Each year, 9 affluent acquaintances meet to compete in a unique game of chance: each creates a fair six-sided die and rolls against each opponent 100 times -- the player with the most total wins in the end receives a cash prize.
- All six side values must be non-negative integers
- The sum of all sides must not exceed 21
Examples of valid dice include [1,2,3,4,5,6], [0,0,0,0,0,21], [3,3,3,4,4,4], etc.
- Each player will compete for 100 rolls against each opponent
- Mr. A rolls 100 times against Mr. B, 100 times against Mr. C, ..., and 100 times against Mr. I for a maximum of 800 wins; Mr. B rolls 100 times against Mr. A, 100 times against Mr. C, and so on)
- In a roll, each player rolls his die once.
- The owner of the winning die takes the win.
- In the case of a tie, both players re-roll until a win is given for the roll.
This year, the host (let's call him Mr. A) has decided to flaunt a new piece of equipment. His 3D printer will create a beautiful fair die when given a list of six integer face-values. However, the host's printer has a small bug -- each player can easily hack the printer to see the most recent input.
When the time comes to create their dice, Mr. A decides to go first, creating a rather boring die with sides [1,2,3,4,5,6].
Mr. B follows, and finding himself clever, discovers Mr. A's input. Mr. B then chooses die values which are best fit to win against Mr. A. In the case of multiple winning-est combinations, Mr. B chooses the combo which wins most against its fellow winning-est combinations. Mr. B then leaves, never having considered that following players might also cheat to beat him.
All players follow likewise, choosing the die best suited to win against the previous player's die.
If all players after Mr. A discover the hack, and each player designs a die best fit to win against the previous player, which player is most likely to win the competition?
In other words, of players A through I, who is most likely to win the most rolls and earn the cash prize?