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 $16, 25, 36, 49, 64,$ and $81$. We can eliminate $16$ and $64$. Let's check each of the next ones.

For $25$, we have $5-2 = 3$, $5+2 = 7$.

For $36$, we have $6-3 = 3, 6+3 = 9$.

For $49$, we have $9-4 = 5, 9+4 = 13$.

For $64$, we have $6-4 = 2, 6+4 = 10$.

We can clearly see that $25$ doesn't work because $7$ would identify it, nor does $49$ because $13$ would identify it, nor does $64$ because $10$ would identify it. We're left with only $36$. Therefore, our answer is $\boxed {36}$.

-Arcticturn