Dungeons, dragons and dice

[krusader74]

My younger son has become interested in the game "dungeons and dragons". He is intrigued by the shapes of the dice that are used to play the game and by the number of faces they have. Besides the standard cubical die with its 6 faces, there are the other regular solids with 4,8,12 and 20 faces. There is also a die with 10 faces, each in the form of a kite, and my informant points out longingly that our local games shop has a die with 100 faces. "Could you have one with an odd number of faces?" he asks.

His innocent question led me to think about many mathematical aspects of dice. This article answers his question, examines the geometry of dice (highlighting some unnamed solids which deserve to be better known to model-makers), considers surprising aspects of the mechanics of tossing dice and describes sets of unusual dice whose probabilities offer scope for many games which can trap the unwary gambler.

Quoted from the article: McLean, K. (1990). "Dungeons, dragons and dice." The Mathematical Gazette, Volume 74, issue 469, pages 243-256.

D&D and its polyhedral dice got me interested in math as a kid, and keeps me interested in math as an adult.

Here is a brief outline for this article:

1. The definition of an isohedron: "Any two faces are congruent and any face may be mapped onto the other by some rotation or reflection of the die onto itself."
2. Euler's formula, V - E + F = 2, and the proof that an isohedral die cannot have an odd number of faces.
3. A complete table of convex isohedra:
   
   • 5 platonic solids
     
   • 13 Archimedian solids
     
   • r-gonal dipyramids (an infinite sequence)
     
   • r-gonal trapezohedrons (another infinite sequence)
     
4. Non-isohedral dice, like the d5, a truncated prism. Can they be unbiased?
5. Non-transitive dice

And here are some comments and criticisms about this article...

RE: The definition of an isohedron and the complete table of convex isohedra

I would have liked to have seen the article talk about the algebra of isohedrons, not just their geometry. Isohedrons are defined by their rotational and reflective symmetries. These symmetries are captured in the idea of a Coxeter group, and may be expressed very concisely in a Dynkin diagram. One could use these diagrams to enumerate all uniform polyhedra.

...

RE: Euler's formula, V - E + F = 2

Euler discovered this relationship studying planar graphs. And each polyhedron is associated with a planar graph. I discussed this in the Appendix: Learn You Some Graph Theory to the answer I wrote to the question posed by the thread Non-Linear Randomly-Generated Dungeon?

...

RE: r-gonal trapezohedrons

An r-gonal trapezohedron is a solid whose faces are quadrilaterals. Applying Euler's formula, it has F=2r faces, E=4r edges and V=2r+2 vertices.

H.P. Lovecraft's 1935 short story The Haunter of the Dark features a "crazily angled stone" of extraterrestrial origin called the "Shining Trapezohedron."

r=3: You can buy fair, non-cubical 6-sided dice with trapezoidal faces, called Skew Dice(TM). (A trapezoid is a quadrilateral with two sides parallel.) I wrote about Skew Dice in the post on Weird But Fair Dice.

r=5: There's a recent discussion of the pentagonal trapezohedron in the thread: When did 10-sided dice start being used in D&D? Its 10 faces are congruent kites. (A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.)

You can make your own origami pentagonal trapezohedron by following the 56 steps on pages 105-109 of the Dover book, A Plethora of Polyhedra in Origami (2002) by John Montroll:



If you prefer to "cheat" -- by cutting and gluing instead of just folding -- the instructions are here.

...

RE: Non-isohedral dice, like the d5

In 2003, a patent was filed for five-sided dice. In 2005, the USPTO granted patent US 6,926,275:



Theoretically, the probability of landing on a rectangular face is a continuous function of the ratio of the lengths of a rectangular side to the triangular side. In the limit, as the prism gets longer, the probability of landing on a rectangular side goes to 100%. And in the limit, as the length of the rectangular side shrinks to zero, the probability of landing on it goes to 0%. So somewhere in between, it must be unbiased. Hence, fairness depends on thickness:

The present invention has been tested for fairness wherein different sizes of dice were included in the test ranging from 13-18 millimeters in thickness... These tests indicated that the dice should be 13.9 millimeters thick in order to yield a fair dice.

