In a recent interview I received the following question (an optimisation/strategy game)...which left me a bit stumped. The rules of play, you start with 0 points, then:
Roll three fair six-sided dice;
Now you have the option:
Stick, i.e. accept the values shown on your dice as the score for your turn. There is a caveat, if two or more dice show the same values, then all of them are flipped upside down - e.g. 1 becomes 6
- reroll the dice. You may choose to hold any combination of the dice on the current value shown (so you can choose to keep 1 dice the same and then reroll the other two). Rerolling costs you 1 point – so during the game and perhaps even at the end your score may be negative.
You can roll an infinite number of times...
- So clearly the best possible score is 18 and is achieved by rolling three 1s on the first roll
- The reroll penalty prevents rolling forever to get 18.
- If the value of the dice is greater than the expected value of rerolling them (accounting for the penalty), then you should stick...
I guess what I am asking is how do I work out the expected value of rerolling them (accounting for the penalty) and how does this fit into the optimal strategy...
Thanks for all help in advance.