That guessing game (which is good for deciding on who will clean up after a party, say) is an example of a non-transitive relationship. Aren't you glad to learn that?
It can also be used as a source for fun thoughts:
Except that fun games shouldn't be subjected to such scrutiny. We can't even be bothered to subject political claims to any real scrutiny!
I always try to play random in that game because most people think the opponent does not. And I hardly ever had to do the worst kind of cleaning after parties*... Though I usually help out, of course, being a kind-hearted goddess who also cares about her pristine reputation.
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*In two wins out of three games, mind you.
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