All of you have played the game of high-low in some variation. You are asked to guess a number between $1$ and $n$ where $n$ is often $100$. Your goal is to determine the number in the minimum number of guesses.

When you guess a number, the responses are "You got it" if you correctly guessed the number, "High" if the number you guessed was larger than the number to be guessed, and "Low" otherwise.

Algorithm X.1: The high-low game

Input: the upper bound $n$, and a response to a question "Is it $m$?"
Output: the hidden number.

Describe an algorithm to guess the number. You should try to make sure that your algorithm never requires more than $\left\lceil \log_2(n + 1) \right\rceil$ guesses, and you should never need to guess the same number twice.