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I need an example of a hand, in riichi, where if you kan, the wait will change, but for the better. Naturally, this would be an illegal move and result in penalty.
In the hand , the waits are .
If you draw and kan the , the wait changes but obviously gets worse ( only), so no one would ever be tempted to make that mistake.
I need to illustrate a hand where the wait improves.
If this is possible, can you help. I have tried.

I would be surprised if you could get more combinations with less tiles to play with.

I suspected as much, after numerous attempts. I guess it's safe to assume that if a player calls an illegal kan during riichi, it's probably due to a lack of full comprehension of the rules, perhaps thinking they might score big with the kan dora/ura dora, and then later finding out it is not a valid win.
Am I correct in saying that this offense is a chombo and is discoverable only if the player calls ron/tsumo or the hand ends in a draw.

Yes, it's exactly that!