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What Do You Do with 120-Sided Dice? (newyorker.com)
107 points by anthotny on April 26, 2016 | hide | past | favorite | 88 comments

When I was a kid, I played RPGs (AD&D, mostly) a lot, and we all loved the oddball dice. 30 sided die and 100 sided got a lot of use. The 100 sided one was a dimpled ball with some sort of pellets inside (to make it stop rolling faster). I'm not sure how fair it was, but it was fun. There were, at the time, several little books of random/funny things that could happen in the game based on a 30 sided die roll; battle results, random encounters, etc. Published by the dice manufacturer, I presume.

I long ago got rid of all of the books and boxed sets. But, I still carry the little faux leather bag of dice from house to house. Not sure why. Something about the feel of the dice has the strongest memory association with that time and those friends.

I don't really have anything to say about this particular die, but if I still played RPGs, I would find something to use it for. I would have bought it in a heartbeat, as a kid.

What with one thing and another I've long since lost the collection of polyhedral dice I accumulated as a teenager. And I don't have time in my life for RPGs. But I have a modest collection of dice stashed away. Including weird-ass ones like big transparent D10s with smaller red D10s inside.

They're just kind of... totems, I guess. Symbols of the kind of nerd I am.

I have absolutely no earthly use for this damn thing. But I ordered one. And a set of these folks' interestingly-shaped d4/8/12/24/60, as well. And their "numerically balanced" d20. I may never use any of these dice. But there is a part of me that is intensely satisfied by owning them.

You captured a particular feeling I have as well - perfect!

Diceware and password generation

If you still played you could create a table with 120 random encounters :) Really more a tool for the DM, I guess.

It functions (with easy math) as a d2, d3, d4, d5, d6, d8, d10, d12, d15, d20, d24, d30, d40, d60, d120. Almost covers all the bases for DCC.

>as a d2, d3, d4, d5, d6, d8, d10, d12, d15, d20, d24, d30, d40, d60, d120

Thank you! Reading the submission title, I was like, "um, replace every other die"?

This same phenomenon, BTW, is precisely why base-12 is far, far, hilariously far better than base-10, which is somewhat sad and pathetic by comparison.

The mathematical properties of the distribution of the numbers on the sides are very pleasing.

They remind me of the (absolutely fascinating) "Go first" dice[1], any subset of which are guaranteed to produce a perfectly fair ranking.

1 - http://www.ericharshbarger.org/dice/go_first_dice.html

Why not use a single d24?

It takes some of the fun away not to give everyone their own die to throw.

Snarky answer is to divide by 20. But I wonder if a die that large is even fair. Presumably a lot of the randomness in a coin or 6 sided die is that each bounce can flip the object multiple times, generating a roughly uniform distribution after a few bounces. If its too large, it may only reach a portion of the neighboring values at each bounce.

only reach a portion of the neighboring values at each bounce

From the picture of the dice[0], it looks like it would still be fair enough due to the fact that there is a large difference in the numbers available 'nearby' any one spot. But since it's big (but not super big), I'd guess that it would probably be rolled towards some sort of back stop, which would allow for additional opportunities for it to bounce.

[0] - http://www.newyorker.com/wp-content/uploads/2016/04/Roberts-...

One of the problems with a d100 that I saw is that it can be rolled along an axis somewhat reliably. If the numbers are well distributed, the average and those above/below average may still be sufficient. Specific numbers would be avoided by putting them at the poles, or others given greater chance by putting them along the equator of the roll.

I have not seen the d120, I don't know if the points would bounce it off its axis or not.

If you want a random value in the range, that's probably fine, but it would be bad for a roulette-style game, where you bet on a single, specific number. If you can indeed roll it on an axis reliably, you can increase your odds significantly.

True. The problem I remember about a friend's d100 was that it was basically a golf ball and the damn thing rolled and rolled and rolled...

Yes, it always was a neat gimmick, but in practice, using two ten-sided dice was a far superior approach. d30s were bad enough, but at least with them simulating (e.g. with a d6 and d10) wasn't as intuitive.

According to the article, every vertex that sticks out has 10 triangles whose sum is 595. So it is as fair as can be managed at any point.

That enforces that the mean value should be close to 60, but doens't say much about whether it is fair in that it should have also have a uniform distribution.

> it looks like it would still be fair enough [if it only reaches a portion of the neighboring values at each bounce] due to the fact that there is a large difference in the numbers available 'nearby' any one spot

what is this, NSA logic day? :-D

(This is my way of saying that your argument actually results in really low randomness. It means the random process has some memory and has far less distribution of states than you would expect. your argument is like saying a good way to get a random number between 1 and 1,000,000 is to flip a coin, then heads it's 170,158; tails it's 587,439. There's a large difference between 170,158 and 587,439 but those two numbers wouldn't be random AT ALL, because all the other numbers couldn't come up. you're saying the die would 'still be fair enough' because the numbers that can come up are far from each other... well, no...)

