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How Gaming Got Its Dice (2013) (playingattheworld.blogspot.com)
51 points by benbreen on June 14, 2017 | hide | past | web | favorite | 26 comments

> There is no intuitive way to extract percentile results from rolling a small number of d6s.

Here is an algorithm for rolling a single decimal digit using on average about 2.19 d6 rolls (which can be done in parallel if you have more than one d6):

  function one_digit
    d = roll()
    while d == 6
      d = roll()
    if roll() <= 3
        return d - 1
        return d + 4
Do this twice to get a percentile. This seems reasonably intuitive to me.

The problem of doing the job of a d10 with a d6 can provide a nice illustration of an application of bases other than 10. Might be useful for an elementary school math teacher trying to explain why the kids should learn this stuff.

Write the numbers from 0.0 through 0.9 in steps of 0.1, but write them in base 6, to produce this table:

  dec base 6
  0.0 0.0000
  0.1 0.0333...
  0.2 0.1111...
  0.3 0.1444...
  0.4 0.2222...
  0.5 0.3000
  0.6 0.3333...
  0.7 0.4111...
  0.8 0.4444...
  0.9 0.5222...
Roll a d6 to generate base 6 digits, which you take as the digits of a base6-imal fraction in [0,1). Roll until you have enough digits to determine which [k/10, (k+1)/10) interval it falls in, using the above table as an aid.

In general this can be used to make a table for simulating a dN with a dM, by writing i/N in base M for i = 0 to N-1.

When M is 2 this gives a nice way to simulate a dN with coin flips.

The above can be made easier to use by representing it in a graph instead of a table, where you start at a root node and follow edges labeled by dM values, and the leaf nodes give the dN values.

That would be easy and fast to use.

I've never heard of this method before! That's a _really_ nifty trick, thanks for putting it here!

Here's another approach you might enjoy. We wish to simulate a dN with a dM. I'll use simulating a d10 (N=10) with a d6 (M=6).

1. let L = a common multiple of N and M. The least common multiple of N and M is a good choice here, so for our example we'll use L = 30.

2. Make a list of the values you are trying to choose from. Repeat each value L/N times, so that the length of the list is L. For our example, L is 30, N is 10, and L/N is three, so our list consists of the values 1 through 10 each repeated 3 times: [1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10].

3. Viewing this list as M chunks of length L/M, select one of the chunks at random via a roll of our dM. For our example that is 6 chunks of length 5. Here is the list again with parenthesis drawn around the 6 chunks: [(1 1 1 2 2) (2 3 3 3 4) (4 4 5 5 5) (6 6 6 7 7) (7 8 8 8 9) (9 9 10 10 10)]

4. Take the chunk selected by the roll of the dM. If all L/M numbers in the chunk are the same, that is the value of our simulated dN roll and we are done. Let's say we've picked the second chunk, (2 3 3 3 4). That contains more than one distinct value, so we go on to step #5.

5. Replace the list with the items from that chunk, each repeated M times. In our example, our new list is [2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4].

6. Go to step #3.

Note that the second time through this is how we chunk the list: [(2 2 2 2 2) (2 3 3 3 3) (3 3 3 3 3) (3 3 3 3 3) (3 3 3 3 4) (4 4 4 4 4)]. 4 of the 6 chunks contain only one distinct value now, so there is a 2/3 chance that we find our d10 value on the second iteration.

If you actually implement this, you don't need to actually make the lists like I did above. You just need a count of how many times each thing appears. So instead of [1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10] you would have [3 3 3 3 3 3 3 3 3 3], meaning you have three '1', three '2', and so on.

To pick the second chunk, you skip 5 items and then take 5. That means we skip all 3 '1's, and 2 of the '2's, and take 1 '2', then all 3 '3's, and 1 '4'. That gives us [0 1 3 1 0 0 0 0 0 0].

BTW, note that this also works for simulating a smaller die with a larger die. For example, simulating a d6 with a d10. There is a really simple way to do that, of course: roll the d10 and discard anything above 6. With that method, you get a d6 output on 6 out of 10 rolls. Let's call this method the trivial method.

