
The first question: Are you
going to lie to one of my first three questions?
Answers:
No (but lying): He is obligated to answer the next 2 questions
truthfully (He's already used up his one lie).
No (truthful): He is obligated to answer the next 2 questions
truthfully.
Yes (but lying): Impossible. By lying, he is saying he isn't lying, a
total paradox.
Yes (but truthful): He is obligated to lie to one of the next two
questions.
If "Yes" to question 1, one of the next two answers is a lie, so:
Question 2: Are you going to lie to either this question or the next
question?
No (but lying): He must answer the next question truthfully (He's used
up his lie).
No (but truthful): Impossible. He has to lie to either this question
or the next one.
Yes (but lying): Impossible. By lying, he is saying he isn't lying.
Yes (but truthful): He will lie to the next question.
From here you can simply subdivide the numbers into halves and in 4
more questions, have the correct answer, keeping in mind based on the
answer from question 2, that the actual answer to the next question
may need to be reversed, if you know he is lying.
e.g. (After a "Yes" to question 2):
3: Is is between 0 and 7? Yes (a lie, meaning it's between 8 and 15)
From now on he must tell the truth:
4: Is it between 8 and 11? No.
5: Is it either 12 or 13? No.
6: Is it 14? No.
It must be 15. Answer guaranteed in exactly 6 questions.
If "No" to question 1, the next two questions subdivide the numbers
(e.g. "Is it between 0 and 7" and based on the answer subdividing down
to 4 possible numbers (e.g. 47)).
The fourth question is then, Are you going to lie to one of my fourth
through 6th questions?
No (but lying): He is obligated to answer the next 2 questions
truthfully (He's already used up his one lie).
No (truthful): He is obligated to answer the next 2 questions
truthfully.
Yes (but lying): Impossible. By lying, he is saying he isn't lying.
Yes (but truthful): He is obligated to lie to one of the next two
questions
If "No", you can get the answer with two more subdivision questions,
to which he must answer truthfully. (Splitting the 4 numbers into half
(is it 4 or 5?) and based on the answer asking about a single number
(is it 6? (with 7 the other possibility)), and you have the answer in
6 questions.
If "Yes", you pull the same trick as above using question 2 as your
fifth question. Based on his answer, you'll know whether he is
required to lie to the 6th question or not. The sixth question then
subdivides the 4 possible numbers in half to two numbers. Based on his
answer and whether or not he had to lie, you are left with two
numbers, and one question. It is impossible to reach this point
without the person having lied once, so he must tell the truth to the
final question which asks between the two possible numbers. You then
know which number he was thinking of in 7 questions.
This was a fun one to figure out! Too many lies!!!
Jason 