You're reading that post wrong.

No matter how you lay out the numbers, a nice hard roll will give you a completely uniform result. The goal here is to make it harder to cheat when you can partially control how the die rolls.

Nobody is pretending there is more entropy than the number of outcomes. What we want is a design that can't be influenced in favor of the roller, even when entropy is a bit low.

Let's design two layouts for a d100. We'll say that you can roll each design two ways. You can roll it properly and get a random number from 1-100, or you can roll it improperly and it will stay within a cluster of 10 numbers, giving you a random number out of that cluster.

The first layout puts similar numbers together. 1-10 are all together, as are 11-20, etc.

The second design is laid out so that every tens digit show up in each cluster, and every ones digit shows up in each cluster.

Both designs can give you equally random numbers. A proper roll gives you a uniform distribution over 100 numbers, and an improper roll gives you a uniform distribution over 10 numbers.

But look at what happens when you apply certain functions to the output. A common use for the d100 might be emulating a d10. If you divide the first die by ten, a proper roll acts like a d10, and an improper roll lets the cheater pick what result they want. With the second die, a proper roll acts like a d10, and an improper roll also acts like a d10.

Fairness is preserved when the imperfections in the randomness don't matter. Your coin method looks like it would be a fair way to pick lottery numbers if it was only used once. But it would be a fair way to pick even or odd indefinitely.

I don't think I'm reading that post wrong! The original poster said " But I wonder if a die that large is even fair." Then the response said it's "fair enough" because even if it only spins part of the way, you'll still get one of a group of dissimilar numbers.

but that's not fair by the definition of what a fair die is. (it has a definition, if you search google you get "In mathematics we say 'fair dice' when we mean that there is an equally likely chance of landing on any face.")

You're basically arguing that dice don't need to be fair. also you completely missed my jab at the NSA breaking RNG's while making them seem "random enough", i.e. getting to quite dissimilar next states while in fact having only a few possible next states.

It's "fair enough" because it gets the important parts right of having a large number of states. Just like your coin flip would get the important parts right if it was used in specific ways.

A coin in the real world isn't fair, because it could land on its side. But it's fair enough.

Statistical sampling doesn't give you guaranteed-accurate results. But the biases cancel out across the metrics you're measuring, so it's good enough.

I didn't ignore your jab at the NSA, but it's pretty weak. The backdoored CSPRNG doesn't give numbers that are worse than any non-backdoored CSPRNG. Depending on your perspective, it can give you uniformly any number, or it can give you only a single number, and the same is true of good algorithms too.

I enjoyed this perspective and it changed my thinking, you've convinced me. It's also unlikely that humans can keep track of the distribution of 120 numbers on a die (in any distribution) so even if the game the cheater is playing is "Pick 20 numbers, then roll. If you roll any of the 20 you win" it is unlikely that they can remember what the die looks like well enough to cheat - except perhaps after massive effort. Fair enough :)

I'm not really sure what the argument here is.

Given the fact that each vertex of the die has 10 triangles that sum to 595, and including a non-malicious roll against a backdrop, I don't see how this die is any more or less fair than a normal d6 (which I believe we can concede is 'fair'). After all, if you want to cheat at dice, then you can cheat at dice regardless of any common understanding of 'fairness'.

In particular, I was responding to the following comment, "If its too large, it may only reach a portion of the neighboring values at each bounce".

My argument was only that if you drop the dice from say a distance of 3", and it doesn't bounce then at a minimum, you wouldn't get a large grouping of numbers that are all near each other (Even if you were able to somehow pick a specific 'vertex' to always land up, you're going to have a total average of 59.5). In other words, you couldn't drop the die such that you could guarantee a range of 110-120 (which you could for example do on a MTG spindown die, which is fair but not uniform). In other words, even if there is no bounce, a typical rolling of the dice would be 'good enough' for say a pen and paper RPG, and even just dropping it without the bounce is likely enough to give non-Vegas levels of 'fairness' even if 'it may only reach a portion of the neighboring values at each bounce'.

mathematically, if your "range" is one of the ten numbers 108, 43, 21, 18, 32, 40, 79, 50, 94, 59 then it is just as broken as if your "range" is 110-120. it's not considered fair. fair means random and evenly distributed. (when it comes to dice and coins.)

You are confusing the definitions of "fair" and "random".

fair means random. (for dice, coins, etc)

not sure how someone can downvote me, if you google "define dice fair" you get "In mathematics we say 'fair dice' when we mean that there is an equally likely chance of landing on any face", i.e. random and equally distributed.

