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additional (absurdly nerdy and pedantic) point:

These layouts keep biased dice more fair by having their average rolls be as close to the expected value as possible.

A fair d20 rolls 10.5 on average, a biased d20 with these layouts rolls 10.5 on average (unless the bias is HUGE)

But that doesn't mean the biased die is now equally likely to get any number from 1-20, it still lands on some faces more often, those faces are just less likely to be always a help or a hinderance.

And depending on the game system you are using, the biased die may still be more likely to give positive or negative results, especially if it has many re-rolls, discarded values, or non-numerical meanings that aren't balanced (eg, four crit successes with only one crit fail - the system itself is skewed but a biased die could skew it more/less for an individual player)

Before I forget, reflected layouts are the same but mirrored in 3D (you can't just flip the 2D net!).

So layout R(C) is what you would see if looking at layout C in a mirror, which I have a photo of!

(the reflection has R(C) layout, but obviously the labels also appear flipped, because, mirror)

hmmm, maybe you *can* just flip the 2D net now that I think about it. It wouldn't look the same as above but it would make a rotation of the reflected layout... I think?

tbh I'm starting to get tired and don't want to share misinfo so I'll stop adding to this thread here. Everything but this point I'm super confident about though.

okay, awake Sophie is thinking clearly, you CAN just flip the net to get a reflected layout.

I've made a new image to be clearer about this (the original is still not wrong, it's just that the reflected nets shown are also rotated - which makes no difference to the layout but is an unnecessary complication for understanding)

@fletchmakesstuff I think that is cool tbh! It might make some things a little less "crunchy" which might disappoint some players, but if the group is happy with it then it's a great tool to have in the pocket. (Though I am also a person who doesn't play often these days, so, grain of salt!)

@Stellar tbh I don't think I have any less balanced d20s that don't use the standard (apart from some gamescience ones, and I don't have the energy to check the math on them rn)

But, how about a gamescience d12 with one half as unbalanced as numerically possible?

@Stellar (technically, a d12 layout can't be balanced in all the ways a d20 can, but this one for sure is almost as bad as it could be without adding extra '1's.)

@tomasekeli honestly it's not something I thought about when working on the problem. I think the discrepencies are almost always negligable and if it truly matteredwe would be doing what casino dice do; filling the pips/numbers with a material of the same density as the rest of the die.

That said, any shift to the center of mass made by void space of the numbers shouldn't change the die's overall average roll if using this layout.

@fencepost my dice app uses layout A by default for icosahedral dice.

@Sophie I pinged Matt Boyd because he's started making and selling dice, many of which have things in them.

@Nocta because of its age, it's the result of a time where you could leave your dice in the sun or the oven and they would become squashed, it was also a verrry common defect depending on who you asked (IIRC part of Lou Zocchi's sales pitch was how his dice were rounder than other makes). Opposite-face balancing is sufficient for those dice. But also it is just really tedious to search for fully balanced layouts (or even discover they exist at all). I had the advantage of a modern computer.

@Nocta and after a while, I think it just became iconic in its own right. A particularly experienced dm will be able to tell you what you rolled even if the number is faded, so long as the surrounding numbers are visible.

@Sophie Makes sense. I still feel like it would be better to update it to one of those but yeah as you said it doesn't really matter.

@Nocta it would probably result in fewer dice thrown across the room in anger, but that's never the fault of the dice. Even if some consistently roll low, they are trying their best!

@Sophie having a dad who's a poor loser, the actual chances of winning the game or anything doesn't matter they have a paranoiac confirmation bias that always make them appear as the victim anyway :b

@Nocta yeah you can try to make randomness more random, but human brains are going to do human brain things regardless

@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.

@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.

@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.

@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.

@60sRefugee @henryseg it’s pretty late for me rn, I don’t think I’ve overlooked anything, but I’ll check tomorrow and let you know :)

@60sRefugee okay so I checked and the six layouts I list share the same properties for numerical balancing that the paper describes (six vertex sums each of 52 and 53, ten face sums each of 31 and 32). As Henry said, their solution wasn’t assumed to be unique, my guess is they just had the search stop once a solution was found, while I went looking for all solutions from the start (after they had proven there was at least one to find, so I already knew I wasn’t wasting my time :) )

@henryseg @60sRefugee I know I’ve definitely claimed to have found all the working solutions, but I can’t remember where I put the code so it’d take me a while to prove it lol.