I came across this page late last night. Being a physicist, I thought of a sophisticated mathematical model to determine the algorithm. It certainly would have worked, but required a lot of input data to fix parameters. Even then, it would only have provided a predictive formula to arrive at the correct answers -- some inductive "zen"
would still be needed to connect the rule to an elegant and explanitory solution that the puzzle's inventor no doubt had in mind.
While thinking about this in bed, I wondered if it would be possible to solve the problem without any data at all. If so, I thought, the only things left to go on would be the even-ness of the outcomes and the name of the game, "Petals around the rose..." The insight came immediately thereafter!
