       Optimization problem for dice game

• To: mathgroup at smc.vnet.net
• Subject: [mg108860] Optimization problem for dice game
• From: Peter Sisak <p-kun80 at hotmail.com>
• Date: Sun, 4 Apr 2010 07:46:08 -0400 (EDT)

```Hello,

I wish to use Mathematica to determine the best strategy applicable to the following game:

1. The game is played with 6 6-sided fair dice.
2. At the beginning of the game, the player rolls all 6 dice at once.
3. Each die may then be either a) put in hold (its value gets fixed) or b) set to be re-rolled. At least one die must be put in hold per roll. If all dice are put in hold, there are no further rolls, scoring follows.
4. On re-roll, step 3 follows again, i.e. the player decides which of the resulting dice are put in hold (at least one must be), and re-roll the rest, if there are dice remaining which are not in hold.
5. Scoring is executed in the following manner: The hand must contain at least one "1" and one "4" die. If either of these cannot be found, then the hand does not qualify for scoring (the score is 0). If it qualifies, the values of the respective dice are summed; the maximum reachable score is thus 29 (four 6s, one 4 and one 1), while the minimum score where the hand still qualifies for scoring is thus 9 (one 4 and five 1s).

Since the dice already put in hold do not change, it is trivial that the following cases must be examined separately:
a) if there are six dice that are still free, then the entire set of possibilities.
b) if there are five dice that are still free, then 1. the case when the die put in hold is a "1" 2. the case when the die put in hold is a "4" 3. the case when the die put in hold is neither "1" nor "4".
c) if there are less than five dice still free, then 1. the case when amongst the dice put in hold both "1" and "4" can be found 2. the case when amongst the dice put in hold "1" can be found but not "4" 3. the case when amongst the dice put in hold "4" can be found but not "1" 4. the case when amongst the dice put in hold neither "1" nor "4" can be found.

It would be helpful furthermore if there would be a kind of generalisation/collation step at the final "best strategy" output, since otherwise, the number of entries in the list would be huge.

Ideas?