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I have 19 polygon shapefile layers, with some occasionally overlapping with one another. As a final output, I would like a shapefile that includes only the overlapping areas between 2 or more of these 19 layers (and if it could keep their respective attributes, that would be fantastic).

The Intersect tool provides an ideal output, yet if I add all 19 layers as the inputs, it searches for areas where all 19 intersect. The only remedy I can currently imagine is doing pairwise intersections, which would mean I'd have to do it 171 times, then merge them all into one shapefile layer.

I cannot seem to find a method for what I want to do that doesn't involve painful pairwise analyses.

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    Unioning all 19 (without attributes) is the best way to do this, since having a single table with FID1,FID2,FID3... allows you to increment a COUNT field for each non-negative FID. At which point you could select out those features with a COUNT of 2 or greater. Probably not more than one page of ArcPy code. Attributes could be accessed through joins. – Vince Jun 5 '16 at 12:00
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As commented by @Vince:

Unioning all 19 (without attributes) is the best way to do this, since having a single table with FID1,FID2,FID3... allows you to increment a COUNT field for each non-negative FID. At which point you could select out those features with a COUNT of 2 or greater. Probably not more than one page of ArcPy code. Attributes could be accessed through joins.

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