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Background

This question is inspired by this 538 "Riddler Classic" puzzle, and the following puzzle explanation below is copied from there:

You have four standard dice, and your goal is simple: Maximize the sum of your rolls. So you roll all four dice at once, hoping to achieve a high score.

But wait, there’s more! If you’re not happy with your roll, you can choose to reroll zero, one, two or three of the dice. In other words, you must “freeze” one or more dice and set them aside, never to be rerolled.

You repeat this process with the remaining dice — you roll them all and then freeze at least one. You repeat this process until all the dice are frozen.

If you play strategically, what score can you expect to achieve on average?

Extra credit: Instead of four dice, what if you start with five dice? What if you start with six dice? What if you start with N dice?

Question

We are interested only in the general $n$ case, and we are interested in a question tangential to the puzzle's solution.

Assuming perfect play, consider the expected value (EV) of your sum when starting with $n$ dice, which we'll denote $E_n$.

The question is: Is it possible that, for some $n$, the following holds?

$$E_{n+1} > E_n + 6$$

Alternatively, should you ever choose to re-roll a six when playing optimally?

Intuitively, it seems like the answer to both of these equivalent questions should be "no". But I cannot think of a rigorous argument to prove it.

RobPratt
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Jonah
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    For what it's worth, I started to leave a comment with hints, and then realized that I was badly underestimating the problem's complexity. Interesting problem. – user2661923 Aug 22 '21 at 01:28
  • I don't yet see a proof for an answer of "no" to either question, but I am pretty sure I can prove that they have the same answer. Would that help as a partial answer, or are you already aware of this fact? – Misha Lavrov Aug 22 '21 at 01:40
  • @MishaLavrov yes that was why i choose to say “alternatively“ meaning it in the sense of “equivalently” – Jonah Aug 22 '21 at 01:45
  • What does "perfect play" mean? – Annika Aug 22 '21 at 02:11
  • @Bey It means that at every decision point (which dice and to hold and which to re-roll) you make a decision that maximizes your expected value. – Jonah Aug 22 '21 at 02:18

1 Answers1

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I think this problem has been discussed before. The top-rated answer by joriki seems to suggest that $E_{n+1}-E_{n}$ could indeed be greater than 6 if you never reroll 6s.

user960549
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