Octagon Counter

First, it took me a few moments to even figure out which rectangles were being used to create the figure, so here's an image with one of the rectangles in green; if you imagine rotating it in $45^\circ$ increments about the green central point, you can see all eight rectangles.

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The three images below show regions with value 1, 2 and 3; the red rectangles are the only ones containing the shaded regions.

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The next three images show regions with value 5, 6 and 7; the blue rectangles are the only ones NOT containing the shaded regions.

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Clearly, the central region is contained by all eight rectangles. Finally, here's an image showing all the region values:

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There are eight regions each with values of 1, 2, 3, 5, 6, and 7, and one region with value 8; so the total value of all regions is

$8 \ (1 + 2 + 3 + 5 + 6 + 7) + 8 = \boxed{200}$

Each $1 \times \sqrt{2}$ rectangle contains 25 regions. Therefore, the sum of the region values is $8 \cdot 25 = 200$ .