The Three Indistinguishable Dice Puzzle
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when a dice question hinges on the same indistinguishable-dice constraint, and as the source of the puzzle itself.
Recommended by 1 people across 1 different months, 2016 to 2019.
What it is
Matt Parker of Stand-up Maths poses a puzzle: given the unordered result of three indistinguishable six-sided dice, find a rule that maps the outcomes to the 36 equally likely results of rolling two dice. A separate solution video is linked in the description.
What people said
Recommends JMoravitz · 2017-09 · Mathematics Stack Exchange · on “Dice equal percentage”
@amd the difficulty comes into play when we are unable to distinguish between a "first die" and a "second die." A video by Matt Parker [this video] describes a similar puzzle (and eventual solution) where we roll three indistinguishable six-sided dice and wish to partition the results into categories to simulate having rolled only two. As such, reference to "first die" or "second die" are often not allowed and we are allowed only to roll the dice once. I'm aware that the OP hasn't explicitly stated those rules, but they make problem more interesting/useful.
Cites Axoren · 2016-04 · Mathematics Stack Exchange · on “Creating $Y \sim U[1, \dots, 6] + U[1, \dots, 6]$ as a function of $X_1, X_2, X_3 \sim U[1, \dots, 6]$”
The Three Indistinguishable Dice Puzzle from standupmaths [this video] The problem therein can be summarized as follows: You have three dice rolls. Each die is indistinguishable from the others. However, using the result of those three dice, emulate the roll of a pair of dice. Stated formally: Let $X_1, X_2, X_3$ be three independent identically distributed uniform random variables $X_i \sim U[1, \dots, 6]$. The objective is to find a function $f$ such that: $$f(X_1, X_2, X_3) \sim U[1, \dots, 6] + U[1, \dots, 6]$$ $$f(X_1, X_2, X_3) = f(X_1, X_3, X_2) = f(X_3, X_2, X_1)$$ That is to say $f$ is invariant in function argument ordering. A weaker alternative objective is to find functions $f, g, h$
Cites Arthur · 2018-08 · Mathematics Stack Exchange · on “Formula for how many possible end states any number of 6 sided dice throw can result in?”
As noted in the comments above, the results aren't equally likely. There are ways around this, but it gets messy quick. See, for instance, this video [this video] posing such a puzzle (simulate 36 equally likely outcomes using the 56 possible 3-dice outcomes), and the proposed solutions [LINK] .
Cites JMoravitz · 2019-08 · Mathematics Stack Exchange · on “Is it possible to simulate a d20 roll using smaller dice?”
A bit off topic, but I remember Matt Parker posing the challenge problem [this video] of simulating a roll of two six-sided dice using three indistinguishable six-sided dice.