The problem with this argument is that the world isn't continuous. In practice, the fairness of such a die depends on the conditions in which it is thrown. You want:

• High initial angular velocity.
  
• Rough, perfectly inelastic surface.

McLean's paper references other studies where these practical dependencies were demonstrated.

...

RE: Non-transitive dice

I posted about non-transitive dice in detail, in the thread on The Most Powerful Dice.

[foxroe]

And I was so proud of myself for entering that Chi square BASIC program from Dragon magazine into my Commodore Vic20 back in the day...

Excellent post, krusader74.

[krusader74]

Dec 1, 2017 19:52:20 GMT -6 foxroe said:

And I was so proud of myself for entering that Chi square BASIC program from Dragon magazine into my Commodore Vic20 back in the day...

Yeah, there were a number of great articles in Dragon about dice that left a lasting impression...

Here is the Dragon magazine index of articles on dice:

Subject					Title							Author			Location	System
Dice:
                                        "Wild, Wild World of Dice, The"				Michael J. D'Alfonsi	182(45)		--
  Alternatives to			"What To Do When the Dog Eats Your Dice"		Omar Kwalish		7(5)		--
  Checking for fairness			"Be Thy Die Ill-Wrought?"				D.G. Weeks		78(62)		--
  Computer				"Electric Eye, The"					Mark Herro		45(56)		--
  Divided rolls				"Same Dice, Different Odds"				David G. Weeks		94(18)		--
  Games of chance			"Games of Chance"					Seth Irvin Williams	346(42)		D&D3			
  Simplifying complex rolls		"One Roll, To Go"					Larry Church		113(74)		--

And here's my review of all these articles...


1. In "What To Do When the Dog Eats Your Dice", Dragon issue 7, pages 5-6, Omar Kwalish lists some alternatives to dice:

• Percentages generated by 2d6
• Chits in a jar
• Calculators
• Cutting cards
• Numbered straws
• watch with a second hand
• Spinners
• Using six siders for higher numbers
• Coin flipping
• Phone book and blind fold
• Lazy Susan dartboard
• Classic Greco-Roman augury method
• Mouse in a maze
• Maso/macho delight
• Numbered jumping beans

2. In "The Electric Eye", Dragon issue 45, pages 56-57, Mark Herro presents

• A dice rolling routine for a Hewlett-Packard HP-41C calculator.
• Quick reviews of two affordable computers:
  
  
  • Sinclair's ZX-80
  • Radio Shack's "Pocket" TRS-80

3. In "Be Thy Die Ill-Wrought?", Dragon issue 78, pages 62-65, D.G. Weeks describes the chi-squared test to check dice for fairness, and he provides a BASIC program to compute chi-square values.

This was a great article. Its only shortcoming, like so many other articles on the subject, is that it doesn't provide an adequate answer to the question, How many times must you roll a die to test for fairness? He merely says,

Calculate the number of times you will roll the die for the test by multiplying N × E, where N is the number of sides of the die, and E is the number of times each side would be expected to appear. Ten is usually a good choice for E. Then for the d8, with E = 10, the total number of rolls (T) is 80. The number E can be more than 10, and should never be less than 5; a higher number will yield a more reliable test, but is more work.

But what's the justification for rolling it 80 times as opposed to 8 times or 8,000 times? I posted about that here.


4. In "Same Dice, Different Odds", Dragon issue 94, pages 18-20, David G. Weeks says sometimes you want a probability roll with a "long tail" in order to model extraordinary events. So he suggests

A divided die roll is created by dividing the result of one roll by another roll. For instance, for a d20/d4 divided die roll, simply roll a d20 and a d4, divide the number on the d20 by the number on the d4, and round the result to the nearest integer. (For certain purposes, you may want to round all fractions up, to avoid a result of 0.)

Here is his plot of this distribution using Palamedes:

palamedes.altervista.org/images/divided_roll.jpg

I used the command:

barchart(p(floor(d20 / d4)))

That rounds fractions down. If you want to round up, so that you don't get "0" as a value, use this instead:

barchart(p(ceil(d20 / d4)))

At the very end of the article, Weeks provides a table in the form dX/dY. For each of these "divided rolls," he gives the

• Average
• Conventional roll with a similar average
• Probability of 0
• 95th percentile

This is a very interesting article, mathematically, since most "real life" distributions have long tails.


