You are asked to guess an integer between 1 and inclusive.
Each time you make a guess, you are told either:
(a) you are too high,
(b) you are too low, or
(c) you got it!
You can guess as many times as you like, but are only allowed to guess too high 10 times and too low 3 times. That is, the time you make a guess and are too low, or the time you make a guess and are too high, you lose the game.
What is the maximum for which you are guaranteed to be able to accomplish this?
Clarification: For example, if you were allowed to guess too high once and too low once, you could guarantee to guess the right answer if , but not for . So, in this case, the answer would be 5.