Tuesday, February 24, 2009

Proposal: Weapon Classes

I rather quickly shot off a proposal for simple weapon vs. armor adjustments a few weeks ago; I wasn't completely happy with it, and I've been meditating on the subject ever since.

I suppose that you can take the subject in one of two directions, depending on your motivating principles. One: You could decide that you really want to work up a simulation of the penetration ability of certain weapons against different armor types, within the context of the D&D combat system. Two: You could instead decide that your overarching goal is to give each weapon some metagame reason to be chosen by characters in different circumstances.

Having mulled it over, I think I'll have to pick option #2 as my priority. I will again point out DMG p. 9, where Gygax makes clear that for D&D, the designer is supposed to put game-design principles ahead of realism-simulation, when a choice must be made between the two ("This is not to say that where it does not interfere with the flow of the game that the highest degree of realism hasn't been attempted...") Taking this longer-view perspective, the answer will sometimes be to-hit-adjustments, and at other times be something entirely different. It will not be systematizable in a matrix of numbers.

So here is a new proposal for distinguishing between different weapon types. First of all, let me be clear that I use the variable weapon damage from Greyhawk, but ignore the separate statistics for L-size enemies (and other modifiers). Hence weapon damage follows this simple rule:

Weapon Damage
  • d4: Dagger, sling.
  • d6: Hand axe, mace, spear, arrow, quarrel.
  • d8: Sword, battle axe, morning star, flail, pole arm, lance, pike.
  • d10: Halberd, two-handed sword.
Now let's divide up the different weapon types into a few recognizable categories, based on how the blow is delivered, and come up with a concise reason why they might be chosen over other types of weapon (melee weapons only for today). As usual, these points should be easy to remember, so we don't have to look up the information while the game is in progress.

Weapon Classes
  • Swords (dagger, sword, two-handed sword): Blades are light and compact. They can be easily carried in a scabbard (half encumbrance), and can stab in constricted spaces where other weapons are unusable (e.g., narrow tunnel, monster gizzard, etc.)
  • Spears (spear, lance, polearm, pike): Pole weapons provide reach. The wielder gets a free attack when readied against an onrushing attacker with a shorter weapon; however, in each later round of melee they permit a free attack on the part of the enemy.
  • Axes (hand axe, battle axe, halberd): Chopping weapons can sometimes cleave through heavy armor. They get a +1 to hit opponents wearing chain or plate.
  • Clubs (mace, flail, morning star): Bludgeons can deliver shocks directly through plate without penetrating the armor. They get a +2 to hit opponents in plate mail.
Now, we could certainly go further and provide specific modifiers or exceptions to each individual weapon in a class (some will already be obvious; notice that the first item in each class is usually throwable, the last item is usually two-handed, and so forth). However, this is the point that I think the advantages still outweigh the disadvantages of such a system, before the DM has to either memorize or look up a dozen different rulings for all the various weapon types. So I'll stop here with a nice amount of mechanical flavor for each of the different available weapon choices.

Saturday, February 21, 2009

Review: M5, Talons of Night

M5: Talons of Night
D&D Master Game, Levels 20-25
By Paul Jaquays

Talons of Night is probably one of the best overlooked gems of the D&D game. It has some of the most imaginative settings and situations I've ever seen in a high-level D&D module, usually depicted in a small number of pages. It's written by Paul Jaquays, who also did all of the interior illustrations (and also authored the famed Judges' Guild adventure Dark Tower, co-developed Twilight Calling, Egg of the Phoenix, Forests of Alfheim, Top Ballista, and the humorous version of Castle Greyhawk, among others).

This M5 adventure is driven by the event of a great peace conference between the competing empires of Thyatis and Alphatia. The PCs are sent from their domains in Norworld to hunt for ancient documents settling the ownership of that region, along with an artifact known as the Peaceful Periapt of Pax. Later, the leaders of the empires are kidnapped by the forces of Chaos into another plane of reality, and the PCs must rescue them. The adventure has 3 distinct sections.

Part 1 is called "The Quest for Peace". Here, the PCs are sent on the initial mission into Thothia on the Isle of Dawn (a territory between Thyatis & Alphatia composed of many minor kingdoms). Thothia is based on ancient Egypt. It's ruled by a decadent pharaoh who is married to an immortal spider-woman, herself the daughter of an evil spider demi-goddess. The PCs likely search the old library in the capital Edairo, then are chased inland up a dangerous river to the ancient Temple of the Dawn. There they meet the pharaoh's vampire ancestor and play him in a novel "Spider's Web" boardgame (based on Mill or Nine Men's Morris); each victory reveals a clue to the future of the adventure.

