## Guess the Number in One Question

There are a lot of puzzles where you need to guess something asking only yes-or-no questions. In this puzzle, there are not two but three possible answers.

Puzzle. Mike thought of one of three numbers: 1, 2, or 3. He is allowed to answer “Yes”, “No”, or “I don’t know”. Can Pete guess the number in one question?

Yes, he can. This problem was in one of my homeworks, and my students had a lot of ideas. Here is the first list were ideas are similar to each other.

• I am thinking of an odd number. Is my number divisible by your number?
• If I were to choose 1 or 2, would your number be bigger than mine?
• If I were to pick a number from the set {1,2,3} that is different from yours, would my number be greater than yours?
• If I have a machine that takes numbers and does nothing to them except have a 50 percent chance of changing a two to a one. Would your number, after going through the machine, be one?
• If I were to choose a number between 1.5 and 2.5, would my number be greater than yours?
• If your number is x and I flip a fair coin x times, will there be at least two times when I flip the same thing?
• I am thinking of a comparison operation that is either “greater” or “greater or equal”. Does your number compare in this way to two?

One student was straightforward.

• Mike, please, do me a favor by responding ‘yes’ to this question if you are thinking about 1, ‘no’ if you are thinking about 2, and ‘I don’t know’ if you are thinking about 3?

One student used a famous unsolved problem: It is not known whether an odd perfect number exists.

• Is every perfect number divisible by your number?

Then, I gave this to my grandchildren, and they decided to answer in a form of a puzzle. Payback time.

• I’m thinking of a number too, and I don’t know whether it’s double yours. Is the sum of our numbers prime?

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### One Comment

1. #### Ivan:

PETE: Mike, if I asked you to raise the biggest number that is not yours to the power of the smallest number that is not yours and answer with exactly that many letters, what would your answer be?