WakeUp wrote: $a_1a_2a_3$ and $a_3a_2a_1$ are two three-digit decimal numbers, with $a_1$ and $a_3$ different non-zero digits. Squares of these numbers are five-digit numbers $b_1b_2b_3b_4b_5$ and $b_5b_4b_3b_2b_1$ respectively. Find all such three-digit numbers. Both squares are in $[10000,99999]$ and so both numbers are in $[100,316]$ and $a_1,a_3\in\{1,2,3\}$ Wlog say $a_1<a_3$ and so : $(a_1,a_3)=(1,2)$ $\implies$ $\overline{1a2}$ and $\overline{2a1}$ and squares are $\le 49991$ and so numbers $<224$ and so $a\in\{0,1,2\}$ and just 3 tests giving $(102,112,122)$ $(a_1,a_3)=(1,3)$ $\implies$ $\overline{1a3}$ and $\overline{3a1}$ and squares are $\le 99991$ and so numbers $<317$ and so $a\in\{0,1\}$ and just 2 tests giving $(103,113)$ $(a_1,a_3)=(2,3)$ $\implies$ $\overline{2a3}$ and $\overline{3a2}$ and squares are $\le 99994$ and so numbers $<317$ and so $a\in\{0,1\}$ and just 2 tests giving no solution. Hence the answer : $\boxed{\{102,103,112,113,122\}}$