# Crazy dice rolls

All this discussion of D&D got me thinking. A typical roll of the dice in such games is denoted d6 — this produces a random number uniformly distributed among {1, 2, …, 6}. Hardly more complicated is nd6: 2d6 results in a tent-shaped function if you graph the probabilities of outcomes, 3d6 gives a piecewise-quadratic bell-shaped function, and so on, until you are approximating a normal distribution by rolling a large number of dice.

Consider, however, d6d6: roll a die, then roll that many dice and add up all the numbers. This crazy mash-up has 6 as the most probable result, then the probability drops and flattens out a bit before tailing off, an effect which is more pronounced if you go to d10d10, d20d6, d20d20, etc. Then we have d6d6d6…

You probably think no one would ever need such weird rolls, and you are probably right, but then there are exploding dice: roll again whenever the highest value comes up.

What kinds of exotic rolls are used in real games?

Fifth Edition D&D has the concept of “advantage” and “disadvantage” – situations or conditions when you have a better, or worse, chance of success than normal.

If you have advantage, you roll 2d20, and take the higher of the two dice results. If you have disadvantage, you roll 2d20, and take the lower of the two results.

Feng Shui uses a mechanic it calls “swerve”. You roll two d6 - one is a positive die, and one is a negative die. You add the numbers together (remember, one is a negative number), and the result is your “swerve”. In addition, both dice can “explode” - if you roll a 6, you roll again and add the result. Again, though, one die is a negative, so you can explode in either direction - or in both simultaneously!

I don’t think this is quite what you’re looking for, but Fantasy Flight Games’ “Genesys” system, which first appeared in their licensed Star Wars RPG, uses a variety of specialized dice with different symbols on the faces, instead of numbers. The symbols interact with each other in various ways, with some symbols paired so that one of the pair cancels out the other.

There are a lot of board games that have specialized dice like this, but in those you’re usually just looking for a particular symbol, not having some symbols cancel out others.

FUDGE, a freeware RPG that led to the popular FATE system from Evil Hat Games, uses another system with odd dice. FUDGE/FATE dice are six sided, but instead of numbers, they have two “+” faces, two “-” faces, and two blank faces. You roll (usually) 4dF, and the + and - cancel each other out, and you use the net + or - as your roll result (-4 to +4).

The old Alternity RPG from TSR used a d20 as a “Control die” - you always rolled a d20. But then, instead of static modifiers, you had a step ladder of die types, going through all the D&D polyhedrals, d4 / d6 / d8 / d10 / d12 / d20 / 2d20 / 3d20. The ladder also went in the opposite direction: -d4 / -d6 / etc, with the die result subtracted from your control die. If you had modifiers, you moved up or down the ladder, and rolled your control die with the modifier die, and added the result.

I’ve seen a couple of indie RPGs (which I can’t remember the names of right now) which used something like the FUDGE/FATE or Genesys systems, but using regular dice. You have an Ability die and a Difficulty die. You roll both, and try to beat the result of the Difficulty die with your Ability die.

Savage Worlds is probably the most prominent example of the exploding die mechanic. It also uses a “Wild Die” - when you make a Trait roll, you also roll a d6, which can also “explode”, and use whichever result is better. SW is also a bit unusual in RPGs in that Traits (attributes and skills) aren’t static modifiers or levels or dice pools, they’re die types. So your Strength is a d4, d6, d8, d10, or d12.

Godlike used pools of d6s, and you looked for matches. But, rolls had two dimensions. The higher the number of a match (the roll’s “height”), the quicker you acted, but the more matches you got (the roll’s “breadth”), the more effectively you acted. So, for example, if you rolled 10 dice, and got results of {1, 1, 1, 1, 3, 3, 3, 4, 6, 6} you could choose to use the 4 x {1}, and act slowly but very effectively, take the 2 x {6}, and act very quickly but barely eek out a success, or split the difference and take the 3 x {3}.

Torg (at least the original version; I’m not certain if the new version works this way) used a d20 for conflict resolution, with a variation on the “exploding dice” mechanic – your d20 would explode if you rolled a 20, as well as if you rolled a 10.

The indie RPG, Mythender, actually uses a mechanic that’s similar to your d6d6. As characters act out heavy metal album covers and kill Gods and Myths, they have two dice pools, Storm Dice and Thunder Dice. They roll fist-fulls of both sets of dice (generally starting with a bunch of Storm Dice and only a couple of Thunder Dice). Every 4, 5, or 6 on your Storm Dice adds another Thunder Die to your Thunder Dice pool for future rolls. Every 4, 5, or 6 on your Thunder Dice gives you a Lighting Token. Lighting Tokens are then spent to actually do things, like erase Mythic Blights, and ultimately kill Gods and end their Myth.

Unlike most RPGs, your Storm Dice don’t really represent how capable you are, but how quickly you’re able to build up your power during the Battle. They generate Thunder Dice, and Thunder Dice generate Lighting Tokens, and Lightning Tokens are where it’s at.

Can confirm, Torg: Eternity uses the same mechanic.

Another one I just remembered: West End Games’ Star Wars RPG from the '80s and '90s, which used WEG’s d6 system – in that system, which only used d6s, you had a larger number of d6s to roll on skill checks as your proficiency with that skill increased.

