TITLE: How many $6$-sided and $8$-sided standard dice exist?
QUESTION [12 upvotes]: Standard means that the sum of the opposite sides is $7$ for $6$-sided die and $9$ for $8$-sided die.

REPLY [6 votes]: Another way to do the counting:
Let's start by counting how many ways we can put numbers $1$ through $2n$
on a regular $2n$-sided die if we fix the position of the die in space (do not allow rotating the die in order to change where the numbers appear).
This is just the number of ways we can permute the $2n$ numbers among the $2n$ places where they can be written;
the number of ways is 
$$(2n)! = (2n)(2n-1)(2n-2)\cdots(3)(2)(1).$$
Now count the number of ways we can orient the die (allowing it to rotate now) so that it occupies the same space as before, but not necessarily with the same numbers facing the same directions.
Call this number $|R_{2n}|.$
This is the number of times each possible numbering of the die was repeated when we numbered the faces without considering the possibility of turning the die.
So the total number of ways to number a regular $2n$-sided die, not counting rotations of the same die as different numberings, is
$$\frac{(2n)!}{|R_{2n}|}.$$
But as pointed out in another answer,
the total number of ways to number a regular $2n$-sided die
is exactly $(2n-1)!! = (2n-1)(2n-3)(2n-5)\cdots(5)(3)(1)$
times the number of standard numberings of the die
(if any standard numberings exist).
So the total number of standard numberings is
$$\frac{(2n)!}{(2n-1)!!\cdot|R_{2n}|} = \frac{(2n)!!}{|R_{2n}|},$$
where $(2n)!! = (2n)(2n-2)(2n-4)\cdots(6)(4)(2).$
There are only five regular polyhedra, for which these numbers come out to:
\begin{array}{rrrrrr}
2n & (2n)! & |R_{2n}| & \text{number of dice} & (2n)!! & \text{standard}\\ \hline
 4 &    24 & 12 &    2 &   8 &  - \\
 6 &   720 & 24 &   30 &  48 &  2 \\
 8 & 40320 & 24 & 1680 & 384 & 16 \\
12 &           479001600 & 60 &           7983360 &      46080 &      768 \\
20 & 2432902008176640000 & 60 & 40548366802944000 & 3715891200 & 61931520
\end{array}
There obviously are no "standard" numberings for a $4$-sided die, since no face has an "opposite" face, but the formulas still count how many ways the die can be numbered without that restriction.