Question

A friend proposes the following guessing game: He chooses an integer between 1 and 100, inclusive, and you repeatedly try to guess his number. He tells you whether each incorrect guess is higher or lower than his chosen number, but you are allowed at most one high guess overall.You win the game when you guess his number correctly. You lose the game the instant you make a second high guess. What is the minimum number of guesses in which you can guarantee you will win the game?

Answer 1

The approach should be to choose a small number so that there is good chance that the chosen number is higher and thus we can keep guessing. The number shouldn't be so small that if the chosen number is very high (say 99) then it will take too many guesses. We would like to maintain a gap between our guesses in such a way that if the chosen number is smaller than we can just guess all the numbers between our preceding and present guess with minimum effort.

Using trial and error to optimise that particular number, the answer can be arrived at 14 guesses starting with the first guess as 14.

Consider the following scenario :

Our first guess is 14. If the chosen number is lower than we will start guessing from 1 to 13. The maximum it will take is 14 guesses if the chosen one is 13.

If the chosen number is higher than our next guess would be 27. If this is high then we will guess between 15 and 26. The maximum number of guesses will happen if the chosen number is 26. This will take 14 guesses in total (14,27,15,....,26).

Similarly, if we continue to guess low then our next guesses would be 39,50,60,69,77,84,90,99 and 100. As it can be seen, even if 100 was the chosen number, the number of guesses is less than 14, thus guranteeing our success.