Another D6 Dice Mechanic -- Equiprobable Choices

When playing a game using D6 -- and only D6 -- we're sometimes confronted with a need to make a selection from equiprobable items.

For example, pick a random day of the week: 7 choices, but only 6 sides to a die.

What can we do?

How can we select fairly among 7 choices using only a handful of 6-sided dice?

See this extract from a Jupyter Notebook: Dice Mechanics: Equal Probability Outcomes from D6.

For direct computation of the exact distribution of a handfull of dice, we need a lesson in multinomials.

See https://towardsdatascience.com/modelling-the-probability-distributions-of-dice-b6ecf87b24ea/. From this we learn the following (amongst other things). For \(n\) dice of \(s\) sides (6 in our case), the probability of getting the target value \(T\) is this:

\begin{equation*} P(n, s, T) = \Bigl(\sum\limits_{k=0}^{\lfloor \frac{T-n}{s} \rfloor}\bigl(-1\bigr)^k \frac{n!}{(n-k)!k!} \frac{(T-sk-1)!}{(T-sk-n)!(n-1)!}\Bigr)\Bigl(\frac{1}{s}\Bigr)^n \end{equation*}

Which has a lot of maths, but is very helpful.

TL;DR

Mechanic Alternatives
Outcomes	D6 Mechanic
2	1D [1-3, 4-6]
3	1D [1-2, 3-4, 5-6]
4	1D with reroll {5, 6} or 2D with weights of 2, 1
5	1D with reroll {6} or 4D [4-10, 11-12, 13-14, 15-16, 17-24]
6	1D Do we even need to include this?
7	Use 8 outcomes with a reroll {7} or 5D [5-12, 13-14, 15-16, 17, 18-19, 20-21, 22-30]
8	4D [4-9, 10-11, 12, 13, 14, 15, 16-17, 18-24] or 3D with weights of 4, 2, 1
9	5D [5-12, 13-14, 15, 16, 17, 18, 19, 20-21, 22-30] or 2D with weights of 3, 1
10	3D [3-6, 7, 8, 9, 10, 11, 12, 13, 14, 15-18] or 5 outcomes combined with 2 outcomes
11	Use 12 outcomes with reroll {11} or 5D [5-11, 12-13, 14, 15, 16, 17, 18, 19, 20, 21-22, 23-30]
12	Use 6 outcomes, combined with 2 outcomes
above 12	Decompose into 2 or more tables; use a chain of rolls