If we use the list method as described above, we get a higher output rate. Here is a little table:

  1 1 1 (1)
  1 1 2   111 111 111 111 111 111 112 222 222 222
  2 2 2 (2)
  2 3 3   222 222 222 233 333 333 333 333 333 333
  3 3 3 (3)
  4 4 4 (4)
  4 4 5   444 444 444 444 444 444 445 555 555 555
  5 5 5 (5)
  5 6 6   555 555 555 566 666 666 666 666 666 666
  6 6 6 (6)
On the left, the first 3 columns just show the initial [1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6] list (N distinct values each repeated L/N times) broken down into groups of L/M, which is 3 in this case.

If our d10 roll is 1, 3, 5, 6, 8, or 10 we count that as a successful d6 roll of 1, 2, 3, 4, 5, or 6, respectively, because those chunks all have only one distinct value. I've marked those in the table with the resulting d6 value in parenthesis. So, on the first d10 roll we get a d6 value 6 out of 10 times, just like with the trivial method.

Where we beat it is on the next roll. For each chunk that does not yield a d6 value on the first row, I've added the resulting list, grouped into chunks. Note that in all 4 of the possible lists, 9 out of the 10 chunks have only one distinct value, and so 9 out of 10 times the second roll yields a d6 value.

If the second roll fails too, the trivial method will again only succeed 6/10 times on the third roll. Again we beat it, with a 9/10 success rate, and so on.

I created an account to tell you "great job".

The implication in the article is that humans will be doing this on the fly. This solution does not fit with that criterion.

That's an easy algorithm to do on the fly. Take two d6s that can be distinguished (different color, for example). Call them A and B.

Roll A and B. If A is a 6, reroll A until it is not 6. If B is even, take A as your result. If B is odd, take A+5 as your result. (I changed to use B odd/even instead of B <= 3 because that might be easier for people).

That gives you a number from 1 to 10. Subtract 1 if you need 0 to 9.

If your two d6s cannot be distinguished, take the one that lands leftmost to be A.

No. I won't do any of that. Most games already have enough essential complexity that all of its components must be able to be interpreted at a glance. D20s and D10s make this easy and intuitive without requiring an algorithm or mathematical post-process.

And that's just how old tables often worked.

In fact, this takes exactly 11/5 rolls on average: 6/5 rolls to get a non-6, and then 1 more.

"Intuitive" in this context means "intuitively practical."

Besides which, that may be intuitive to programmers like us, but it's definitely not intuitive to the average person.

Interesting that once there was a trend in tabletop wargaming towards "realistic" probabilities expressed as a percentage (e.g. "these riflemen have a 70% of hitting the target"). I think this isn't as common anymore.

Nowadays, for whatever reason, I see a trend back to the humble d6, both in historical wargaming and fantasy skirmish games. Many wargames use d6 in disguise (see: Bolt Action, SAGA, Warhammer). I think it has to do with two things: the d6 is readily available (though, one has to wonder, why is this an issue when everything else in wargaming isn't as easy to find, like the miniatures and tokens!), and the other fancier dice didn't pan out in practice.

> and the other fancier dice didn't pan out in practice.

For wargaming I think what happened is that “special dice and more detailed simulation” moved into “computerized simulation” (which allows a lot more depth with less player-managed overhead) in the late 1980s through the 1990s, and most of that whole end of the hobby left the tabletop world.

Yeah, although the more complex simulations have largely have largely faded away from the computer space as well. (With a few exceptions.) And very niche board games are probably more viable than very niche computer games. Modder communities for some old computer games notwithstanding.

Personally, I confess to finding complex simulations (whether of the board game or flight/combat genres) more appealing in theory than I do in practice.

They're also a pain to make. I work on complex simulations for a living, and the number of parameters and data sets you need to get good results is staggering.

If you end up just going "We think it's about 100 meters" that's a lot of effort for not much reward.

Agreed. In support of this theory, do note the tabletop world (which still exists and apparently makes money. See: Games Workshop and its many current competitors) is moving away from complex simulations, and striving to streamline and simplify the gaming experience, within reason.

D6 are also common today because they help simplify the user experience. Charts, tables, and fancy dice get in the way of learning the rules and having fun for the average person. Most people need to perceive a low barrier to entry and don't want to invest in learning a large ruleset.

Games you mention are still made, but they're a niche product in a niche market.

Needle moving growth for the tabletop industry comes from introducing new people to the hobby, not from selling another Space Marine model to an existing gamer. It's difficult to build a sustainable tabletop business without one or two big hits (ex: Days of Wonder's "Ticket to Ride"). It's even harder for companies that sacrifice usability for realistic simulations.