When I was a kid, I played D&D once or twice, and noticed on one of the dice, that the edges were somewhat rounded off, and the faces had different areas. I don't know if it was by intent, but I was pretty sure that those dice would favor certain faces.

Oh, nice.

I used to make up RPG rules for fun and one of my systems used a die scale that went: d20, d40, d60, d80, d100, d120.

Those are just die sizes that are easy to make with the commonly available dice: a d20 is a single die, a d40 is a d4 and a d10 (reading units on the d4 and tens on the d10) and so on.

The thing is that d[20,40,60,80,100,120] is a nice, smooth scale, unlike what you get if you use the d[4,6,8,10,12,20] on their own.

And that works alright, but it's always nice to have the exact sizes.

"What do you do with 120-sided dice" indeed! Roll, them, duh. What else?

We went through rolling d100 using three d10. One die determined which of the other two was the tens die. It was odd, but some people thought it gave them an edge and why argue with a harmless tweak?

That seems odd, maybe because I am used to the "conventional" way of rolling d100: roll two d10s, where the first die is concatenated with the second. A roll of 0 on both dies indicates 100. Some d10s are even sold with a zero tacked on, so you actually see 10, 20, 30, etc, instead of 1, 2, 3, etc... just so there is no question as to which is the "tens" die.

> 10, 8

Me: "Sweet! I rolled 18." Friend: "Er, that's 9." Me: "... shoot."

I'd have noticed it sooner if it had been one of the other sides. :)

We usually used different colours and specified just before rolling which was the tens.

We would always roll two d10 for 1-100. You would call one as the "tens" die before you rolled, but inevitably, this would be a great opportunity to fudge things a bit.

Rolled a 19? Oh.. well actually I meant that the RED one was the tens. Now it's a 91! :)

A bag of dice lives in my backpack, as it serves as the Ultimate Decision Making Tool.

Sometimes there is just no obvious good decision, even after all the options have been analyzed and considered.

So whenever I -- or my team -- gets stuck at one of those no-idea-which-road-to-follow points, we roll the dice.

Because a bad decision is usually better than no decision.


I'm going to pick one up and head straight over to The Embarcadero: "Guess the number and roll the dice! Pays $10! only $1 to play!"

You could pay $100 and still have an edge.

Careful, the Lottery commission might come after you ;)

Buy a Dice Labs' 120-sided dice (d120) here: http://thedicelab.com/d120.html

Thank you! I read the whole article but couldn't find the "here's where to buy it" link. (Maybe I just missed it.) I have no need for a d120 but I have the WANT for one.

EDIT: check out other DiceLabs creations, crazy stuff in there. Link:


There was no link to buy it! Just a mention of Dice Labs. That or I missed the "buy it here" link myself.

I found it a little strange since the purpose of the article seems to be to try and sell some pre-orders of the die but there wasn't a link to go order it.

Here is the store page: http://www.mathartfun.com/DiceLabDice.html

Click on 'Add to Cart' for 'd120 Disdyakis Triacontahedron'

Thanks - but I was just replying that they hadn't missed a link. I've already placed an order for a few dice. :)

You're right, the "skew dice" look awesome... :-)

Couldn't you just use a dimpled ball to get however many numbers you'd like?

No, because the group G of rotations of the 2-sphere that preserve the set of all "dimples" X (so for all g in G, g(X) = X) on the sphere has to have a member g(x,y) in G for all x, y in x such that the action of g(x,y)x = y; that is, any point can be transformed into any other point by a rotation that nonetheless preserves the whole lattice. This is called face-transitivity* and there are a finite number of recursively enumerable sets of face-transitive solids: the bipyramids, the trapezohedra, the prisms and antiprisms (as log dice), and the Catalan solids. Bipyramids and trapezohedra are quite impractical at high face counts because they take on a biconical shape whereas spheroids are ideal. Log dice are fair if they roll a lot, but they're not as cool.

Yes, but the currently-manufactured ones tend to be unfair: https://en.wikipedia.org/wiki/Zocchihedron

The advantage of polyhedral dice is that congruency of faces implies fairness.

You could also make a cylinder and roll it, or a top and spin it.

Or a bag filled with numbered tiles, but a dice is a dice not the general category of random generators.

I have seen both your suggestions used in games so fair enough, and the cylinder reminds me of the inelegance of the 10 sided dice, but that had no end face to land on like a cylinder.

You start looking for a really flat, hard surface. It should ship with a polished granite tile and a level.

Technically I think you could make any n-sided die for even n > 2:

1) Take an n-gon cylinder.

2) Smooth off the two ends to be spherical so that it never lands on either.

Now all sides have equal probability of facing up after a roll.

You can even make it polygonal. Even more interesting, you can make it non symmetrical, contradicting the article's claim that dice must be symmetrical to be fair.

