Narrow It Down
A guess is worth what it destroys, so cutting the middle of what is left beats a bold guess or a lucky one.
A number is hiding inside a block of numbers. You do not type a guess here — you slide a cut along the block and press it. Everything on the wrong side of the cut is thrown away, and you watch it go.
That is the part people miss. A guess is not good because it feels close or because it is daring. A guess is worth exactly the number of hiding places it destroys. Cut at the very end of the block and you throw away one number. Cut down the middle and you throw away half of everything, whichever answer comes back.
There is a bar of guesses beside the block, with a mark printed on it. The mark is how many more halvings the block still needs. While your bar is longer than the mark you can still win, no matter where the number is hiding. Waste a guess and the bar drops under the mark, and you can see the trouble coming before you run out.
Three rounds. Sixteen numbers in six guesses, then fifty in seven, then a hundred in seven — which is exactly enough, and only if every single cut halves what is left.