## Sunday, December 25, 2016

### Cloud Giant Damage

A general method for converting range notation to dice notation.

My first copy of the game was a Holmes edition with geomorphs.

The rules have staying power. This summer I met a Holmes player born after they went out of print!

And so the questions we asked thirty-five years ago are still getting asked. A classic is how to roll the 6–63 damage that cloud giants deal out:

The experienced referee makes this roll with a 3d20+3, but—and this is another example of the ambiguity of range notation—a 19d4-13 will also work.

One wonders whether there are other ways to make the roll. Also, is there an easy way to find those ways?

If we use a single type of die, then we must use a d20 or a d4 as above. If we allow more than one type of die in the roll, then 6d10+d4-1 works.

In fact, if we use more than one type of die, there are 164 ways to roll cloud giant damage.

To find them all, observe a fact about the difference between the largest and smallest value of each die; a d6 produces results in the range 1–6 and has a difference of 5. When multiple dice are summed, the difference of the sum is the sum of the differences of each die used; 2d6 has a range of 2–12 and a difference of 10; 3d6 has a range of 3–18 and a difference of 15.

The rule holds if we use more than one type of die: a d4 has a range of 1–4 and a difference of 3. A d4+d6 roll has a range of 2–10 and a difference of 3 + 5 = 8.

To find out how many ways there are to roll 6–63, we ask ourselves how many ways there are to sum to the difference of 57 given the differences we have available: 3 (d4), 5 (d6), 7 (d8), 9 (d10), 11 (d12), and 19 (d20). This is the unbounded knapsack problem where we are only interested in solutions which fill the "knapsack" completely.

The complete list of ways to roll cloud giant damage is here.

Code for finding ways to roll an arbitrary range is here.

## Saturday, December 24, 2016

### Binomial Parameter Estimates

More on estimating class qualification probabilities.

## Choosing a Character Attribute Generation Method

On p. 11 of the DMG four alternatives for generating character attributes are given. How should a referee decide which one to use?

The referee might have a sense for how easy it should be to roll up a prestige class. Does he want the players all playing paladins and rangers instead of fighters? If so, he should consider the chances listed in the previous post. The 4d6kh3 with re-arrangement method gives the best chances overall.

## Did the Designers Know the Odds?

If you asked the designers forty years ago to guess the chance of rolling a paladin, how close would they have been? The calculation requires writing code and running it on a modern computer, so the exact chance was probably unknown in the 1970s. Still, the designers could have made a reasonable guess just by rolling lots of characters.

## Monte Carlo Method

To estimate the chance of rolling a class, the designers could have rolled 100 characters and counted how many of them qualified. If 35 of them qualified, the estimated chance is 35%. This is the Monte Carlo method applied to the binomial parameter estimation problem.

The Monte Carlo method is slick! It is easy to understand. Complicated mathematics is avoided.

Monte Carlo can be used to verify complicated mathematics. After writing code to do the exact calculation, I wrote code which uses Monte Carlo to make sure the exact code is in the right ballpark.

Often it isn't enough to get an estimate; we also have specify how accurate it is. We can increase our accuracy by doing more rolls. For example, we will get a better estimate of the chance of a paladin if we roll 1000 characters instead of 100 characters. It is natural to ask how many characters must we roll to get an answer which is accurate enough for our purpose.

## Normal Approximation Interval

One way to represent the accuracy of an estimate is with a confidence interval. For binomial parameter estimation, one way to get a confidence interval is with the normal approximation:
$$p \pm z_{1 - \frac{α}{2}} \sqrt{\frac{p \cdot (1 - p)}{n}}$$Here n is the number of characters generated, p is the observed fraction, and α is the chance of being outside the confidence interval. When α is 5%, z1-α/2 is 1.96.  In R one can compute z1-α/2 with

qnorm(1-alpha/2)


The normal approximation suggests that if we increase the number of simulations 10-fold, the length of the confidence interval will shrink to about 0.32 of its previous size.

The normal approximation doesn't work well when the observed fraction p is close to zero or 1.  For example, when n is 100 and p is .01, the 95% confidence interval is [-.95%, 2.95%] even though we know a negative value is not possible. This is a consequence of using the normal distribution as an approximation for a distribution with non-negative support.