Playing devil's advocate, I'm not sure the D6 provides a simpler user experience. Why is rolling a D6 to determine whether you kill something in an RPG or wargame "easier" than rolling a D10? A single D10 seems more intuitive to me than a single D6 (that is, a score "from 1 to 10" seems more natural than "from 1 to 6").

An argument could be made that the D6 is more readily available, but this is probably irrelevant in the context of miniature gaming: miniatures themselves are NOT "readily available", and you can get the specialized dice from the same store you get your miniatures!

The difficulty is when you have to go hunting for the right die to use, and mistake, say, the d12 for the d20. Or get confused that, for d4s, you have to read the downward face instead of the top face.

That being said, I fell deeply in love with the classic D&D polyhedra the moment I first saw them. Platonic solids are cool! They're delightfully arcane and geeky. They complement each other in a set. I wish the weird d8 and d12 throws came up more often.

Perhaps there's a lesson here for UX in general: sometimes there's a tradeoff between being _simple_ and being _interesting_, and (particularly in the world of games) it's a tradeoff worth taking seriously.

I thought with d4s you read the value at the point, not the face on the bottom? Maybe my dice are different.

I prefer the d12 to the d6 for generic rolls, as it allows for the least number of sides with the most human-usable fractions (25/50/75%, 33/66%, and "a little less than 10%").

Well, wargames and RPGs are so complex enough already that surely mistaking the type of die is the least of a player's problems. And D6 used to require complex table lookups (which current tabletop game design seems to eschew), so not sure it was easier to begin with... just the most common shape of dice.

I share your love of D&D dice :)

> There is no intuitive way to extract percentile results from rolling a small number of d6s.

Shadowrun 5e -> Roll 21d6 defense test against a 18d6 attack.

In all seriousness, good article. I always wondered where the need of the d20s came from, and always wondered why d10s and the likes didn't catch on. I always had shadowrun to blame for an ungodly amount of d6's laying around my house

Is 5e any good? I loved the flavor update in 4e, but felt like they threw out a lot of what was good with the mechanics from 3e.

Yeah, besides the huge amount of errata and bizarre inconsistencies between what you would expect a roll to be vs what it actually is 5e is ok.

The amount of flavor is sad, but at lest deckers are cool now since [everything] is wireless.

Does anyone know where I could find the Gygax article mentioned, "Four & Twenty and What Lies Between"? I loved his explanation of probabilities with dice and cards in the 1989 "Advanced Dungeons & Dragons, Dungeon Masters Guide" (apologies for my book-scanner errors):

As the DM, the tools of your trade are dice — platonic solid-shaped or just about any other sort. The random numbers you generate by rolling dice determine the results based on the probabilities determined herein or those you have set forth on your own. In case you are not familiar with probability curves, there are two types which are determined by your dice: linear (straight line),'which has equal probability of any given integer in the number group, and bell (ascending and descending line), which has greater probability towards the center of the group of numbers than at either end. The two curves are illustrated thus:

[Image of linear curve for 10-sided dice, with each number having a 10% chance of coming up.]

Linear probability develops a straight line of ascending probability when used as a cumulative probability as shown above.

Bell distribution, when used to delineate the probability of certain numbers appearing, develops o curved line like this:

[Image of a bell curve for a 3d6 roll, with 3 and 18 being the least probable and 8 through 12 being the most probable.]

A single die, or multiple.dice read in succession (such as three dice read as hundreds, tens and decimals) give linear probabilities. Two or more dice added together generate a bell-shaped probability curve.

Before any further discussion takes place, let us define the accepted abbreviations for the various dice. A die is symbolized by "d", and its number of sides is shown immediately thereafter. A six-sided die is therefore "d6", d8 is an eight-sided die, and so on.-Two four-sided dice are pressed by 2d4, five eight-sided dice are 5d8, etc. Any additions to or subtractions from the die or dice are expressed after the identification, thus: d8 8 means a linear number grouping between 9 and 16, while 3d6 -2 means a bell-shaped progression from 1 to 16, with the greatest probability group in the middle (8, 9). This latter progression has the same median numbers as 2d6, but it has higher and lower ends and a greater probability of a median number than if 2dl2 were used. When percentage dice are to be used; this is indicated by d%.

