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?

Write the number as a Hamming word i.e.
inserting parity bits in positions 1,2,4,8.

The procedure is described here: http://users.cis.fiu.edu/~downeyt/cop3402/hamming.html

After getting 14 answers the decimal value of
the “parity word” will specify the bit to be corrected (zero for ok, 15-more
than one error)