In the extreme cases where p is exactly 0 or 1, the normal approximation gives an obviously invalid interval with zero length. The usual advice is to never use the normal approximation when there are fewer than 5 successes or 5 failures. That is, there should be at least 5 characters which qualify as a paladin and 5 characters which don't.

## Wilson Score Interval

At the cost of some extra complexity, there is a formula which works better when p is near zero or one:
$$\frac{1}{1 + \frac{1}{n}z_{1-α/2}^2} \bigg[p + \frac{1}{2n} z_{1-α/2}^2 \pm z \sqrt{\frac{1}{n} p (1 - p) + \frac{1}{4n^2} z_{1-α/2}^2} \bigg]$$

## Beta Distribution

We can look for a distribution which describes the possibilities for the parameter we are estimating. This is more informative than a confidence interval. Bayesian statistics gives us a way to do this, but we must start with an initial distribution, the prior, which is somewhat arbitrarily chosen. We use the data we observe and the prior to compute the posterior distribution, which is the distribution we seek.

When estimating a binomial parameter, the beta distribution is convenient because it makes computation of the posterior easy. The beta distribution has two parameters. If the prior has parameters α and β and we observe m successes in n trials, then the posterior is a beta distribution with parameters α + m and β + n - m. We can think of the parameters as tallies of the number of successes and failures we observe.

How should α and β be chosen for the prior? If we set them both to 1, then the prior is identical to the uniform distribution. That is, we are saying that before we look at the data we think each possible value for the parameter is equally likely. For those who know information theory, the beta distribution with parameters (1, 1) is the beta distribution with highest entropy. Such a prior, incidentally, was how Laplace solved the sunrise problem.

Suppose we decide to start with a beta (1, 1) prior. We roll 100 characters and 35 of them qualify. Then the posterior parameters are (36, 66). We can use R to compute the chance the actual probability is within any interval. For example that chance that the true probability is inside [.32, .38] is 47.4%:

> pbeta(.38, 36, 66) - pbeta(.32, 36, 66)
[1] 0.4739478


If we think we already know the mean and variance for the distribution, we can use them to set the parameters of the prior. Here is how to use Mathematica and the formulas for the mean and variance to solve for the parameters:

Solve[{μ == α/(α+β),
σ^2 == α β/((α+β)^2 * (α+β+1))},
{α,β}]


Here are the formulas for the parameters, given mean μ and variance σ2:
$$α = \frac{μ^2 - μ^3 - μ σ^2}{σ^2}$$ $$β = \frac{(-1 + μ) (-μ + μ^2 + σ^2)}{σ^2}$$

## Hoeffding's Inequality

Hoeffding's Inequality is invaluable when doing Monte Carlo estimates. It directly tells us how much data we need to ensure our estimate is near the true value with a given probability:
$$\mathrm{P}\big(\big|p - \mathrm{E}[p]\big| \geq t\big) \leq 2 e^{-2nt^2}$$
Here p is the estimate, E[p] is the true value, t is half the length of our confidence interval, and n is the number of observations. If we know the probability and interval we want, we recast the inequality as:
$$\frac{\mathrm{ln} \frac{2}{\mathrm{P}(|p - \mathrm{E}[p]| \geq t)}}{2 t^2} \leq n$$ If we want to be 95% certain the estimate is within a tenth of a percent, we get this lower bound on n:
$$\frac{\mathrm{ln} \frac{2}{.05}}{2(0.001)^2} \approx 9.22 \times 10^5 \leq n$$ The number might seem discouragingly large, but generating a million characters is feasible on modern computers; just write code to do it!

## Friday, December 16, 2016

### Chance of Rolling a Class

e.g. how hard is it to roll up a paladin?

It's an obvious question. What is the chance of rolling a character that qualifies for a given class?

