Problem 1 · Squares hiding another square
Scanning the three-digit squares and reading their outer digits.
Integer answer, at most 4 digitsHow many three-digit numbers $\overline{abc}$ are perfect squares such that $\overline{ac}$ is also a perfect square?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: there are few three-digit squares: from $10^{2} = 100$ to $31^{2} = 961$. We just scan them and read the number formed by the first and last digits.
The possible two-digit squares for $\overline{ac}$ are $16$, $25$, $36$, $49$, $64$ and $81$. Scanning the list of three-digit squares:
$$196 \to 16 = 4^{2}\ \checkmark, \qquad 225 \to 25 = 5^{2}\ \checkmark, \qquad 841 \to 81 = 9^{2}\ \checkmark.$$
No other square in the list ($100, 121, 144, \dots, 961$) yields a square $\overline{ac}$. Three numbers in total.
Answer: 3