The following addition operation shows the sum of the two-digit positi

\(X,Y\,\,\, \in \,\,\,\,\left\{ {1,2, \ldots ,9} \right\}\)
\(Z \in \,\,\,\,\left\{ {0,1,2, \ldots ,9} \right\}\)
\(X,Y,Z\,\,\,{\text{distinct}}\)

\(? = Z\)


\(\left\langle {XXZ} \right\rangle = \left\langle {XY} \right\rangle + \left\langle {YX} \right\rangle < 99 + 99 < 200\,\,\,\,\,\,\, \Rightarrow \,\,\,\,X = 1\)

\(\left\langle {1Y} \right\rangle + \left\langle {Y1} \right\rangle = \left\langle {11Z} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,Y = 9\,\,\,\,\,\,\left( {18 + 81 < 100} \right)\)

\(\left\langle {11Z} \right\rangle = 19 + 91 = 110\,\,\,\,\, \Rightarrow \,\,\,Z = 0\,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.