## Minimum Attributes

The chance depends on the minimum attributes for the class. Let's use the first edition values:

minimum attributes per Players Handbook (1978)

Str Int Wis Dex Con Char
cleric:       3   3   9   3   3   3
druid:        3   3  12   3   3  15
fighter:      9   3   3   3   7   3
paladin:     12   9  13   3   9  17
ranger:      13  13  14   3  14   3
magic-user:   3   9   3   6   3   3
illusionist:  3  15   3  16   3   3
thief:        3   3   3   9   3   3
assassin:    12  11   3  12   3   3
monk:        15   3  15  15  11   3
bard:        15  12  15  15  10  15 

## Odds by Class: 3d6

The original method of rolling up attributes is to use 3d6 for each attribute. The attributes are generated in the order of strength, intelligence, wisdom, dexterity, constitution, and charisma. No rearrangement is allowed.

The odds of a character qualifying for a given class by this method are:

  assassin:      7.031%
bard:          0.002%
cleric:       74.074%
druid:         3.472%
fighter:      67.215%
illusionist:   0.429%
magic-user:   70.645%
monk:          0.040%
ranger:        0.176%
thief:        74.074%

We knew that rolling up a paladin was a long shot, but rolling up a monk is harder still. Your chance of rolling up a bard are 1 in 58,140.

## Odds by Class: DMG Methods

Some prestige classes might seem pointless, given how unlikely it is to roll a character that qualifies to be one. However, the DMG allows, at referee discretion, one of four methods for rolling attributes, each producing higher attributes on average. For example, in method I the player rolls the six attributes with 4d6kh3 and then rearranges the attributes as desired.

The chances of rolling up a character according to these four methods are:

(METHOD I) 4d6kh3 PLAYER-ARRANGED:
assassin:      93.616%
bard:           1.581%
cleric:       100.000%
druid:         78.245%
fighter:      100.000%
illusionist:   35.782%
magic-user:   100.000%
monk:          13.641%
ranger:        30.470%
thief:        100.000%

(METHOD II)  3d6 BEST-OF-12 PLAYER-ARRANGED:
assassin:      96.180%
bard:           1.887%
cleric:       100.000%
druid:         68.206%
fighter:      100.000%
illusionist:   24.302%
magic-user:   100.000%
monk:           9.231%
ranger:        33.984%
thief:        100.000%

(METHOD III) 3d6 BEST-OF-6 EACH ATTRIBUTE:
assassin:      87.053%
bard:           3.572%
cleric:        99.970%
druid:         41.544%
fighter:       99.970%
illusionist:   10.936%
magic-user:    99.970%
monk:           8.487%
ranger:        29.788%
thief:         99.970%

(METHOD IV) BEST-OF-12-CHARACTERS 3d6:
assassin:      58.309%
bard:           0.021%
cleric:       100.000%
druid:         34.562%
fighter:      100.000%
illusionist:    5.024%
magic-user:   100.000%
monk:           0.475%
ranger:         2.097%
thief:        100.000%

Overall, method I is the best, though method II is best for an assassin or ranger and method III is best for a bard.

## Code

The code for calculating the probabilities is on GitHub.

## Sunday, December 11, 2016

### The Arch-Mage Casts a Fireball

Fireball damage: its calculation and its distribution.

This is what the game is about. I cast the spell:

Look at all the damage! But how much damage, exactly? My enthusiasm is tempered by the prospect of adding up the pips.

Fortunately, as a wizard I am familiar with the arcane rule of tens. Prestidigitation ensues and the hidden number is revealed:

Apropos of nothing, I timed myself on a few more rolls:
• 65 hit points: 22.2s
• 63 hit points: 19.6s
• 56 hit points: 28.3s
• 63 hit points: 28.3s
• 62 hit points: 24.9s
An average of 24.7s spent to get a number that was usually within a couple hit points of 63.

## A Shortcut?

When younger I was tempted to roll a single d6, multiply it by 18, and be done with it. That is, use a d6 × 18 roll in place of the 18d6. However, even then I couldn't escape the feeling that this was not an equivalent roll. I could see that the latter method always produces a multiple of 18, but I wondered if it wasn't close enough all the same.

Both methods cause 63 hit points of damage on average, so in that respect they are the same. But if the mean is all we care about, why roll dice at all?

