# Is there a pattern for d8 dice?

I've just bought a d8, and noticed that the number arrangement is different from my other one. My new die has only even numbers on one half, and only odd on the other half. My old die has 1,8,5,4 on one half and 2,3,6,7 on the other.

Is there any correct pattern for it? Does it make any difference?

• There are three (obvious) ways to divide a d8 into two halves. It's perfectly possible for both of your descriptions ("only even / only odd" vs. "1,8,5,4 / 2,3,6,7") to correctly describe the same die. Commented Apr 11, 2014 at 21:10
• It's also possible that if you re-orient the die so that your "half" mid-plane falls along a different axis, you might see all even and one half all odd - but, with the numerals turned in a different direction. There are six "ups" on an eight sided die, depending which of the six corners you call the top and bottom, and the direction of the numerals only match one (or two) of those six ways. Your die might already be configured the way you're asking, except for the orientation/rotation of the numerals. Commented Aug 30, 2018 at 21:28
• @KRyan Please avoid what is essentially a personal accusation of ill-intent. You can disagree with an opinion and teach the difference, but I hope you see how (what reads to me as essentially) doom mongering doesn't really help anyone. All comments like that achieve is reducing everyone's desire to state why they're voting to close, and that would eliminate our ability to communicate and teach. Commented Aug 12, 2021 at 17:05
• This is not inviting opinions for answers. If there's no specific answer, an answer can factually say so. Commented Aug 12, 2021 at 19:21

There's no correct pattern, and in a fair die it makes no difference.

In practice, most standard dice sets are not entirely fair, because of how they're tumble-polished after removal from the molds.† How an unfair die is numbered can make it more unfair if the manufacturer decided to cluster all high (or all low) numbers near each other. Few manufacturers do this though, exactly for that reason—both your dice mix high a low numbers in each half, just in different ways.

However, the imperfect shape caused by tumbling affects the "rounder" dice more (such as the d20), and it's unlikely for a d8 to suffer noticeable unfairness.

† Due to demand for fairer dice, manufacturers such as GameScience and Chessex make dice that haven't been tumble-polished.

• Not sure how to update this comment, but Chessex no longer appears to sell raw dice. Commented Mar 6, 2020 at 3:06

There is no "correct" pattern, but a very common one, seen on 6-sided dice, and sometimes extended to higher-sided ones is to pair up numbers on opposite sides.

For 6-sided dice it is normal to have sides paired 1,6 / 2,5 / 3,4, each pair adds to 7. There are two different ways of having this arrangement. If you take a large number of 6 sided dice from different sources it is very likely you will find you can group them into two sets that both follow this arrangement, but which vary in how you can rotate them to match.

For 8-sided dice, it seems common to have sides paired 1,8 / 2,7 / 3,6 / 4,5 but I have seen exceptions (which are generally fine, you should not be concerned if it is not true for a die you own). There are 16 possible to arrange those pairings, see below for a full list.

For why this arrangement is common, many sources claim that it helps to keep averages correct for imperfectly-shaped dice. If your 'cube' is shorter on one edge, it would favour two opposite sides - if those sides don't give you an overall advantage, this is seen as fairer. Although it may also have been driven by numerology, and the design kept by tradition.

Identifying D8 variations (those with opposite sides adding to 9).

Rotate the die so that you are looking down onto a point with the 1 at "12 o'clock" and with the 2 at either "3 o'clock" or "6 o'clock" (one of these positions will be possible, if not then you have already got 1 and 2 on opposite sides, which of course makes the total 3). Read the numbers, starting from the 1, in clockwise order.

The following variations are possible:

• 1-2-3-4
• 1-2-3-5
• 1-2-4-3
• 1-2-4-6
• 1-2-5-3
• 1-2-5-6
• 1-2-6-4
• 1-2-6-5
• 1-3-2-4
• 1-3-2-5
• 1-4-2-3
• 1-4-2-6
• 1-5-2-3
• 1-5-2-6
• 1-6-2-4
• 1-6-2-5

These variations are the complete set for opposing sides summing to 9. It is not possible to take one of them and then rotate the die so that it exactly matches another one.

If you see one of the above variations, it does not prove that the opposite sides sum to 9. If you are not sure, then you need to check that also. And worth repeating: It is not a problem if you find a d8 that does not have one of the above patterns or does not have opposite sides sum to 9.

