You are asked to guess an integer between 1 and N 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 4th time you make a guess and are too low, or the 11th time you make a guess and are too high, you lose the game.

What is the maximum N 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 N=5, but not for N>5 . So, in this case, the answer would be 5.

To solve this problem, we need to find the maximum value of N for which we can guarantee finding the correct answer within the given constraints.

Let's analyze the number of guesses needed to find the correct answer for different values of N.

If N = 1, we can guess 1 and immediately get the correct answer without any additional guesses. So, the maximum value of N in this case is 1.

If N = 2, we can guess 1 and if it's too low, we can guess 2 to get the correct answer. This requires a maximum of 2 guesses.

If N = 3, we can start by guessing 2. If it's too high, we can then guess 1. Otherwise, if it's too low or the correct answer, we can guess 3. This guarantees we will find the correct answer within 3 guesses.

If N = 4, we can follow a similar strategy. Guess 3, if it's too high, guess 1. If it's too low, guess 4. Otherwise, if it's the correct answer or too low, guess 2. In any case, we will find the correct answer in a maximum of 4 guesses.

Now, if N = 5, we can guess 3. If it's too high, we guess 1. If it's too low, we guess 5. If it's the correct answer or too low, we guess 4. Finally, if we haven't found the correct answer yet, we guess 2. This guarantees we will find the correct answer within a maximum of 5 guesses.

By analyzing the patterns, we can conclude that the maximum value of N is equal to the number of guesses we are allowed to make when we guess too high (10) plus the number of guesses we are allowed to make when we guess too low (3), minus 1 (to account for the first guess). Therefore, the maximum value of N is 10 + 3 - 1 = 12.

So, the maximum N for which we are guaranteed to find the correct answer is 12.