Plotting the probability mass distributions is a good way to show how different rolls are. Here the 18d6 distribution is in red and the d6 × 18 distribution is in blue:
The standard deviations are 7.2 and 30.7, a large difference.

## Chebyshev's Inequality

At this point, I'm going digress in an attempt to make the standard deviation seem useful.

According to Chebyshev's inequality, the chance of being more than k standard deviations from the mean can never be greater than 1/k2. Thus, in the case of the 18d6 distribution, the chance the value is less than 49 or more than 77 is no more than 25%. The same probability for the d6 × 18 distribution is 66⅔%.

Chebyshev's inequality is a rough upper bound that works for any distribution for which we know the mean and standard deviation. If we know the distribution, we can do a summation or integration to get the exact probability; for the 18d6 distribution the exact probability is less than 5%.

## A Better Shortcut?

Back to rolling fireball damage.

What if we allow two rolls of a d6? I wrote some code which searches for an expression with the same mean and the nearest standard deviation. To keep things simple the first d6 always gets multiplied by a non-negative integer and the second d6 is taken as is. With those constraints here are the best approximations:


   1d6: d6 × 0 + d6 +  0
2d6: d6 × 1 + d6 +  0
3d6: d6 × 0 + d6 +  7
4d6: d6 × 1 + d6 +  7
5d6: d6 × 2 + d6 +  7
6d6: d6 × 1 + d6 + 14
7d6: d6 × 2 + d6 + 14
8d6: d6 × 3 + d6 + 14
9d6: d6 × 2 + d6 + 21
10d6: d6 × 3 + d6 + 21
11d6: d6 × 2 + d6 + 28
12d6: d6 × 3 + d6 + 28
13d6: d6 × 4 + d6 + 28
14d6: d6 × 3 + d6 + 35
15d6: d6 × 4 + d6 + 35
16d6: d6 × 3 + d6 + 42
17d6: d6 × 4 + d6 + 42
18d6: d6 × 3 + d6 + 49
19d6: d6 × 4 + d6 + 49
20d6: d6 × 5 + d6 + 49
21d6: d6 × 4 + d6 + 56
22d6: d6 × 5 + d6 + 56
23d6: d6 × 4 + d6 + 63
24d6: d6 × 5 + d6 + 63

Here is how well the approximation works for the arch-mage; the 18d6 distribution is in red and the d6×3 + d6 + 49 in blue:

## Wednesday, December 7, 2016

### A Nod to the Original Dice

We've been running a campaign using the Holmes rules for several years now. I've put some effort into getting the feel right.

An important touch is the dice. I have a few sets of the Creative Publications dice, but I don't want to use them and wear them out!

Here's what I've been able to put together in the way of a proxy:

## Sunday, December 4, 2016

### Dice Notation

What our group sees as the essentials of dice notation.

Here are examples of dice notation: d4, d6, d8, d10, d12, and d20.

The notation refers to a die, but more often a roll of a die. The rules might say that a weapon causes d8 of damage, which is equivalent to saying it does 1–8 hit points of damage.

The notation has been extended in a couple of ways: 3d6 means to roll three 6-sided dice and sum them, generating a number in the range 3–18. 1d4+1 means to roll one 4-sided die and add one to it, generating a number in the range 2–5.

## The Introduction of Dice Notation

The notation isn't used in the original box set or the Monster Manual.

The Players Handbook (June 1978) was the first TSR publication to use it. Jon Peterson suggests the PHB was written as if players were already familiar with the notation, but the occurrences I've found are used in a parenthetical manner. For example, consider this spell description:
A fireball is an explosive burst of flame, which detonates with a low roar, and delivers damage proportionate to the level of the magic-user who cast it, i.e. 1 six-sided die (d6) for each level of experience of the spell caster. Exception: Magic fireball wands deliver 6 die fireballs (6d6), magic staves with this capability deliver 8 die fireballs, and scroll spells of this type deliver a fireball of from 5 to 10 dice (d6 + 4) of damage.
The November 1978 printing of the Holmes rulebook appends the following text:
In some places the reader will note an abbreviated notation for the type of die has been used. The first number is the number of dice used, the letter "d" appears, and the last number is the type of dice used. Thus, "2d4" would mean that two 4-sided dice would be thrown (or one 4-sided would be thrown twice); "3d12" would indicate that three 12-sided dice are used, and so on.
The blurb suggests that dice notation is used elsewhere in the Holmes rulebook, but it isn't!

