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Sophie 🎲🏳️‍⚧️

I've been meaning to make an updated version of this for a while so here it is, every numerically-balanced d20 layout!

Feel free to share, especially with any dice makers you think may be interested!

There is a more detailed explanation video by @henryseg for The Dice Lab (which use layout C, if you're curious!) youtu.be/Nh2H_4g6evc

An illustration of 12 icosahedron nets where each face is numbered from one to twenty. The nets are labelled from A to F and R(A) to R(F). Below is the text: "There are only six number layouts (and their reflections) for a d20 that are optimally fair. The most common layout (where the 20 is surrounded by 2, 14, 8) is not numerically balanced, apart from by opposite-face sum (Which only helps negate bias from a squashed shape). d20 dice with the above layouts should roll the expected value (10.5) on average, even if their shape is squashed (common for very old, cheap, or melted dice), their center of mass is offset (common in dice with inserts, multiple materials, air bubbles), or they are not fully isohedral where faces/edges are slightly different (common for handcrafted gem/glass/metal dice, and dice not evenly tumbled). Key features: • Opposite-faces always sum 21 (two faces with an average value of 10.5) • Faces around each vertex always sum 53 or 52 (five faces with average value of 10.6 or 10.4.) • A face’s three adjacent faces always sum 32 or 31 (three faces with average value of 10.666… or 10.333…) Feel free to use any of the layouts above for your own dice to make them fairer overall!"
#dice#ttrpg#DnD
Jigme Datse

@Sophie @henryseg I've really wondered about this... Thank you. I mean... It seems as the number of faces increases, it's harder to have a really fully balanced die.

Sophie 🎲🏳️‍⚧️

@jigmedatse from what I recall from my searches, it’s not necessarily the number of faces that is the issue (a d120 can be perfectly balanced numerically, a d12 cannot), though it definitely effected the search time it takes to find those layouts.

IIRC, I was starting to conject that the key thing to whether it was doable was the relationship between how many faces surround a vertex relative to how many faces are adjacent to a face. I have no real idea though tbh.

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Jigme Datse

@Sophie Yeah, mentioning the d12, was bringing up what I was thinking about that, about how a d4 really can't be, but probably doesn't matter. Or... It is intrinsically, depending on how you think about it.

Sophie 🎲🏳️‍⚧️

@jigmedatse any d4 is simply as balanced numerically as it can be, there is only one possible layout (and its reflection). The vertex sums will always be 6, 7, 8, and 9, and it doesn’t have opposite faces. There just isn’t much to work with.

A fun thing though is any d6 is also optimally balanced so long as the opposite faces sum 7. There’s only one layout (and its reflection) that match the criteria- so even most ancient dice use the same layout as modern dice.

Jigme Datse

@Sophie Yeah, like I was trying to say... The d4 really can't be any different so you either call it balanced, or not balanced, because there's no different ways to handle it. Wasn't really aware of the d6, but I guess thinking about it, yeah that makes sense.

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