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I learned the following dice game from another forum. It was not solved there. The dice game is as follows.

You start tossing six dice. After each toss you must put aside at least one of the dice tossed. You continue to toss until you have no dice remaining. You cannot reintroduce a die once it has been put aside.

In order to get a score in this game you must have retained both a 2 and a 4. You get no score for them but without them you get no score at all. Your score is the sum of the remaining 4 dice.

The question is what is the maximal expected score and how to act in order to maximize your expected score. I would guess it is optimal to put each time exactly one die aside as long you have not obtained both a 2 and a 4.

Jeff Snider
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user123981
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1 Answers1

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I don't think it is easy, but here are some thoughts. Let's first ignore the 2 and 4 requirement and think about throwing four dice for maximum sum. On the next to last throw, you throw two dice. You must keep at least one, which will obviously be the higher. Should you keep the lower? On average, it will be a $3.5$, so you should keep it if it shows four or above. Now imagine you are throwing three dice. You will keep the highest. The question is whether to keep the second. You can do a more complicated calculation in this vein to get the answer.

My guess would be to start by keeping a 2, a 4, and whatever sixes you get. When you get to three dice, keep fives if you already have the 2 and 4. This is clearly not the final answer-if you throw all sixes on the first throw you can't keep them all.

Ross Millikan
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