The Mind Reader

I am thinking of a number between 1 and 9. You can ask me two yes/no questions and I will answer them truthfully.

You can't ask me any open questions, I will only answer “Yes” or “No”! However, if for some reason I cannot answer it, I will tell you “I don't know”.¹

What two questions should you ask me to find the number I'm thinking of?

¹ Let's pretend I'm a genius and that the difficulty of your question does not stop me from answering it. Also, me not knowing the answer is not the same as making me deal with an invalid answer! So having me divide by zero will do you no good. That's just cheating out of a valid yes/no question. Encoding the numbers as yes/no/don't know? Same story!

Solution

This puzzle is a whole lot easier to solve if you break it up. First, aim to solve a reduced version: I'm thinking of a number between 1 and 3, and you can ask one question.

That's still no picnic! Somehow, you'll have to make sure that each number (1, 2 and 3) has an accompanying answer Yes, No or Don't know (in some order). What kind of question could accomplish this?

There's more than one solution, but a particularly good question would be:

• I too am thinking of a number. It is either 1½ or 2½. Is your number higher than my number?

Try it for 1, 2 and 3. It works! That's a perfect question to yield a different answer for each number.

So now it's time to scale upwards. If one question can find a number between 1 and 3, then an additional question can find a number between 1 and 9 (3 × 3: three groups of three). But the second question depends on the answer to the first.

Now the two questions you should ask me, are clear:

• If I was thinking of either the number 3½ or 6½, would that number be higher than yours?
  
  • Yes: If I was thinking of either the number 1½ or 2½, would that number be higher than yours?
  • Don't know: If I was thinking of either the number 4½ or 5½, would that number be higher than yours?
  • No: If I was thinking of either the number 7½ or 8½, would that number be higher than yours?