• I believe there are actually 16 distinct d8 patterns that satisfy the "opposite sides sum to 9" constraint. Specifically, the sides numbered "1" and "2" can share either an edge or just a corner. (They can't be opposite due to the constraint.) Either way, rotate the die so that "1" is facing you and "2" is to the left of it; that's enough to fix the orientation of the die, and leaves four choices for the placement of the "3" side and, once that's determined, two choices for the placement of the "4" side, for a total of 2 × 4 × 2 = 16 unique configurations. Commented Apr 12, 2014 at 21:19
• @Ilmari Karonen: That's good logic. I'm going to have to double-check now, my original thinking was just finding an N!/M! style formula that made sense to me and fit 2 for a 6-sided die :-) Commented Apr 13, 2014 at 6:33

After a lot of research, I drew my own conclusions — it seems that there are only two patterns available:

• 1,3,5,7 / 2,4,6,8 where the opposites sides sum to 9. I think this is the correct distribution, and it is used in brands like Chessex and GameScience.
• 1,4,5,8 / 2,3,6,7 where opposite sides don't sum to 9. This distribution, though, is used by the Chinese brands. So that's one of the reasons why it is so common to see dice with this pattern.

I tried to find some extra information related to why this is happening because this is the only die size that has this happen. All the other dice have standard distributions: D6 sums to 7, D10 (starting with 0) sums to 9 (as well as the percentile dice that sum to 90), D12 sums to 13 and D20 sums to 21. This is on any brand or manufacturer.

The current common choice is one of the configurations which implement the opposite-faces convention: values on opposite faces sum to one more than the number of faces. In the case of the d8, the sum is nine.

## The mathematical possibilities

Bosch, Fathauer, and Segerman offer a thought experiment to explain how the opposite-faces convention preserves averages for imperfectly shaped dice.

There are 8!=40320 ways to map numbers 1...8 on the regular octahedral die (d8), but many of these configurations are rotationally indistinct. The size of the symmetry group |G|=24. There are 8!/|G|=40320/24=1680 rotationally distinct configurations. This count includes mirror images. If we conflate the mirror images, then the count reduces by half to 840 symmetrically distinct configurations. Rolling a fair d8 in a perfect world means that dice makers may choose any of the configurations. But, the world is not perfect. In our imperfect world, dice makers have chosen a few configurations, and the chosen few have appeared on the market and later in dice collections.

As Slater listed above, there are 16 configurations:

• There are 8 ways to place "1" on an arbitrary face; this placement determines the opposite face.
• There are 6 ways to place "2" on a remaining face; this placement determines the opposite face.
• There are 4 ways to place "3" on a remaining face; this placement determines the opposite face.
• Finally, there are 2 ways to place "4" on a remaining face; this placement determines the opposite face.

So, there are 8!!=8×6×4×2=384 configurations implementing the opposite-faces convention, but many of these configurations are rotationally indistinct. Again, divide out the symmetry: 384/|G|=384/24=16. These 16 include the mirror images. If we conflate the mirror images, then the count reduces by half to 8 symmetrically distinct configurations.

## Actual dice

Even so, dice implementing configurations other than the opposite-faces convention are available. Earlier this year, I bought a 1-8-5-4 d8 from GameStop. A visual inspection of the faces on the six square pyramids reveals that this die does not implement the even/odd split; each pyramid has two even and two odd numbers on its faces. Nor does it implement the high/low cluster.

opposite faces
------------------
8         7                  1/2, 3/4, 5/6, 7/8
5   1     2   6     opp sum:    3    7   11   15
4         3

Alea Kybos’ Dice Collection illustrates variants of the 1-8-5-4 d8. The variants involve keeping the 1-8-5-4 pyramid fixed while rotating the 2-3-6-7 pyramid by 90 degrees:

8         6                  1/7, 2/4, 3/5, 6/8
5   1     7   3     opp sum:    8    6    8   14
4         2

8         3                  1/6, 2/5, 3/8, 4/7
5   1     6   2     opp sum:    7    7   11   11
4         7                  high/low cluster

8         2                  1/3, 2/8, 4/6, 5/7
5   1     3   7     opp sum:    4   10   10   12
4         6

More details appear here.

It sounds like you have a dice designed for tracking numbers (life totals, mana, money, etc.) moreso than for rolling.

i.e. A 20-sided dice that goes from 1 to 20 sequentially.

I've haven't seen an 8 sided die before that wasn't the "1,8,5,4 // 2,3,6,7" combination before as life total die are normally higher using d10s and d20s. When you have a die that doesn't have the smallest paired up with the highest values, there is the chance to increase the odds of a better than average roll depending on how you throw it. That said, if you're not trying to cheat with it and throwing it intentially so it rolls to one side over the other it shouldn't matter.