The Dungeon Masters Guide (August 1979) also explains the notation:
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 expressed by 2d4, five eight-side 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 2d12 were used. When percentage dice are to be used, this is indicated by d%.
As Jon Peterson has discussed, essentially the same notation, albeit with a capital D, was being used in the fanzines for several years before TSR embraced it. It appears, amazingly, in the first issue of Alarums & Excursions from 1975.

## The Old Notation Isn't Good Enough

Looking through the older texts, you can see a couple of different ways for specifying dice rolls: "5 + 1", "3-8 sided", and "2–24".

This notation is inferior in various ways. The first doesn't make clear which die to use: a 5d6+1 roll is intended. Of course, ambiguity can sometimes be advantageous. The way hit dice are specified in the Monster Manual might have made the book more appealing to players still using d6 hit dice for monsters.

The third example, range notation, looks like a concise way to specify rolls, but it also can be ambiguous. For example, 3–12 can be either d10+2 or 3d4. The first method is uniform, whereas the second starts to approximate a bell curve.  If you roll it the first way, the chance of getting a 3 is 10%; if you roll it the second way the chance of getting a 3 is 1 in 64 or about 1.6%.

3–12 is the smallest range which can be ambiguous, and it is used in the PHB! A bardiche inflicts 3–12 hit points of damage on large opponents.

## The New Notation Isn't Good Enough

On p. 10 the DMG explains dice notation and on the following page it describes a method for rolling attribute scores which can't be expressed with that dice notation: rolling 4d6 and dropping the lowest die!

The site roll20.net has some notation for this. The roll can be written as 4d6d1 to indicate the lowest die is dropped, or 4d6k3 to indicate the highest three dice are kept.

There is alternate notation which make it explicit that the lowest die is dropped: 4d6dl1 and the highest three dice are kept: 4d6kh3.  One could call for the highest die to be dropped: 4d6dh1 or the lowest three dice are kept: 4d6kl3.

The lowercase L seems like an opportunity for confusion with a numeral one, so we just use 4d6kh3 for a 3d6 with negative skew and 4d6dh1 for a 3d6 with positive skew.

## Code

We don't like laptops, ipads, or even phones at the table. Nevertheless it was convenient to implement a command line tool which understands dice notation—including the "keep high" and "drop high" extensions:

$roll 6d6 16$ for i in $(seq 1 6); do roll 4d6kh3; done 14 8 7 14 15 10  The code is on GitHub. ## Factor Rolls One can use the dice to generate other ranges of integers:  1–2 ⌈d6/3⌉ 1–3 ⌈d6/2⌉ 1–5 ⌈d20/4⌉ 1–10 ⌈d20/2⌉  In case the notation on the right is not clear, one rolls the indicated die, divides by the following number, and then rounds up. The most practical notation is d2, d3, d5, d10. I'm not aware of a standard term for this type of roll; we've been calling them factor rolls. ## Product Rolls Percentile dice are an example of what we've been calling a product roll. We could use two d6 to create a d36, for example. This is not multiplication, but more like working with base 6 numbers. The percentile dice make the process easier by using zeros and distinguishing the tens die from the ones die. One formula for getting a range 1–36 is 6×(d6 - 1) + d6 Dice of different colors are needed for a product roll. Our convention is to use a white die for the ones die. If dice of different colors are not available, a single roll is not possible; roll the most significant die first. The dice do not have to have the same number of faces. If you wanted the use the 30 sided dice gaming tables published by the armory, you could generate d30 rolls with a d6 and a d10. If factor rolls and product rolls are allowed, then the only numbers we cannot generate are ones containing a prime larger than 5 as a factor. ## Re-rolling Thus there is no way to generate d7 uniformly using a single roll. One could roll a d8 and re-roll if the result is 8. Simply writing d7 is the best notation. Seven sided dice have been manufactured. One design is a pentagonal prism, and another "is based on spacing points as equally as possible on a sphere and then cutting planar slices perpendicular to those directions." It would be interesting to test these dice and see whether the distribution is uniform. ## Saturday, November 26, 2016 ### Tray or Cups? How to roll dice. Jay insisted on today's topic. The problem is if players cast dice on the open table, the dice land cockeyed or go off the table entirely. Is there a better way to roll dice? ## Tray In a way we've been there. We would just flip a Holmes set lid over. The dice popped out all the time, though. Chaosium used to make a 2" deep box which was better at containing the dice. However, the players lean in to see the result and it feels like playing over a craps pit. The dice trays I see for sale online are 1.5" deep, which might be about right. Trays aren't how the Bookhouse Boys roll, though. ## Cups ### Smonet Traditional Professional PU Leather Dice Cup The idea of using cups suggested itself after playing a game of liar's dice. We used plastic cups initially, but they are noisy, so we upgraded to felt lined cups. Cups are great for storing dice. Cups aren't great for rolling lots of d6, since the dice tend to stack. Another downside to cups is players who can't resist playing up the drama with excessive shaking. Jay's catchphrase in this situation is "hurry up and roll, dice bag." ## Friday, November 18, 2016 ### Testing Dice for Fairness Are Chessex dice fair; are Koplow dice fair; how to test dice. Testing dice for fairness is tedious but simple: roll the dice a large number of times and tally how often each face comes up. Here is an example where a set of Chessex dice are rolled 100 times each: We don't expect the numbers to be exactly the same, even if the die is fair. Any outcome is possible from a fair die, though some outcomes, such as seeing a 6 a hundred times and the other faces not at all, are vanishingly unlikely. How do we recognize implausible results from a fair die? My time in the statistics department at Ohio State acquainted me with a test statistic which can be used to answer the question. Let n be the number of sides the die has. Let Oi be the number of times we observe the i-face to come up. Let Ei as the number of times we expect the i-face to come up, assuming the die is fair. The test statistic is: $$χ^2 = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}$$ The test statistic has a Chi-squared distribution with - 1 degrees of freedom. It is used to assign a p-value to the result, which is the chance that a fair die would produce results as or more extreme than what we observed. If you are interested, here is some code for making the calculation. The closer the p-value is to zero, the stronger the evidence that the die is not fair. I took a set of Koplow dice and a set of Chessex dice and rolled each die 100 times. The only die which had a p-value less than .05 was the Koplow d6. However, if we compute p-values for 5 fair dice, there is a 22.6% chance that one of the dice will have a p-value less than .05. To account for this, I applied the Bonferroni correction, which raised the p-value of the Koplow d6 to 0.11. So