Guessing number between 1-100 always can always be guessed in 7 guess. Why? I'm no good at probability and statistics as this question will soon demonstrate. My question is this, I have an instructor who proved that he could guess a number between 1-100 using the divide and conquer approach. For example, I choose 7.

He: 50

Me: High

He: 25

Me: High

He: 12

Me: High

He: 6

Me: Low

He: 9

Me High

He: 8

Me High

He: 7

See, its not possible for it to take more than 7 guess if your halfway smart. My question is whats the math behind this, and how could I figure the number of guess for like 1-1000, whats the formula?

Thanks!
Hint: $2^7=128&gt;100$.  That is, you can simply guess in the middle and bisect until you get to the solution.

In general, just find the smallest power of $2$ greater than the max number in your set.  For $1000$, you should be able to get it in $10$ guesses, because $2^{10} = 1024$.
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Hint: 27=128>10027=128>100. That is, you can simply guess in the middle and bisect until you get to the solution.

In general, just find the smallest power of 22 greater than the max number in your set. For 10001000, you should be able to get it in 1010 guesses, because 210=1024