11. A two-digit number, when added to the number written with the same digits but in reverse order, results in the square of a natural number. Find all two-digit numbers with this property.
11. Solution. Let \(a\) be the tens digit, \(b\) be the units digit. Then for \(k \in N\) according to the condition

\[
10 a+b+10 b+a=k^{2} \text { or } 11(a+b)=k^{2}
\]

Obviously, \(a+b\) must be a multiple of 11 and \(a+b \leqslant 18\). Therefore, \(a+b=11\). Consequently, the conditions of the problem are satisfied by eight numbers: \(29,38,47, \ldots, 83\) and 92.