Clever idea with cute presentation! I think having the list of what the rules *could* be would make it more enjoyable, since otherwise it's just too hard to figure out what the patter is from 3 plays alone. Maybe adding more players to the table would help on that front as well.
But the idea really is good and the time pressure for figuring out the rules is challenging and fun.
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Yes! Because, like, from 3 examples alone, there are more than one reasonable/possible rule that could fit the that cards were played. If there were one or two more players at the table, you would get a few more examples before your turn, which in turn could help you eliminate some more possible rules, and find the correct one.
You could argue that it would make the game too easy, but I think the time pressure would still make it quite a challenge.