Guessing a number knowing only if the number proposed is lower or higher?

Question

I need to guess a number. I can only see if the number I\'m proposing is lower or higher. Performance matters a whole lot, so I thought of the following algorithm:

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Answer 1

In general, binary search starting at the middle point of the range is the optimal strategy. However, you have additional specific information which may make this a suboptimal strategy. This depends critically in what exactly "close to the average of the previous results" means.

If numbers are close to the previous average then dividing by 2 in the second step is not optimal.

Example: Previous numbers 630, 650, 620, 660. You start with 640.

Your number is actually closer. Imagine that it is 634.

The number is lower. If in the second step you divide by 2, you get 320, thus losing any advantage about the previous average numbers.

You should analyze the behaviour further. It may be optimal, in your specific case, to start at the mean of the N previous numbers and then add or substract some quantity related to the standard deviation of the previous numbers.