# Guess the secret number

I'm thinking of a 2-digit secret number XY, where X is the first digit (non-zero) and Y is the second digit. You can make guesses to find it. If your guess is correct then the game finishes. Otherwise after a guess AB, I will tell you the result of $$|A-X|+|B-Y|$$. What is the minimum number of guesses you need to guarantee finding the secret number?

I can do it in

three guesses. Two to determine the number, and then the third to finish the game.

The first guess is

$$99$$, and then the second guess is $$90$$.
Let the answers to these guesses be $$d_1$$ and $$d_2$$ respectively.
We have
$$d_1=|9-A|+|9-B|= 18-A-B$$
$$d_2=|9-A|+|B|= 9-A+B$$
From these two equations we can always deduce the two variables $$A$$ and $$B$$. In particular, $$d_1-d_2=|9-B|-|B|= 9-2B$$ allows you to determine $$B$$. Once you know $$B$$, you can substitute it in either equation to get $$A$$ itself.

For example, if the answers were $$d_1=7$$ and $$d_2=6$$, then $$9-2B=7-6$$ gives $$B=4$$, and then $$6=9-A+4$$ gives $$A=7$$.

The reason this works is that the two guesses are in adjacent corners of the rectangular search space. Only for those points are the signs of $$A-X$$ and $$B-Y$$ known regardless of the values of $$A,B$$. With any other points the absolute value signs would allow multiple possibilities, and then the two equations would not always have a unique solution. Of course any pair of adjacent corners could be used: $$(10,19)$$, $$(90,99)$$, $$(10,90)$$, or $$(19,99)$$.

As 2012rcampion points out in their answer, it is impossible to do it in fewer guesses. The result given is a whole number in the range $$[1,18]$$, so cannot dstinguish between all $$89$$ unguessed numbers.

• Correct and well done! Jul 4, 2022 at 14:24