In Part 2, "Against Aran", the PCs have learned that the Periapt of Pax is guarded by the Night Spider somewhere in the uplands north of the temple. This region of forest, hills, and plains are populated by contentious clans of phanaton and aranea. In order to search all the aranea villages, the PCs are encouraged to muster the phanaton into mass fighting units using the War Machine rules. When the PCs finally find the capital Aran and attack the Night Spider in a miles-long webbed pit underground, they find themselves forced to play the Spider's Web game again, this time for their lives and the artifact prize.

Part 3, "Journey into Night", is one of the most dizzying adventure sequences I've seen in a D&D game. The PCs go to the peace conference to find the emperors kidnapped and themselves framed for the deed. They return to the Night Spider's pit to find a gate to the world of Thorn. This world is a giant splintery thorn-bush in distant space; it is the original homeworld of the phanaton and phase spiders who dwell in peace there. (Two side-effects occur in Part 3: every new world has its time passing at 10 times the previous one, and the life-force is sucked out of the PCs, leaving them as zombie-like creatures.) On Thorn the PCs battle the extraplanar home fortress of the vampire pharaoh, his queen, and the Night Spider. Then they find another gate to "Chasm ", a frightening world of evil clinging to the side of a colossal cliff. Here there is a giant volcano shaft filled with evil slime, and a cube-world suspended on a black ray of life-sucking energy. In this setting the PCs must ascend into the cube world and play a tessaract-like variant of "Spider's Web". Victorious, the PCs find another gate to the "Isle of Night", an extradimensional shrunken toy-version of the Isle of Dawn, where the zombified-emperors fell in love and founded a kingdom some 400 years in the past. Finding their crypt, the PCs can recover sealed packets of flesh, return home by the power of the artifact, and make use of a clone spell to recover them.

Phew, got that? This is all in the course of 6 pages of text for Part 3 alone. (!) But we're not done yet…

At the end, there is an "Epilogue" which is a unique game unto itself. Here the PCs role-play all the various factions in the peace conference, with points awarded to each depending on their individual priorities and goals. If everyone ends happy (sufficient points), then peace prevails. If not, the D&D world is plunged into war for decades thereafter.

As you can tell, Talons of Night represents a true D&D auteur at the height of his powers of imagination, game-mechanics mastery, and even artistic flair. The games-within-games (such as the Spider's Web boardgame, the phanaton-versus-aranea War Machine campaign, and the Epilogue peace conference rules) seem to be extremely solid and well thought out. Jaquays' interior illustrations do more to capture the flavor of his wild settings than most other adventures I can think of.

Now, here are the few critical notes I can think of. One, I'd wish the adventure were longer to more fully expand on the worlds of Thorn, Chasm, and the Isle of Night (which only get 1 or 2 pages each as written). Two, the author perhaps over-uses the Spider's Web game (using the basic theme and layout in at least 5 different encounters that I can count). Three, the module also has one of the most spectacular covers in D&D publishing history, with an atomic-powered Cthulhu-like monster blasting an entire kingdom to ruins. However, that image has nothing to do with the adventure inside. (Similarly, there are references to Alphaks and Night – chief hierarch of the immortal Sphere of Death – but neither makes any appearance in the actual adventure.)

Those quibbles aside, Talons of Night stands as a truly impressive adventure that stands head-and-shoulders above others in the same publishing line. I'd like it to serve as a model for what high-level and extra-planar D&D adventures should look like. It's highly motivated me to seek out other adventures by the author Paul Jaquays.

Monday, February 16, 2009

Interesting Choices

A good game is a series of interesting choices. -- Sid Meier

Just wanted to post this quote up for posterity (I've heard it before, it was quoted in each of the last 2 issues of Game Developer magazine, and it's been on my mind anyway lately). Even if I'd never heard of it, I'd probably still say something similar, just not so concise.

Saturday, February 14, 2009

Review: M4, Five Coins for a Kingdom

M4: Five Coins for a Kingdom
D&D Master Game, Levels 28-32
By Allen Varney

Five Coins for a Kingdom is something of a recovery from the feeble module M3. It definitely has some interesting ideas and a creative spark by author Allen Varney. The cover and interior artwork is excellent and unique, by the team of John and Laura Lakey. I'm not sure that all of the ideas work in the context of a D&D game; I'll discuss those in detail below.

