At first it seems like Alice and Bob can do no better than random chance. If Alice is told 20, for instance, there is no way to know if Bob has 19 or 21. But since Alice can limit Bob’s number to 2 possibilities, she can at least have a 50% chance of guessing correctly.

Bob has the same issue. If he is told a number *N*, then he cannot be sure if Alice was told *N* – 1 or *N* + 1. If Bob guesses between the 2 possibilities, then he also has a 50% chance of guessing correctly.

It would seem Alice and Bob are stuck. Neither person has can do better than random chance, so regardless of who guesses, it would seem they are limited to a 50% chance of winning.

But remarkably they can do much better than random chance! They can actually increase their odds of winning to 100%. That is, they can win the game for sure! The trick is that they can use logic and the ringing clock to coordinate which player guesses.

**The strategy**

The answer lies in the subtle rule that the clock rings every minute. The clock essentially serves as a signal between Alice and Bob that allows each person to reason inductively.

One key detail is the two are given positive consecutive numbers. When Alice gets a number *N*, she usually has to consider Bob has *N* – 1 or *N* + 1. But this is not always true. Suppose that Alice gets the number 1. She would have to consider that Bob got 0 or 2. But since 0 is not positive, she knows that Bob must have gotten 2.

So if Alice gets 1, then she would know Bob has 2 for sure, and she would answer on the first ring of the clock. Similarly, if Bob got the number 1, he would know Alice must have 2, and he would answer after the first ring of the clock.

Now consider instead that Alice was given 2 and Bob was given 3. Alice would be wondering if Bob has 1 or 3. But Alice would think, “If Bob has 1, he surely will answer after the first ring of the clock. Therefore, if the clock rings and he does not answer, he must surely have 3 instead.” So the clock will ring once, and then after it rings a second time Alice will answer and guess Bob has 3. (If instead Bob was given 2 and Alice was given 3, then Bob would answer after the second ring and guess Alice has 3.)

This reasoning can be extended inductively. If Alice and Bob are assigned *N* and *N* + 1, then the player with the lower number will answer in exactly *N* rings of the clock and correctly answer the other person has *N* + 1.