In the revised version of the game, there was a “wild die” mechanic – whenever you rolled your d6s, you needed to have one die which was a different color, and which was your “wild die.” If you rolled a 6 on the wild die, it exploded (as the typical “exploding die” mechanic), but if you rolled a 1 on your wild die, you removed that die, as well as the highest die among your other dice, from your result. (It made for a very swingy system, and I wasn’t a huge fan of it.)

I’m teaching a probability lesson tomorrow to fifth graders, who last week figured out the probability distribution for 2d6. If anyone has a cool probability set that is super basic (that is, would be a good “next lesson” for kids who just found out that 7 is the most common 2d6 roll and that 2 and 12 are the least common), I’m all ears! I’ve got an idea for tomorrow but it’s not amazing.

I see that anydice.com actually has a list of a few articles about various dice mechanics, including both “exploding dice” and also interpolating between distributions by rolling extra dice.

Birthday problem?

The birthday problem is great, but I am concerned that it is a little too advanced to explain. I don’t know, maybe it would work.

I was thinking about doing a Monty Hall simulation with them, see how that goes.

Maybe go with powers of two. That’s a bit more advanced, but it’s an easy way to understand how quickly exponents grow. 2^4 is over 10, 2^7 is over 100, and 2^10 is over 1000 for example, so you get to a large number very quickly. You could simulate it with coin flips or even/odd dice rolls.

Well, the obvious (to me) next step is 3d6, where unlike 2d6, the mode {10, 11} is different from the mean {10.5} and median {10.5}.

You could also introduce them to the full wonder of polyhedrals - d4, d6, d8, d10, d12, and d20, etc. It may seem obvious, but a few of the other students in one of my high school math classes seemed to have their minds a bit blown by 1) the existence of dice with different numbers of sides, and 2) the fact that probabilities for a d6 are only a specific application of universal probability rules. That last one really seemed to be a revelation for some of them - that the bell curve for 2d6 isn’t a special rule for “regular” dice, but any size of die will follow the same sort of bell curve. Actually physically seeing 2d12 really seemed to make the whole thing a lot more concrete to some of them.

Fudge dice aren’t actually all that complicated: A single Fudge die is just 1d3-2, and so (for instance) 4dF = 4d3-8.

A lot of games have a “dice pool”, where your aptitude in a skill is represented by some number of dice (usually exclusively d6s, because most gamers have a lot more d6s than the other sizes), and any given task’s difficulty is set both by a required number, and by the number of dice that must meet that requirement. So a task that required eight 3s, for instance, someone with only seven dice to roll couldn’t possibly succeed, someone with eight dice would have a chance, but a very small one, and someone with sixteen dice would have a very high chance.

I was actually thinking about this, and getting them to come up with a general rule: to find the mode (median, mean) roll of 2dx, just find the mean of the highest and lowest possible rolls.

Torg:E has a d20 that can explode on a 10 and also on a 20 but only exploded on a 20 if you are skilled in whatever you’re rolling.

They also have “up” conditions where you roll and add 2 dice, and “favored” conditions where you can roll a second die if you don’t like the first roll, but have to take the second roll whether it’s better or not.

Also they have “bonus” dice usually used to figure out damage totals, which are D6. The dice are worth 1-5 points. If you roll a 6, it’s only worth 5 but you roll a second die and add it. That second die can also be a 6 and give a further die, potentially to infinity.

There are also “perks” that can affect bonus dice, like “trademark weapon” that lets you reroll any 1 you roll on a bonus die if you’re doing damage with said weapon, but if you reroll a single die and roll a 1 again you keep it. Which can lead to a scenario where you roll a bonus die, roll a 1, reroll it and get a 6, roll a second die and get a 1, reroll it and get a 6, roll a third die…

If you’re looking for weird dice rolling with standard dice Torg:E gives you lots of great examples.

This seems to have to do with the symmetry of the distribution rather than the specific 2dx shape: symmetry forces the mean and median to coincide and it is easy to see why, and the fact that the mode is also at the same place then follows from the fact it is unimodal with a single peak.

Totally–but these are kids who are new to distributions and to the idea of probability. This pattern is one they can probably discover with a little bit of work, and explain with a little more. And if any of them play D&D, it’ll help them figure out average damage for their attacks

You could go with average dice. which are numbered 2,3,3,4,4,5. rolling two of these have the same mode, but if construct a table and graph the occurrences they will hopefully notice that the curve is “thinner”. 5th grade is probably too early to formally introduce standard deviation – but maybe introduce the general concept? (what if the dice were numbered 3,3,3,4,4,4 ?)
“all of the graphs have the same peak, but notice how likely the average will be rolled”

I like the idea of different sided dice as well.

Brian

What about how to use a single 6-sided die to generate random numbers between 1 and 7 (uniformly distributed). It’s a little mind-bending until you realize you can throw it twice and only accept, for example, the seven possibilities (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) and (6,1) and reroll any other outcomes.

The old puzzle is to re-label two dice in a nonstandard way so that rolling them gives the same probabilities as 2d6 , but that is more of a challenge for advanced students than an easy problem for typical fifth-graders.