as far as I can tell, my Koplow dice and my Chessex dice are fair. It doesn't mean yours are too, but you can test them. ## Saturday, November 12, 2016 ### Buying Dice What dice you need; whose dice to buy. Knowing most readers would sooner give advice than receive it, we state our opinions softly. ## Which Dice You Need The original edition of the game called for two sets of polyhedral dice and 4 to 20 six-sided dice. Polyhedral dice aren't hard to get these days, so no reason to limit yourself to two sets. Also, modern gamers will want percentile dice. As a general principle, our group likes to resolve a character's fate with a single cast of the dice. If you find yourself rolling a die repeatedly and summing the numbers in your head, you don't have enough dice! Or so we think. Another group custom is we expect the host to provide dice for everybody. We stopped carrying dice bags around almost thirty years ago. Possibly related: Jay sometimes calls other players "dice bags" when he doesn't like their style of play. With that said, here is what I set out for the players: For the record—and for the search engine—the image shows eighteen d6 with pips, two d100, three d4, three d6 with digits, three d8, three d12, and three d20. I also keep a set of polyhedral dice behind the screen for sealing the players' fate in secret. ## Whose Dice to Buy ### Chessex My fave manufacturer right now is Chessex, founded in 1987. They sell dice in sets of seven like the one above. Chessex sells dice with swirls and speckles. The speckled dice were introduced in 1995. They also sell transparent dice. None of these are to my taste. Sometimes I have to poke around to find the solid-colored, opaque dice with high-contrast paint that I like. I want dice that are easy to read. Chessex dice are made from a hard thermoplastic. They are cast from molds, painted, and then polished in a rock tumbler. The polishing removes paint from everywhere except the indented numerals. The polishing also rounds the edges. By the way, note the distinct way Chessex numbers the d4. Not many manufacturers use this style. There are some other traits that can be used to recognize Chessex dice. One is that opposite sides of a Chessex die sum to a consistent number. In the case of a d20, opposite sides always sum to 21. If you take a Chessex d20 and look at the 1-face, it shares edges with the 7-face, 19-face, and 13-face. I'm not sure if Chessex dice have always had this arrangement, but molds are expensive, so manufacturers don't change them often. Finally, I'll mention some traits which Chessex dice have and which in my mind all good dice should have. 6s and the 9s should be distinguished. Chessex dice use underlines. However, the underline on the Chessex d20 is so thin it doesn't take paint well. The d20 should be numbered from 1 to 20, and not 0 to 9 twice. In a set of percentile dice, the tens die should be distinguished from the ones die with an extra zero digit on each face. ### Koplow I have several sets of Koplow dice. Koplow used to sell sets where each die was a different color. I like these sets because they make it easier to find the die I want. The dice in the above picture were sold as two sets: a set of 6 polyhedral dice, and a set of 4 "value dice". Koplow dice are manufactured the same way as Chessex dice. The edges are more rounded than Chessex dice. Koplow Games was founded in 1974. They make more than dice and I'm not sure when they started making polyhedral dice—I bought a few sets in 2005. Traits for recognizing Koplow dice: opposite faces on Koplow dice don't always sum to the same value. In my set, the d6 and the d10 do, but the rest of the dice don't. Oddly, among the "value dice", the ones die does, but the the tens, hundreds, and thousands dice don't. On a Koplow d20, the 1-face shares edges with the 6-face, 13-face, and 10-face. 6s and 9s are underlined, except on the d20, where a lower right dot is used. I bought the "value dice" because I wanted the percentile dice. The ones, tens, and hundreds dice together make a set of per mille dice. Is such a set useful? The Treasury of Archaic Names contains a table with 1000 entries, but most players will find both the hundreds and thousands die to be superfluous. ## Collecting Dice I'll mention a few more manufacturers. Most of these are no longer in business, but good to know about if you are interested in old dice. ### Creative Publications Creative Publications started selling polyhedral dice in 1972. They were the only supplier of polyhedral dice—at least in the United States—when the original edition of the game came out in 1974. TSR resold them at prices ranging from \$1.49 to \\$3.00. The dice were included with the Holmes Basic set. In fact, most dice were probably sourced by TSR directly from manufacturers in Taiwan Hong Kong.

