Determine the Number

Solution

As so far it was our hardest contest - we've received just two answers. Below we show these winning solutions.

Solution 1 by Kaj Braek

Number can under normal circumstances be determined using 4 yes/no questions (e.g. thinking of 11):

Q 1. Which group of eight (0-7: "No")
Q 2. Which group of four (8-11: "Yes)
Q 3. Which group of two (10-11: "yes)
Q 4. Which of the two (10 - "No" - we now know it was 11)

Solution:
1. Q 1
2. Q 2
3. Q "Did you lie so far?" 
- If "yes" the lie must have been used. Start from Q 1 - Q 4 again = total 7 questions
- if "no" ("No" has to be true - otherwise it would be second lie):
4. Q "Will you lie on the next question?" 
- If "yes" the next answer will have to be false - otherwise it would represent two lies (Remember to reverse the next answer) 
- if "no" the next answer will have to be correct - otherwise it would represent two lies
5. Q 3
6. Q "Will you lie on the next question?" 
7 Q 4

If you know what I mean...

Solution 2 by Jason Meyers

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. 4-7)).
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