The guiding plot of the adventure is that on another plane, a group of five good wizard-kings are at war with an evil archmage named Durhan. This evil wizard has managed to imprison the good wizards and cast a spell to make their entire capital vanish and fly into space. The PCs will have to travel to this foreign realm and save the day.

The adventure has five chapters. Chapter 1 is "A City Vanishes", in which the PCs' beloved city (presumed to be their capital in the Norworld campaign setting) initially vanishes, as a ripple-effect of the evil wizard's spell in the other dimension. However, at this time they see five magic coins shoot down from space (sent as a last act of warning from the good wizards, and keys to enter their dimension).

Chapter 2 is called "Hard Work for Small Pay", wherein the PCs recover the five coins from several wilderness sites around the location of their vanished city. Here they'll meet a sphinx family, a watery ghost, an underwater devilfish tribe, and a mercenary mountain giant on a mission to kill a blue dragon.

In Chapter 3, "Arrival in Eloysia", the PCs use the coins to travel to the other planar realm. The capital has vanished except for the protected residences of the five wizards, where they can find assistance and clues to their hidden prison, and finally free the good wizards and escape from immediate danger.

Chapter 4, "Against Durhan" has the PCs move against the evil ultra-wizard himself; here, they need to marshal and lead the good wizards' army against his invasion forces. This includes an entire War Machine mass-combat scenario (which you may or may not like, depending on your taste for those rules).

Chapter 5 is called "Into the Sun". After victory in the realm of the good wizards, they return home to find that their own city is still in danger, specifically, having been flung into their own Sun. They must travel to the heart of the sun and negotiate with the ruler there in order to return their home city.

Now, I've avoided the overall structure of the planar realms until this point, but you may have just gotten a clue from the title of the last chapter. My main dispute with this adventure is that the otherworldly locations don't feel like high fantasy -- they feel more like sci-fi, and are detailed and explained in that way. For example, the good wizards' plane of Eloysia is an inverted solar system -- space outside the solar system is not a vacuum, but solid matter. Within the open sphere of the solar system is a sun, surrounded by thousands of small flat asteroid-nations facing that sun. The closest region is for mining; a middle region used for agriculture; the outer realms for cities and high culture. Rainstorms and giant fishlike creatures cycle through the space. If you fall off the edge of a realm, the solar wind propels you to fall thousands of miles to the solid boundary of the system.

So, overall, this is fairly imaginative, but the whole setup feels a bit too mechanical and scientific for my tastes. The evil wizard's spell has both vanished the good wizards' city and sent its chunk of rock flying towards the sun, so the events in Chapter 3 occur as a hot sun grows larger in the sky minute-by-minute. The mass combat in Chapter 4 is set around the evil wizard having snared his dark realm to the good wizard's realm via miles-long cables and grappling hooks (so the armies are actually fighting on top of the cables, with hosts falling off into the abyss, which is a pretty cool idea). In Chapter 5, numerous physical details of moving through outer space and the physics of the Sun are presented (such as density, pressure, sunspots, and darkening wavelengths of light as PC's travel to the center). Is it reasonable to have the Sun be a more dangerous setting than the Elemental Plane of Fire, for example, in a D&D campaign? I'm not so sure that's the right move.

There are some good things in this adventure. The theme of the five coins (the five monetary types in D&D; the five wizards using what tokens they have, being imprisoned in their own treasury) is very effective. The different wizard characters are pretty well developed in the text (although I'm not sure how effective that will be in the course of play). There's a great scene where the PCs have to wrest leadership of the good wizard's army from a group of painfully, aggravatingly too-Lawful Archons.

There are also some things here that don't work well, starting with the sci-fi environments mentioned above. One thing is that there's several action scenes where the DM is encouraged to "let the PCs act if they wish", but in the end is told that absolutely nothing they do will have any effect on the proceedings (for example, the fairly long initial city-vanishing scene). Personally, as a gamer-centric DM, I think that's bad form. There are numerous places which exhort everyone to do some character-oriented roleplaying, but it never really has an affect on the adventure itself. There are suggestions for "random encounters", without any specified encounter chances, intended to slow up the PCs if they're doing too well (and likewise help that shows up at certain points just in case they're failing).

There's also an evil artifact being used by Durhan which is a belt that shoots hundreds of golden caps, connected by a network of wires, to control lesser wizards and suck their power into the evil wizard. The whole concept seems incredibly clunky and, again, more like a science fiction idea than magical fantasy.

