In the extended experiment, if a child receives three Y cards, he returns the three Z cards. In any other case, he returns exactly the cards that he receives.
Using the above strategy, the children either win on the three decks simultaneously (if exactly one child receives three Y cards), or loose on the three decks simultaneously (otherwise). It can easily be verified that the probability of winning is 18/27 = 2/3, exactly as in the original experiment. Hence achieving a success rate of 65/100 is normal.
The puzzle illustrates the difficulties involved in answering questions of the type: How many parallel repetitions are needed in order to reduce the probability of simultaneous success below one in a million? Though not obvious at first, answers to similar questions play an important role in proving that for certain NP-hard optimization problems, finding near optimal solutions is as hard as finding the optimal solution.
A survey on this topic is available as technical report CS95-32.