Almost 0

Probability Level 5

Consider a 2017×2017 2017 \times 2017 square grid. Each cell is filled in with a real number with absolute value at most 1, such that the sum of any 2×2 2 \times 2 square grid is exactly 0.

Let SS be the sum of all of these real numbers. What is the maximum value of SS?

Note: Below is a 3×33 \times 3 grid which satisfies the requirements above (although its sum is not necessarily maximized).

0.30.20.60.40.30.70.50.60.2\Large \begin{array}{|c|c|c|} \hline -0.3 & 0.2 & -0.6 \\ \hline 0.4 & -0.3 & 0.7 \\ \hline 0.5 & -0.6 & 0.2 \\ \hline \end{array}

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