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Four states each of six pairs of which have a common boundary

If this question as I wrote it was originally unclear to some people, I expect that the visual image added to the question in the edit by Steven Kay should make it clear and justify reopening it.

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If the question were about how to use a particular GIS software to test a polygon feature class to determine if any polygons within it meet your criteria then I think it would be ready to re-open.

If I were to try and answer the question it would need to be about using ArcPy to do it, but I am sure there are many other GIS software products and those with expertise within them, that would be able to provide an answer too.

By not specifying a particular product, or asking for an algorithm, you are effectively asking multiple questions (how to do in any of many products) which I think makes it too broad and/or unclear (because you have not specified a product).

If you are only interested in this as a general geography question then please be aware that this is the GIS Stack Exchange and it may or may not be on-topic for the Earth Science Stack Exchange (review their on-topic page before asking).

A Meta GIS SE discussion that I believe to be relevant to your question as currently asked is Is geography trivia on topic?

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