It's not clear exactly what you are after or where the 2.7 comes into play, but I'll try an answer anyway.
Suppose you want a high probability that in your sample you observe values that are 2.7 or more standard deviations away from the mean. For a sample of 431 observations, you will have a probability > 0.95 that you see a value 2.7 or more standard deviations from the mean. To see a value at or beyond 4 standard deviations from the mean with greater than 95% probability, you will need a sample of at least 47,293. And for 5 standard deviations, the sample needs to be 5,225,389.
These sample sizes are based on assumptions that the population is normal and your sampled values are independent.
The answer is that $2^7=128>100$.
To be more precise:
Say that you start by guessing $50$. Either you're right, in which case you're done, or you're wrong; if you're wrong, then you know that the number is either in $[1,49]$ or $[51,100]$, so that there are only $49$ or $50$ possibilities left.
Now, assuming that you aren't already done, you have a collection of $49$ or $50$ possibilities left; guess the middle one. That is, if you are on $[1,49]$, guess (say) $25$; if you're on $[51,100]$, then guess (say) $75$. If you got it right: great. If not, then you find out whether it should be higher or lower; in particular, if you previously knew it was in $[1,49]$, then you either now know it is in $[1,24]$ or you now know it is in $[26, 49]$. If you previously knew it was in $[51,100]$, then either you know that it is in $[51,74]$ or that it is in $[76,100]$.
In any case, you have either already guess right, or you have narrowed down the possibilities to one of $[1,24]$, $[26,49]$, $[51,74]$, or $[76,100]$. In any one of these cases, there are at most $25$ possibilities left, and it must be one of them.
Continue in this way: cut your current interval of possibilities in half by a guess, so that you are either right or you can discard roughly half of the possibilities based on the announcement of "higher" or "lower".
By continuing this process, in the third step you narrow it down to at most $12$ possibilities; in the fourth, to at most $6$; in the fifth, to at most $3$; in the sixth, to at most $1$; and voila! In your seventh guess, assuming that you're unlucky enough to have not guessed it yet, there's only one number that could possibly be it.
Best Answer
Hint: $2^7=128>100$. 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 $2$ greater than the max number in your set. For $1000$, you should be able to get it in $10$ guesses, because $2^{10} = 1024$.