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?
If you ask 10 yes/no questions, there are exactly 1024 different
possibilities for answers, and each of the 1000 numbers must correspond
to a unique set of answers.
If up to one lie is allowed, the total number of possible sets of
answers is still 2^N, where N is the number of questions.
However, the 1000 numbers must not only each correspond to a unique set
of answers, but each pair must differ by more than two answers.
So there are at least a thousand possible sets of answers. A
thousand of them correspond to the numbers 1 to 1000, but none of these
sets are two or less answers away from each other. In a way, each
number corresponds to N+1 sets of answers. Therefore, at most, N
questions can correspond to floor(2^N/(N+1)) numbers. This
sequence looks like this: 1, 1, 2, 3, 5, 9, 16, 28,...
case, it turns out that this is not the correct sequence, since with 4
questions, you can still only determine between 2 numbers.
Hmm... I'm stuck! I shall now consult the other comments.
Edit: I think it might be relevant to put a lower limit on the
answer based on the above sequence. Though the sequence is
incorrect, the true sequence cannot be any higher than it.
The lower limit is 15, which can correspond to 2^14/15 = 16/15 * 1024 = 1092
Edit: Oops! I said the lower limit is 15 when it is really 14. The rest of the calculations are correct.
Edited on April 14, 2005, 11:34 pm
Edited on April 15, 2005, 10:22 pm
Posted by Tristan
on 2005-04-14 23:26:22