arcpy select points - chessboard style

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Could you please help with selection problem. I have about 1000 of shapefiles. Each shapefile has points located in a rectangular net. But size of that nets (and as a result number of points) differ in every shapefile. Only thing all shapefiles have in common are distances, net size is 10 meters. Is there any way to select points in chessboard like pattern? Image below there is an example, I need to select all rows (not only first three as in image) in that pattern.





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