I sure don't! But I also couldn't find an easy way to express the relationship between (1) the average of two numbers being even and (2) the third plate. That's when I disappeared down the rabbit hole of the difference between a and b being divisible by 4.
I think we're in the same boat, though. We just can't come up with a way to express what's in our heads.
Whenever I'm in a situation like this (which happens pretty often), I sit back and watch the rest of you all solve where I have struggled.
Godspeed, my friends! Should anything pop into my head, I will be certain to post it.
EDIT: What if we rename the plates to a, a+x, and a+y (or a+x+y)?