If I think of a number between 1 and 1,000, guessing it in 10 yes-no questions is easy... so that's not the puzzle!

Guessing it in 10 yes-no questions, that must be all asked in advance, is also relatively easy... so that's not the puzzle either!

How many questions would you need to guess my number, if you had to ask all questions beforehand, and I also could lie once?

The easy puzzle is satisfied by expressing a number in binary and asking if the 1st, 2nd, ..., 10th binary digit is a 1.

There are 10 possible places where the clue giver might lie, and to find the answer, we must figure out which one it is.  To determine one of ten possibilities, you need 4 bits (yes-or-no answers), bringing the total to 14.  But now since there are 14 places which might be erroneous (or a lie) we have to be sure that we have enough to cover 14 possible places for the misinformation.  But indeed the extra 4 bits would be enough to cover in fact 16 possibilities.

So information-theoretically, 14 yes/no questions should suffice.

Posted by Charlie on 2005-04-14 19:17:58