The d4 can be used to generate 25% incremental probabilities, random numbers from 1 to 4, with 1 it generates a linear 2-5, etc. It can be used to get 1 or 2 (1 or 2 = 1, 3 or 4 = 2) or in conjunction with any other dice to get linear or bell-shaped probability curves. For example, 2d4 = 2-8, 3d4 ^ 3-12, d4 d6 = 2-10, d4 d20 (as dlO) = 2-M.'when rolled in conjunction with another die, the d4 can be used to determine linear number ranges twice that shown on the other die, thus: d4 reading 1 or 2 means that whatever is read on the other die is the number shown; but if the d4 reads 3 or 4, add the highest number on the second die to the number shown — so if d8 is the second die 1 to 16 can be generated if a dl 2 is used 1 to 24 can be generated. If a d20 is used either 1 -20 (assuming the use of a standard d20 which is numbered 0-9 twice without coloring one set of faces to indicate that those faces have 10 added to the number appearing) or 1-40 (assuming that one set of faces is colored) can be gotten by adding 0 if 1 or 2 is rolled on the d4 and 10 or 20 (depending on die type) if a 3 or 4 is rolled. Linear series above this are possible simply bv varying the meaning of the d4 number; 1 always means add 0, but 2 can be interpreted as add the value (highest number) of the second die, 3 can be twice value, and 4 can be thrice value. Thus, a d4 reading 4 in conjunction with a d8 (linear curve 1-32) would mean 24 d8, or 25-32.

What applies to d4 has similar application with regard to d6, d8, dl2, and d20. The d6 has 16 2/3% intervals, d8 has 12 1/2% intervals, and d20 can have (1-2 = 1,3-4 = 2,5-6 = 3), while 1 to 5 can be easily read from a d20 (1-2 = 1, 3-4 = 2,5-6 = 3,7-8 = 4,9-0 = 5).

The d20 is used often, both as dlO and d20. The bell-shaped probability curves typically range from 2-20 to 5-50, i.e., 2, 3, 4 or 5d20 added together. Also common is the reading as above with one decimal place added to the result to get 20-200, 30-300, etc. In the latter case, a roll of 3 on one die and 0 (read as JO) totals 13, plus one place, or 130.

Non-platonic solid-shaped dice are available in some places. The most common of these is a ten-sided die numbered 0-9. As with the d20, this con be used for many purposes, even replacing the d20 if a second die is used in conjunction to get 5% interval curves (1-20). Also, the die can give 0-9 linear curve random numbers, as the d20 can.

Other dice available are various forms of "averaging" dice. The most common of these has six faces which read: 2, 3, 3, 4, 4, 5. The median of the curve it generates is still 3.5, that of a normal d6, but the low and high numbers, 2 and 5, are only half as likely to appear as 3 or 4. There is a 33 1/3% chance for either of the two latter numbers to be rolled, so the probabilities of absolutely average rolls are far greater. Other such dice have zeros on them, several low numbers, and so on. These sorts of dice, along with poker dice, "put & take" dice, or any other sort can be added in order to give you more flexibility or changing probabilities in random selection or event interpretation. For example:

The author has a d6 with the following faces: SPADE, CLUB, CLUB, DIAMOND, DIAMOND, HEART. If, during an encounter, players meet a character whose reaction is uncertain, the card suit die is rolled in conjunction with 3d6. Black suits mean dislike, with the SPADE equalling hate, while red equals like, the HEART being great favor. The 3d6 give a bell-shaped probability curve of 3-18, with 9-12 being the mean spread. SPADE 18 means absolute and unchangeable hate, while HEART 18 indicates the opposite. CLUBS or DIAMONDS can be altered by discourse, rewards, etc. | Thus, CLUBS 12 could possibly be altered to CLUBS 3 by offer of a tribute or favor, CLUBS 3 changed to DIAMONDS 3 by a gift, etc.

In closing this discussion, simply keep in mind that the dice are your tools. Learn to use them properly, and they will serve you well.

[Edited for some scan errors.]

Gary Gygax writes an article for Lowrys Guidon entitled "Dice...Four & Twenty and What Lies Between", in which he expounds on the application of various polyhedral dice in wargaming--specifically, Tractics.

From: http://www.commonroomgames.com/timeline.html

Don't know where to source a copy however.

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