Mathophobia

The Devil challenges you to a dice game he calls Six-Six-Six. The stakes? If you win, you get an answer for the MIT Mystery Hunt. If you lose, the Devil takes your soul. Before committing yourself, you ask the Devil to explain the rules. He does:

When a game of Six-Six-Six starts, you each pick an integer, which is referred to as your "base number." The Devil will pick 666 as his base number, of course. You may pick any integer you like as your base number.

Each round, three dice are rolled. Each of you uses the three resulting numbers as three digits that you may order in any way you wish. You each add your newly-constructed three-digit number to your own base number. Your score for the round is the absolute value of the difference between your sum and 1,000. Likewise for the Devil. When the game is over, all of your individual round scores are added together. The person/being with the lowest total score wins.

Example round: Your base number is 536. The three dice read 5, 2, 5. You may interpret this as 255, 525, or 552. The resulting scores would be 209, 61, and 88, so you chose 525, and score 61 for that round. The Devil would interpret this as 255, add this to his base number of 666, and he would score 79 for that round.

But wait . . . it's a tad more complicated than that. If on any round the dice show three sixes, then you are penalized for rolling the Devil's number, and your score for that round is 700 points. The Devil is never penalized -- he uses the usual scoring rule no matter what the dice say. You comment on the unfairness of this, but the Devil points out that you will still beat him in the long run IF you pick a good base number.

Something about the way he says that makes you suspicious. "Long run?" you ask. "Could you be a little more precise?" "Sure," the Devil replies. "A game of Six-Six-Six consists of six times six times six rounds." You protest, "But I don't have time to play that many rounds! I have a Mystery Hunt to finish."

The Devil thinks for a moment, then offers an alternative. Instead of playing a whole game, you may compute the expected results for perfect play, as follows:

Determine the best base number (B) to pick.
Determine your expected score (S) for a ROUND if you were to use this base number.
Determine your expected margin (M) of victory over the Devil if you
were to use this base number for a GAME of Six-Six-Six.
Reverse the digits of M and square the result to get N.
Reverse the digits of B and square the result to get C.
Reverse the digits of S and square the result to get T.
Take the first two digits of T to get U.
Write out the three numbers C, N, and U (in that order) digit by digit. There will be 13 digits in all. Add each digit to its respective number in the table below, then correctly interpret the results.

1. 8
2. 5
3. 11
4. 11
5. 17
6. 15
7. 9
8. 5
9. 10
10. 20
11. 0
12. 3
13. 17

So there you have it. Do a bit of analysis and computation and you'll have another Mystery Hunt answer. Make a mistake and you'll be spending a good percentage of eternity (98.32%, to be precise) with the Devil. Hope math doesn't make you nervous...