# How can you guess that?

**I am an honest person, and am thinking of one of three numbers: 1, 2 or 3. You may ask me EXACTLY one yes-no question, I will answer truthfully, and if you chose the right question, you will know which number I’m thinking of!**

**Communicated by Rob Andler.**

**First Thoughts:**

Although we could probably skip to the answer a little quicker than usual on this question, we will walk through a longer process to make sure everything is clear. To start out, we can begin to ascertain the path to a solution by asking a few yes-no questions. For example, “Is the number greater than 2?”. The answer to this question would be “no” if the number is 1 or 2, but “yes” if it is 3. Unfortunately, this does not seem to hold enough power to determine the number, since there is confusion between 1 and 2. After trying a couple more questions, we may start to notice a pattern. It seems that in any question where the answer is yes or no, we can only separate our set of three numbers into two categories. That is, we cannot create a category to uniquely determine each number. We have to think of something else.

**Second Thoughts:**

Is there any way that we can think of a yes-no question that has three answers? This seems to be what is required to distinguish the three numbers. With a little thought, we come to an idea. Perhaps one of the answers to our yes-no question can be “maybe”. If we can think of a question such that the answer for one of the numbers is 1, the answer for one of the numbers is 2, and the answer for the last number is 3, then we will have a solution. All that remains is to think of one of these questions.

**Third Thoughts:**

How do we get a “maybe” answer? We have to ask a question that causes uncertainty. How can we be uncertain about whether a property applies to a given number? We have to find a property that sometimes might be true, and sometimes not. One such example is the following: if we are targeting 2 as the uncertain number, we can try to think of a way to ask if 2 is greater than 1 or 2. (2 is greater than 1, but 2 is not greater than 2.) In order to do this, we can ask the question “I am thinking of a number that is 1 or 2. Is your number greater than my number?”. We know that this works for 2, but we still have to check it for 1 and 3. 1 is equal to 1 and less than 2, so the answer is “no” for 1. 3 is greater than both 1 and 2, so the answer is “yes” for 3. Therefore, the solution we have found is the correct solution and we have solved the riddle.

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