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I've had the same problem, what helped was starting to feed the cat semi-regularly. Cats quite literally don't shit where they eat!

Sure fire solution is to get your own cat. Preferably big one.

I appreciate the suggestion, but I don't think that will solve my problem of too many cats on my property.

Sure would -- the number of cats would be capped at one.

Is there really no way to have zero cats?

Cats are not things. That aside, house cats usually only shit in sand-ish environments - is the cat by any chance shitting in an uncovered playground?

No, I don't have a playground on my property, and I dispute your assertion that cats will mostly shit only on sand. Also, a cat is a thing.