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So suppose I only have 6 sided dice, or D6. I want to play an RPG where a D20, or D10 is required. Can I simulate a D20 or D10 or any other sided die, using one or more D6?

abzullah
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  • You can simulate a D5 by always rerolling on one result, say $6$. So you can do a similar thing for D10 or D20, except that you have to have a way to make the results be of equal probability. One way to do this is to use two dice at a time and treat the results as being base 6, which gives a D36 if you can differentiate the two results, say with different-colored dice. – abiessu Mar 23 '19 at 03:25
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    Yea you can. Clearly, you can simulate $D_{36}$ via two rolls, so just reroll if the result is above 20. – Rushabh Mehta Mar 23 '19 at 03:26
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    Presumably you are hoping for a uniform, independent simulation... – copper.hat Mar 23 '19 at 03:29
  • I’ve provided an answer below. I conjecture, but do not know, that my technique minimizes the expected number of rolls necessary to obtain a result. Does anyone know for sure? – Robert Shore Mar 23 '19 at 03:30

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To complete the answer, there is no way to use D$6$ to emulate D$10$ with a guaranteed finite number of rolls. Suppose you roll $N$ times. There are $6^N$ possible results. However, since $6^N$ is not a multiple of $10$, you cannot divide the results into $10$ equal subsets.

antkam
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  • While guaranteed finite rolls is not possible, the finite (and, in practice, little) expected number of rolls is trivially possible. – peterh Jun 11 '21 at 18:13
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If you’re willing to roll the dice as often as necessary to get a result, you can use a D$6$ (or any other die you’d like) to emulate any die you’d like. Just think of a roll of the die (minus $1$) as the next digit in a fraction written in base $6$. There is a probability of $1$ that you’ll get a determinate result after finitely many rolls.

Robert Shore
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    You write in another comment: "I conjecture, but do not know, that my technique minimizes the expected number of rolls necessary to obtain a result. Does anyone know for sure?" Well, imagine using a d100 to simulate a d99. A very efficient way to do that is to just re-roll if you roll a 100, and otherwise take the result as-is, for an expected number of rolls very close to 1. But if I understand your technique correctly, it will require 2 or more rolls 98% of the time. – Tanner Swett Mar 23 '19 at 05:58