# Maximizing score in dice rolling game (part 2)

I am trying to figure out how to get the highest score in a dice rolling game. Every roll adds to your score. The player has has 3 dice. Rolling a one on any of the dice resets the score to zero and ends the game. They can be rolled in any order. They are rolled in groups/turns meaning that once all three are rolled, then they can be re-rolled, (this will be relevant later). This can be repeated any number of times and the player can end the game at any time. For example, if there are three dice, dice a, dice b and dice c, here are some examples of valid rolls:

$abc$ (can be rolled in order)

$acb$ (out of order)

$cab, cba$ (can be rolled multiple times)

$bca,b$ (you can stop at any number of rolls)

But $abb,c$ is not possible because every dice can only be rolled once in a turn.

With that established, you use one regular six-sided dice which is mandatory, but the other two dice are special and are picked from three options:

1. a dice that rolls 2-6
2. a dice that rolls 1-6 but doubles the value of the roll (rolling a one still resets the score)
3. a dice that rolls 1-3 and multiplies the score of that turn by that amount (rolling a one in this case does not reset the score)

So my question is which combination of dice and how many rolls would result in the highest average score?

• So die #1 has only 5 sides? Are the faces equally likely? Same question for die #3. Also, could you give a couple examples of play to make sure I understand your description? It is not clear to me what the commas in your examples of valid rolls mean (i.e., how is bca,b different from bcab?) Cheers! – Matthew Conroy Jun 9 '17 at 20:22
• @MatthewConroy yes technically it would be 5 sided. Each face is equally likely. The comma indicated that you've rolled all of your dice and you pick them up to roll again. You can't roll bcab because after rolling dice a, you've rolled all your dice so you have to pick them all up to roll further – Ryan Jun 9 '17 at 20:34
• @MatthewConroy consider rolling a 4, 5, x2, then rolling 2, x3, 6. The x2 multiplier applies to the 4 and 5. The x3 applies to the 2 and 6. This is why it matters that the dice are separated in threes. – Ryan Jun 9 '17 at 20:38
• Do you roll the dice one at a time, or all three at once? If one at a time, can you stop after any one, or do you need to roll three, then get the option to stop? What would be the score of the example you give in the comment above? With each group of three, can you change which dice are thrown, or do you have to pick a set for the whole game? – Matthew Conroy Jun 9 '17 at 21:23
• @MatthewConroy good question, you can roll one at a time and stop at any point. The score would be 18 for the first set, 24 for the second, totaling 42. You have to stick with the same 3 dice the whole game. – Ryan Jun 9 '17 at 21:38