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 1

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

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2022 MMATHS Individual Round Problems/Problem 4

Problem

Solution 1