicians can still follow the story. I cannot guarantee that all your doubts about the validity of the solution will be laid to rest, but I’ll try my best. The puzzle has a deeply counterintuitive solution and raises philosophical questions about randomness and information. It’s great to see the enthusiastic reader response to our first Insights puzzle.
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