Will single-player Dice Farmer be solved by end of 2024?
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resolved Apr 23
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Dice Farmer is a game where you start with 3d6 and repeatedly roll dice and add and remove dice based on the rolls, as described in the webcomic Leftover Soup.

http://www.leftoversoup.com/archive.php?num=198
For the purposes of this question, assume that rolling all 1s is an immediate loss and the goal is to maximize the probability of getting 3d20 in at most 20 turns.

Will the optimal strategy that maximizes the probability of winning (getting 3d20) in 20 turns be known by the end of 2024?

For the sake of this question, we'll add two additional rules not stated in the webcomic:

  • If there are multiple ways to group your dice for purchase, and one is strictly better than another (strictly more and/or bigger dice) then you are not allowed to take the strictly worse option. If there are multiple options, none of which is strictly worse than another, you are free to choose any of them (this is where "strategy" comes into play). Realistically, this rule doesn't change the game, since there's no incentive to ever choose a strictly worse option, it's just to make things technically cleaner.

  • Example: If you roll two 4s, you can choose 1d8 or 2d4, but you can not pair them to buy a d6, because that is strictly worse than 1d8.

  • Winning is considered strictly better than anything else. If you have high enough dice rolls to win immediately, you must do so.

The combination of these rules means that the state space is large but finite, no matter what a player does. In particular, it is impossible to ever have more than 43 dice, which could happen if you roll 29 2s and then buy 13d4+d6. If you ever roll 30+ non-1s, then you immediately win because you can buy 3d20s no matter what.

An optimal strategy requires determining what the best option is for any (set of dice, set of rolls) that is reachable from the starting configuration in <20 turns by following the chosen strategy, no matter how unlikely. Additionally, since the objective is to maximize win probability in <=20 turns, the optimal choice in a given state may differ depending on the turn count elapsed.

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In light of Manifold's "pivot", I don't feel comfortable using this site, so I'm leaving and resolving all my open markets to current probability. Sorry everyone.

@ThisProfileDoesntExist what do you not like about the pivot (don't have to answer if you don't want to).

Some minor clarifications

For the sake of this question, we'll add two additional rules not stated in the webcomic:

  • If there are multiple ways to group your dice for purchase, and one is strictly better than another (strictly more and/or bigger dice) then you are not allowed to take the strictly worse option. If there are multiple options, none of which is strictly worse than another, you are free to choose any of them (this is where "strategy" comes into play). Realistically, this rule doesn't change the game, since there's no incentive to ever choose a strictly worse option, it's just to make things technically cleaner.

  • Example: If you roll two 4s, you can choose 1d8 or 2d4, but you can not pair them to buy a d6, because that is strictly worse than 1d8.

  • Winning is considered strictly better than anything else. If you have high enough dice rolls to win immediately, you must do so.

The combination of these rules means that the state space is large but finite, no matter what a player does. In particular, it is impossible to ever have more than 43 dice, which could happen if you roll 29 2s and then buy 13d4+d6. If you ever roll 30+ non-1s, then you immediately win because you can buy 3d20s no matter what.

An optimal strategy requires determining what the best option is for any (set of dice, set of rolls) that is reachable from the starting configuration in <20 turns by following the chosen strategy, no matter how unlikely. Additionally, since the objective is to maximize win probability in <=20 turns, the optimal choice in a given state may differ depending on the turn count elapsed.