2022 MMATHS Individual Round Problems/Problem 4
Problem
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit perfect square!"
Claire asks, "If I picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there a chance I'd know for certain what it is?"
Cat says, "Yes!" Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn't know my favorite number!
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
Solution
It would be helpful to list some two-digit perfect squares. These are and . We can eliminate and . Let's check each of the next ones.
For , we have , .
For , we have .
For , we have .
For , we have .
We can clearly see that doesn't work because would identify it, nor does because would identify it, nor does because would identify it. We're left with only . Therefore, our answer is .
-Arcticturn