You make an (n-2)-gon cylinder. This has n faces (the sides, the top, and the bottom). If you make the top and bottom very small, the probability of landing on them is smaller than the probability of landing on the sides. If you make the top and bottom huge (compared to the sides) the probability of landing on a side is smaller. Somewhere in the middle, the probabilities are balanced.

I'm boycotting the New Yorker for failing to mention D&D even once in the article.

I would like one the size of a softball. It's pretty cool looking to put on your desk and roll when you get stuck.

The article states no higher sided fair dice can exist:

"“ultimate fair die allowed by Mother Nature (i.e., mathematics!),” since a die couldn’t, practically speaking, possess more sides or more symmetry, and dice must be symmetrical to be fair."

Mathematicians is this true ? Or only "practically speaking".

Yes, that is true. See here for a full explanation: https://www.youtube.com/watch?v=uAnCL3vhVIs

As noted elsewhere, bipyramids and trapezohedra can have unlimited (even) numbers of sides - not very practical though.

Extend the magic item table?

In 4 dimensonal space, the 120-cell would make a d120 that's completely regular, like a platonic solid:


If you are going to go 4 dimensional, you'll be wanting the dual of an omnitruncated 120-cell, giving a d14400:


I had a modem like that once.

It seems to me that the fewer sides, the more "perfect" the die can be, and hence the more fair.

Unless of course near sides have almost equal effect in the particular game of interest, in which case imperfections will have not a big impact.

The bigger issue is that once you go past the 5 platonic solids (4,6,8,12,20 sided) you have to play games with the shape and placement of faces.

You can also use the duals of (convex) uniform polyhedra (as here) - uniform polyhedra have equivalent vertices, their duals have equivalent faces.

I've always imagined using a cylindrical die to get arbitrary odds. E.g. a 17-face cylinder that you roll. The numbers could be printed on the ends, you take what appears at the bottom (flat with the table).

These are moderately common. Pointed ends (without numbers) and skinny triangles that approximate a cylinder (with numbers) roll nicely and, allow unusual face counts. But I've actually seen it most often for the humble d4 which has a reputation, in some circles, for not rolling great in its usual tetrahedral form.

I have a small plastic container of different kinds of dice, which I use to generate passwords from my own Diceware-style word lists (see http://world.std.com/~reinhold/diceware.html). I like the 10 sided dice so that I can roll 4 and pick any number from 0000 to 9999. For an RPG that needs a "random" percentage, two of these (in different colors) would suffice. Any board game store has an assortment of crazy dice for sale.

Generally we would just use die combinations for anything over a d20. D10s are great for this. We had a d100, but it would never stop rolling. Much faster to roll two d10s.

We had one player that would roll a d100 in a bowl. It was the ultimate cheater's die. We banned it after two sessions.

Why? I don't see how a d100 in a bowl would make it easy to cheat.

The sides are so small that its easy to bs the result or play with the bowl to change the result. 100 sides make for hard tell on the top.

Buy it, of course. Though it will be more of a novelty item on my desk to go alongside other oddities rather than see any "real use". I don't play any tabletops and the board games I do play don't warrant any die with more than 20 faces.

Wonder what the practical use for gamers would be. There's obviously good reasons for having 20-sided dice (e.g. more granular results in fights). Would 120-sided dice have added benefits, or is there a point at which more sides don't matter?

"The d120 can be used as a dn, where n is any proper factor of 120. For the standard seven-dice polyset, a d120 is the ideal "multi-die", as 120 is the least common multiple of 4, 6, 8, 10, 12, and 20."

- http://thedicelab.com/d120tables.html

So you could whip one of these out instead of the usual Crown Royale bag full of a zillion different dice if you wanted to have a minimalist flair at your regular gaming session.

Mostly though I think it's just a cool thing to add to that bag full of dice. You're playing these games in part to fantasize about collecting lots of pretty gems, and this is like adding a big diamond the size of your fist to that collection of little rubies and sapphires.

Fun, not practical.

Reducing game mechanics to one or more d6s is within the power of the game makers, but they use weird effects to differentiate themselves.

There's an aesthetic difference in rolling different sorts of dice, they have a different look and feel, so game developers could reduce all rolls to some standard die like d6, but rolling a variety of dice is "more interesting" to players.

There are systems that are all d6s, there are systems that are all d10s. There are systems that like to pick dice based on the whims of what seems most interesting to players.

Games are typically more about fun than practicality.

Knowing my luck, I will be using it to roll natural 1s on the social saving throws.

Wow... I just learned that the singular of "dice" is "die".


Asymmetrical. You'd need to divide the pentagons in a regular dodecahedron or the rhombi in a rhombic dodecahedron into 12 congruent parts, which isn't possible.

Wait around for it to stop rolling?

It must be very difficult to figure out which number is up.

I will let you know when I get mine. :)

Just rolling

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