5. In "One Roll, To Go", Dragon issue 113, pages 74-75, Larry Church describes how to replace lots of d20 rolls with one d% roll by using the binomial distribution:

You're running a game in which the party is ambushed by 45 archers. Arrows begin flying; you resign yourself to rolling your 20-sided die over and over (and over and over and . . .). Anxious players drum their fingers as they await their turn. Is there a quicker way? Obviously there is, or you wouldn't be reading this article.

He provides a concrete example of how his system works...

Example: In the above scenario, the DM determines that 10 arrows are being aimed at Francis the Cleric. The DM consults the 10-Roll Binomial Table. The archers need, say, a 12 to hit, and the DM rolls a 39 on percentile dice. The largest number less than or equal to 39 in the "12" column is 27. This corresponds to 4 hits tread across to the leftmost column) out of the 10 attempts.

And here is the table he's referring to...

palamedes.altervista.org/images/10_roll_binomial_table.jpg

Church provides us with tables for

• A 5-roll binomial table
• A 10-roll binomial table
• A 20-roll binomial table

He computes the values in these tables from the cumulative distribution function of the binomial distribution.

I like the idea here a lot, but I have two peeves with his tables:

• There's a chance that none of the 10 archers hit, but 0 is missing from the table!
• Inconsistent rounding

Despite these problems, I'd have to say this is probably the most useful Dragon article about dice!!!

You could generate the "12" column in Palamedes by inputing:

cdf(binom(10, (21 - 12)/20))//pct//table

The output looks like this:

palamedes.altervista.org/images/12_column.jpg

Use the "key" if the percentage you rolled on a d% is less than or equal to its corresponding "value" but greater than the "value" above it (and assume there is a 0.00% above the first "value" in the table). These "values" are expressed in 100ths of a percent. You can always roll your percentile dice (d%) a couple extra times to generate 100ths of a percent, if need be.

TODO: Maybe in the future, I'll whip up a quick JavaScript fiddle to print an entire n-roll binomial table for a given s-sided die, not just a d20.


6. In "The Wild, Wild World of Dice", Dragon issue 182, pages 45-46, Michael J. D'Alfonsi says,

Dice are the heart and soul of role-playing games... What follows is something of an anecdotal history of my experiences with dice owners and their quirks.

• Superstitions - The player who puts aside any dice that produce low rolls, because they're cursed.
• Specific dice - The player who always uses one d20 to roll high, and another to roll low.
• Strange habits - The player who sleeps with their dice bag under under their pillow.
• Dice bags - The player who uses a bottle of Crown Royal as a dice bag.
• Too many dice - The player with 200 dice.
• Cheater, cheater! - The player who uses loaded dice.
• A few last thoughts.

7. The blurb for "Games of Chance" by Seth Irvin Williams in Dragon issue 346 reads:

New card and dice games for you and your character, including "Rat, Bug, Spider" and "Wyrm Poker".

...

Has anyone else used any of the dice advice in these articles? Let me know if I missed any dice articles from Dragon, and I'll add it to my review!

[krusader74]

An index of dice-related discussions

(For probability-specific discussions, see also the index of prob & stat related discussions. For an index of dice-related articles in Dragon magazine, see this.)

#
3d20 advantage/disadvantage mechanic [1], [2]

A
Alternatives to dice [1]
Art and dice [1]

B
Buying dice [1]

C
Carcosa dice [1]
Chi-square test See Fairness, testing
Chits [1]
Conversions:

• Converting 3d6 to d% [1]
• Converting d20 to d10, d% [1]
• Converting d6s to other dice throws [1]
• Converting lots of d20 rolls with one d% roll by using the binomial distribution [1]

D
d5 [1]
d6, skewed [1]
d10 [1]
d20 [1]
d20, numbered 0-9 twice [1]
d20, replacing lots of rolls with one d% roll by using the binomial distribution [1]
d34 [1]
d60 [1]
d120 [1]
dF [1]
David Wesely on the introduction of polyhedral dice to RPGs [1]
Diceless [1], [2]
Divided die rolls, e.g., d20/d4 [1]
Dragon magazine articles about dice [1]

