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Benquo comments on Rationality Games & Apps Brainstorming - Less Wrong Discussion
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Hmm... perhaps you could just tighten the time/# of moves limit? To give fast, appropriate updating an advantage over a brute-force approach?
The problem would need to change a bit each time you tried it, so that you had to learn within the round.
Zoombinis actually had some great examples of this, where you had to learn what test was being applied, and how to pass the test, by induction. The exact criteria were randomly determined, so you had to solve the problem anew each time. And you were penalized for taking too long to learn, so being good at the game involved coming up with an efficient learning algorithm, rather than just an adequate one.
Oh man, I played Zoombinis when I was a kid, I loved that game and haven't thought about it in forever.
Zoombinis pro tip: The game allows you to create up to two zoombinis of each hair/eye/nose/transport configuration. You can make things way easier by making your party consist of 8 twinned zoombini pairs as opposed to the usual 16 distinct zoombinis. (For puzzles that depend on zoombini features, which is pretty much all of them, the solution for a given zoombini and it's twin will be the same.)
I love how the game's wikipedia page has a fairly detailed explanation of every puzzle...
So did I!
And, as far as I can remember, not only were the puzzles well designed, but there was a reasonably good overarching story with a definite goal, which meant the puzzles had purpose and there was a little sense of adventure too.
(I should really find the disk and play it again... hehe.)