The Creative Publications dice were made from a soft plastic that didn't wear well.  The dice would lose their color and the d20 would become nearly spherical. Players took to calling them "low impact" dice when harder dice became available.

The Creative Publications d20 was numbered 0 to 9 twice. Opposite faces have the same number. The 1-face shares edges with the 5-face, 9-face, and 6-face. Creative Publications dice are easy to identify because the same color was always used for each shape.

Creative Publications was located in Palo Alto, CA. They became a supplier of dice to gamers by accident, as they were primarily a publisher of educational titles. One of these is "Polyhedra Dice Games: For Grades K to 6" from 1978. The book contains 40 games, each of which requires one of the polyhedral dice for play. Each game requires a skill such as counting, adding, or distinguishing odd from even. Otherwise they are simple games of chance and not interesting to adults. The cover shows a sea of dice from the collection of Dale Seymour; a few of the Creative Publications dice are mixed in.
TSR also sold sets of two d20s numbered 0 to 9 twice. They were used for percentile rolls—the pink die was the tens digit. I'm not sure if Creative Publications ever sold sets with a pink die in them.

### Gamescience

Gamescience has been run by Lou Zocchi since 1973. They are still in the dice-making business. Most of their dice are sold unpainted; an ultra fine Sharpie works well for inking in the numbers.

Gamescience dice are not polished, so they may have burrs left over from the molding process. The molding industry calls these burrs flash. Some people sand them down with fine sandpaper.

Gamescience made a name for itself by introducing dice made out of harder materials. In fact, if I'm not mistaken, they popularized the terms "high impact dice" and "low impact dice". They also introduced dice in a number of new shapes, the most important being the d10, which they starting manufacturing in 1980. Update: In a 2015 interview, Lou Zocchi credited TSR with introducing the first d10 circa 1980. These were the dice in the Moldvay basic box set. Zocchi also says these were the first sets with a d20 numbered 1 to 20. Gamescience was the first company to introduce a d10 numbered 00, 10, ..., 90 for the tens digit in a percentile role. The die was advertised in the April 1990 issue of Dragon magazine:

Gamescience was an early distributor of TSR products. Zocchi and Gygax were acquainted; both wrote for Guidon Games. As a reseller of TSR games, Zocchi started ordering dice from Creative Publications. Zocchi was dissatisfied with the price and the availability, so he started making his own dice—initially just a d20—in 1974.

The original Gamescience d20 was numbered 0 to 9 twice. Early gamers would ink half of the numbers with a different color so a number from 1 to 20 could be generated. In 1980 Gamescience invested in a mold where half of the faces had + next to the numbers. The dice with + signs aren't common; d20 dice numbered 1 to 20 became the standard soon after.

Some traits useful for identifying Gamescience dice: the d4 are truncated tetrahedra. The d20 has a small capital G on the 1-face in addition to the numeral 1. The d20 1-face shares edges with the 11-face, the 19-face, and the 18-face.

Opposite faces on the Gamescience d20 sum to 21, and opposite faces on the d6 sum to 7. The other dice don't have faces that sum to a consistent number.

### The Armory

The Armory was a gaming shop in Baltimore, better known as a dealer in miniatures. I've read that the Armory got started in the dice business by inking or painting Gamescience dice and re-selling them. This was back in the 1970s.
Eventually they had their own molds made. The earliest Armory dice used the letter A instead of the numeral 1 on the 1-face of the d4, d8, and d20. As far as I know this was not done on the d6 and the d12.
The Armory introduced a d30 die around 1982. The older d30 is numbered 0 to 9 thrice, each digit appearing with a plus sign, minus sign, or no modifier. The newer d30 is numbered 1 to 30. The dice were supported with a book called "The Armory's 30 Sided Dice Gaming Tables". It could be used with any of the fantasy RPGs of the time.

In later years, they seem to have outsourced manufacturing to other companies. In 1998 they merged with Chessex. Dice aren't being sold under the Armory brand anymore.

### TSR

TSR resold Creative Publications dice for many years. Update: the dice in the above picture come from two different molds. The blue dice came with the first printing of the Moldvay basic set. The green and orange dice came with the first printing of the Mentzer basic set. For a guide on how to distinguish them, see this post and this post.

By 1981 TSR was having their own dice manufactured. Pictured on the left below are dice from the Moldvay basic set; on the right are dice from the Cook expert set.

The dice did not come painted or inked in. Crayons were provided to color in the numbers.

The dice seem too small to me. The dice in my expert set are a mix of colors, but I think that is atypical. I've seen all-brown and all-orange sets.

The dice in the first printing of the Mentzer basic set were sold separately in a blister pack as "Dragon Dice". They are unrelated to the collectible dice game, also called "Dragon Dice", which TSR introduced in 1995.

## Saturday, November 5, 2016

### Welcome to Athenopolis

Announcing a new site.

Lew wanted a site, and now we have one. The time has come for those beer-inspired ideas to be written up and posted. Let's hope they hold up to the scrutiny of the discerning public we are sure to attract.

For topics I propose one: how to play the game well. I also move to ban any and all in-world adventure narratives. On the latter point I see nods of approval all around, so I take the motion to be seconded and approved.

Comments will be open to the public. We'll see how that goes. We can just delete anything that isn't polite or on-topic.