Professor Egghead participates today in a local game show. There are six jewels (a diamond, an emerald, a moonstone, a ruby, a sapphire, and a topaz), and there are six boxes that carry the numbers 1, 2, 3, 4, 5, 6. Each of the boxes contains exactly one of the jewels. The assignment of jewels to boxes is totally random (and of course unknown to the professor), and each assignment is equally likely.
According to the rules of the game show, Egghead announces three guesses for each of the jewels:
- I guess that the diamond is in one of the three boxes D1, D2, D3.
- I guess that the emerald is in one of the three boxes E1, E2, E3.
- I guess that the moonstone is in one of the three boxes M1, M2, M3.
- I guess that the ruby is in one of the three boxes R1, R2, R3.
- I guess that the sapphire is in one of the three boxes S1, S2, S3.
- I guess that the topaz is in one of the three boxes T1, T2, T3.
After the guesses have been announced, the game show host opens all six boxes. The professor wins a fantastic prize, if each of his six guesses turns out to be correct.
What strategy should the professor apply?
What is his winning probability under an optimal strategy?