So, in summary: M4 has some interesting ideas and a clear creative spark. Overall, however, I don't think it would work in the kind of D&D campaigns that I have personally run in the past.

Proposal: Weapons vs. AC

So for a few weeks I've been orbiting around the idea of ironing out a set of weapon-vs-AC rules that I'd be happy to use in conjunction with OD&D. At the same time, James Maliszewski, on his excellent Grognardia blog, put out a call for ideas on the same topic ( http://grognardia.blogspot.com/2009/02/request-for-assistance.html ).

Having considered it today, I think now that I would only want to use an extremely simple couple of rules to put this into play in any game of mine (basically nowadays I want any mechanical rule to be outright memorizable). From a gaming perspective, almost the only thing I need is some boost against plate armor from maces and flails (and whatever else concensus opinion holds as being specialized for cracking plate).

A little research seems to suggest the following qualities for the different types of armor (note that I am discarding shields as creating any distinct types):

Defense
LeatherChainPlate
SlashGoodGoodExcellent
PierceGoodGoodGood
CrushExcellentPoorPoor
And this I would think to turn simply into the following:

Weapons
LeatherChainPlate
Slash00-2
Pierce000
Crush-2+2+2


Notes:
  • Slashing: Hand axe, sword, battle axe, 2-handed sword.
  • Piercing: Dagger, spear, pole arm, lance, pike, arrow.
  • Crushing: Mace, morning star, flail, halberd.
What I'll mostly leave unanswered at this time is the thorny question of how much of this should be applied against monsters. In general, it seems to me that most OD&D monsters can be assumed to be either wearing actual armor or natural hide of the same type as their AC (humanoids, giants, dragons, horses, chimerical beasts, slimes and molds, etc.). The significant exceptions would be creatures who appear to be semi-incorporeal or invisible in some way (wraiths, spectres, stalkers, and elementals). Those I'm not sure what I'd do about.
Now, I'm not planning on actually using this right now. But if I did want to give a taste of different weapon effects (this in combination with variable weapon damage, and the effects of reach for pole weapons, giving a nice cross-matrix of capabilities), it's about how I'd want to do it, unless there's something really glaring that it turns out I'm missing.


Update -- Weapon Classes

Friday, February 13, 2009

Links to Armor/Weapon Articles

Frequently I find myself researching armor & weapon details at Wikipedia, and I wind up poking around for the most appropriate articles (especially for a few oddball cases). Collected here are links to useful Wikipedia articles, with the same content and order as the OD&D equipment list:

Melee Weapons

Missile Weapons

Armor

Notes: I've used the Francisca to represent a throwing-style "hand axe"; the German Zweihaender as the big "two-handed sword"; the Arbalest for the "heavy crossbow"; and the padded Gambeson for "leather armor".

This last choice might be controversial, but in a medieval context the most likely cheap armor for infantry (and also, underlying mail/plate) would be the padded/quilted armor of this sort. In this sense, the OD&D listing is in some sense erroneous, but it's possible to interpret it as a leather-fronted "padded jack" (see article link above). If you prefer to go by the text of the 1E DMG ("shaped cuir bouli... cuirass and shoulder pieces", p. 27) then you might consider it Lamellar armor, but that was used mostly in earlier eras or by non-Europeans (e.g., Sumerians, Egyptians, Byzantines). Alternatively, you might consider it the same as a leather Buff coat -- but that's a later European development of the 17th century. Or see here for another take on quilted cloth, leather, and cloaks.

The one other thing that sticks out anomalously for me in OD&D is the Halberd, that being given a significantly more specific identification than anything else on the list. Personally, I let this item stand in for any two-handed poleaxe-type weapon (including a bardiche, lochaber axe, etc.; I'd be happier if it was just identified as a "poleaxe" in the original rules).

Sunday, February 8, 2009

Follow-Up: Testing Balanced Dice

When I wrote the previous blog on how to statistically test for balanced dice, I hadn't actually used the test on any of my own dice (as was requested in at least one comment). But shortly thereafter I started to get worried that I would be compelled by curiosity to (a) test all the dice in my house, and (b) wonder what testing dice-manufacturers use, if any, for quality assurance purposes.

For part (b), I'm broadly guessing that the answer may be "none" in any sense other than an eyeball-check. If anyone has other information I'd be interested in hearing it.

For part (a), I haven't checked all my dice, but I did check all my d20's, and the results were rather interesting. Again, the process outlined earlier is this -- roll 100 times, count the frequency of each face, subtract 5 and square each, then add them all up; at the end you have the SSE "sum-squared-error" (higher results indicating a more unbalanced die).

