Cheryl versus Computer

We will call Cheryl’s number $\overline{AB}$ , and the computer’s number is $\overline{CD}$ . Then, the computer’s first hint can be written as $10A+B+10C+D=10n$ . Since 10A and 10C are multiples of 10 already, we can remove them. Then, either B and D are both zero, or they sum to 10. The former can be eliminated quickly, so we are left with the latter. The other two facts are not as helpful as equations, but must be kept in mind.

So where do we go from here? If you are like me, you start plugging in for D, then looking for values of A and C that satisfy the other two facts. This is annoying, but it led me to a helpful fact: if C is 9, then regardless of the value of D, there is an A which satisfies both of the computer’s hints (not necessarily the nine difference rule). This is because 9 times n ends in the digit 10-n.

Now, we could just run through the values of D, checking the third rule as we went, but we can once again save ourselves from a little bit of tedium by using divisibilty rules. Since Cheryl’s number is, under this assumption, 9D, it is divisible by 9. For the difference to also be a multiple of 9, the computer’s number must also be divisible by 9. For C=9, this gives 90, which would make Cheryl’s number 0; or the correct answer of 99, which makes Cheryl’s 81.

Is this the only solution? Well, yes; the problem implies there is only one solution. For thoroughness, I should have looked as well for solutions where C was not nine. There are numbers like this which satisfy the first two (44 and 16), but none that satisfy all three requirements. The nine trick wasn’t necessarily going to produce the answer, it only produced a set of possibilities to quickly check, one of which happened to be correct. I’ve made use a little bit here of a logic trick where I rely on the fact that there is a unique solution in order to find that solution faster. This is occasionally useful in puzzles like sudoku; if you can show that a certain action leaves you with multiple, equally valid solutions, you know that the original action is not correct. In other words, it’s a proof by contradiction, where you assume that your initial action preserves the solution’s uniqueness. /tangent

Anyway there is only one solution, and it’s Cheryl: 81, and Computer: $\boxed{99}$