E
Earthdawn dice [1]
Expensive dice [1]
Exploding dice [1]

F
Fairness [1]

• Fairness, testing [1], [2]

Film and dice [1]
Fudge dice [1]

G
Gamescience [1]
Gamescience, pound of dice [1]
Geometry of dice [1]

H
History of dice [1]

I
Inventor of dice [1]
Inventor of dice, images [1]
Isohedral dice [1]

K
Knucklebones [1], [2], [3], [4], [5], [6]

L
Long-tailed dice distributions See Divided die rolls

M
Maximum of 2d6 [1]
Music and dice [1]

N
Non-isohedral dice [1]
Non-transitive dice [1], [2]

O
Oldest dice game See Knucklebones

P
Palamedes See Inventor of dice
Patents, d10s et al. [1]
Patents, d5 [1]
Poetry and dice [1]
Pools [1]
Predictability of dice [1]
Probabilities [1]

Q
Quirks of dice owners [1]

S
Skew Dice(TM) [1]
Statistics for common dice rolls [1]
Step die [1]
Sums of two dice [1]
Symmetry of dice [1]

T
Tali See Knucklebones
Trapezohedrons, pentagonal See d10
Trapezohedrons, r-gonal [1]
Trapezohedrons, trigonal See d6, skewed

U
Unbiasedness See Fairness

[krusader74]

I put together this web page to print out an N-roll binomial table. It's inspired by the binomial tables in Larry Church's article "One Roll, To Go", in Dragon issue 113 (September 1986), pages 74-75.

It differs from Church's tables in a few respects:

• The number of attackers is variable. You can set this to be any integer from 1 to 100. Then press the 'Calculate' button to see the table.
• There is a row for 0 hits.
• The probabilities of the cumulative distribution function for the binomial table are formatted in 100ths of a percent. You can always roll your d% a couple extra times to generate 100ths of a percent, if need be.
• Roll your d%. Then look at the column corresponding to the value needed to-hit on a d20. You want the value of the 'No. Hits' in the leftmost column corresponding to the smallest percentile that's greater than or equal to what you rolled. In other words, if your d% roll is between two consecutive rows, use the 'No. Hits' in the upper of the two rows. If that still sounds confusing, there's an illustrative example with numbers given on the page.
  
• The numbers agree with Church's except: (1) he rounds fractions of a percent, and (2) his table is organized so that you want the first row from top to bottom whose percentile is less than the d% you rolled.

It's a single-page web app -- all the JavaScript, CSS and HTML are together in one file -- no external libraries required. Feel free to download it and modify it to suit your needs, no permission needed.

Here is a screen shot of what it looks like after I hit the 'Calculate' button...


palamedes.altervista.org/images/n-roll_binomial_screenshot.jpg [http images no longer display] "N-roll binomial table screenshot"

[derv]

This is excellent krusader74! I'm glad you reviewed the article.

[krusader74]

I put together a simple web app to help check dice for fairness. It's inspired by the David G. Weeks article, "Be Thy Die Ill-Wrought?" in Dragon issue 78, pages 62-65.

Here is the sample calculation, which Weeks provides in the article:

palamedes.altervista.org/images/Weeks.jpg (image no longer displays) "D. G. Weeks sample chi square calculation from Dragon magazine"

And here is a screen shot of the same calculation using the web app:

palamedes.altervista.org/images/dice_fairness_checker_screenshot.jpg (image no longer displays) "Same calculation using the web app in the link above"

The web app uses public domain code to compute the chi-square values. Feel free to download and modify it, no permission needed.

[Zenopus]

*bump*


Creative Publications dice (front) versus Holmes Basic dice (back). Source: Playing at the World

increment has a very informative video and blog post about identifying dice from the 1970s:

Playing at the World Episode #2: Identifying 1970s Dice (video)

Playing at the World: Identifying Dice of the 1970s (blog)

[Deleted]

A gut check - those dice bring back memories!

[angantyr]

Dec 1, 2017 19:52:20 GMT -6 foxroe said:

And I was so proud of myself for entering that Chi square BASIC program from Dragon magazine into my Commodore Vic20 back in the day...

Excellent post, krusader74.

How primitive. I used a Commodore 64...