Solution to Riddle of the Week: Forehead Numbers

Laura Feiveson
Photo credit: Kory Kennedy using Illustration Copyright csaimages.com

From Popular Mechanics

Photo credit: Kory Kennedy using Illustration Copyright csaimages.com

This is a solution to Figure Out What's on Her Forehead, part of our Riddle of the Week series.

Figure Out What's on Her Forehead

Photo credit: Illustration Copyright csaimages.com

Jaya's number is 50.

To figure this out, let’s go back to what Jaya initially observes: She sees Julian has 20 on his forehead and Levi has 30 on his. That means she can either be 50 (their sum), or 10 (their difference).

Let’s suppose Jaya were 10. Then Julian would have seen 10 (on Jaya) and 30 (on Levi), thus thinking he was either 20 or 40, and would say he doesn’t know what number he is. Now it comes to Levi, who would see 10 on Jaya and 20 on Julian. He would think, then, he’s either 10 (the difference) or 30 (the sum).

But wait! Levi can’t be 10, because Cecilia told everyone all three numbers are different from one another, and Jaya is already 10. So Levi would know he was 30, and would say so. Since he said he didn’t know his number, Jaya can’t be 10. Thus, she knows she’s 50.

For completeness, we need to confirm if Jaya were 50, Julian and Levi would respond as they did: If she were 50, Julian would have seen 50 (on Jaya) and 30 (on Levi), such that we wouldn’t know if he were 20 or 80. Then it would come to Levi, who would see 50 on Jaya and 20 on Julian. He could therefore be either 30 or 70, and he wouldn’t know which one. (For extra completeness, and quite tediously, we need to make sure all answers are also consistent from Julian’s and Levi’s perspectives. For instance, if Levi were 70, is it the case that Julian or Jaya couldn’t have figured out their numbers previously? It is.)

Extra credit: I love these problems in which it appears on the face that no information has been exchanged, and yet there was sneakily enough exchanged to pinpoint a solution. Often, these solutions rely on constraints that are subtly introduced in the body of the problem. In this solution, we rely on the tidbit that Cecilia shared: All the numbers are different from one another. However, given the other constraints already embedded in the problem, we don’t actually need that piece of information. Why?

Solution to extra credit: The bit of information we still need is all of the numbers are positive, and therefore can’t be zero. Once again, we consider what would happen if Jaya had a 10 on her forehead. As before, when it comes to Levi, he’d see 10 on Jaya and 20 on Julian, and think he’s either 10 or 30. This time, he eliminates the possibility he’s 10 by thinking to himself, “If I were 10, then Julian would have seen 10 on Jaya and 10 on me, and therefore would have been able to figure out he was 20, since we all know he can’t be zero. Since he said he didn’t know what his number was, I can’t be 10!” The rest of the solution follows as before.

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