MathDB

Pablo and Silvia play on a

2010 \times 2010

board. To start the game, Pablo writes an integer in every cell. After he is done, Silvia repeats the following operation as many times as she wants: she chooses three cells that form an

L

, like in the figure below, and adds

1

to each of the numbers in these three cells. Silvia wins if, after doing the operation many times, all of the numbers in the board are multiples of

10

. Prove that Silvia can always win.
\begin{array}{|c|c} \cline{1-1} \; & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \end{array} \qquad \begin{array}{c|c|} \cline{2-2} \; & \; \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \end{array} \qquad \begin{array}{|c|c} \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \cline{1-1} \end{array} \qquad \begin{array}{c|c|} \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \cline{2-2} \end{array}

4

Problems(1)