I started by testing all my "newer" sets (at least a couple years old), mostly translucent, pre-inked digits, with slightly softer edges (many by Chessex, I believe). The SSE results I got for them were 150, 148, 106, 100, and 98. Recall that we were going to reject a die as clearly unbalanced if the result was more than 150; the first two d20's I tested (which I was already suspicious about, because of prior in-game behavior) just barely squeaked through the test. The P-values for these dice would be 0.05, 0.06, 0.33, 0.39, and 0.42 (you can sort of think of these as the probability that each die is in fact balanced; that's not exactly correct, but a fair analogy for what P-values represent -- a higher value is better news for the "balanced die" hypothesis). For the first two dice, we've got at least moderate-level evidence that the die is unbalanced.

Now at the end, I tested what I presumed would be the weakest die in my collection: an older translucent red d20, with sharp edges, that I had to color in myself with a crayon. The other dice in this set still show the tab from where it was snapped off the molding sprue (although I can't see it on the d20 itself; these dice are probably from Gamescience). Well, unexpectedly to me, this d20 had the lowest error of the bunch: SSE = 80, significantly lower than anything else I had in the house, and clearly the fairest-rolling die of anything I tested (P-value = 0.66).

So my theory now would be that a die that has sharp edges is more likely to roll fairly than one that has rounded edges, even though I've been avoiding this "sharp-edged" set for years now because to my eye it looked less professional.

Back to item (b) above. Interested in seeing what, if anything, the dice manufacturers have to say about their QA process, I landed at the Gamescience website ( http://www.gamescience.com/ ). And the fascinating thing is that the home page has a couple of videos by the long-time owner Lou Zocchi, where he rants for a total of about 20 minutes on exactly this subject. Namely: (a) he intentionally produces dice with sharp edges, similar to requirements for casino gambling dice; (b) people complain about the cost, the lack of inked digits, and still seeing the sprue-tab, and (c) his competitors solve these problems by mass-painting the whole die and then throwing them into a tumbling sand grinder (removing excess paint, the sprue-tab, and rounding all the edges in an irregular fashion).

So I hadn't known about Zocchi's position before, and I hope I don't come off as a shill for Gamescience in this paragraph. (Zocchi doesn't use formal statistics in his presentations; he uses more user-friendly demonstrations, like stacking up competitors' dice and seeing that you get obviously different total heights from different directions). But at this point in time, I'm basically sold on that position; if I wanted dice that I really knew would roll in as fair a distribution as possible, I would go out of my way and get sharp-edged dice from Gamescience.

This is not to say there's no downsides. Frankly, their dice are more expensive. Moreover, I must say that the rounded-edged dice are a little more pleasant to look at and hold in the hand without the really sharp corners (and even more so to step on in stocking feet if you leave them on the floor; Gamescience dice are like deadly little caltrops).

But I'm glad I found Zocchi's videos, because an older gentleman devoting his life to fairly-rolling dice, and delivering a really angry rant about correct probability distributions, really warms and delights my heart.

I see on the Gamescience web site that Lou is selling the business at this time to Gamestation, Inc., which makes me a little sad, just at the time when I discover the advantages to his product. I'll salute Lou Zocchi for his years in the industry and hope that his dice stay available with the same level of quality in the future.

Saturday, February 7, 2009

Review: M3, Twilight Calling

M3: Twilight Calling
D&D Master Game, Levels 30-35
By Tom Moldvay

This is not a good module.

Here's an initial taste: The best part of the module is the title and the cover art -- a massively elaborate castle/citadel, floating on an asteroid-like crag over an endless abyss. It has scores of towers, turrets, and buttresses, each dozens of stories tall; a truly amazing fantasy locale. This is the location of the climax of a dimension-spanning adventure, where the PCs finally have a showdown with a millenia-old imprisoned evil sorcerous alien race.

Once you get there, you find that the front gate is a normal, locked wooden door -- and the module assumes that the only resource the PCs have to get in is to pick the lock. When you get inside, you find that the entire setting has just three (yes, 3) encounter areas. To quote the module text, the rationale for this is that, over the years, "As their numbers dwindled, they sealed off most of the castle (magically filling in the open areas with solid rock)." Man, what a gyp.

Okay, now let's start at the beginning. We establish that there's an ancient sorcerous dinosaur-like race called the Carnifex who were so dangerous that the gods removed them to a prison plane thousands of years ago. There's a hazy setup whereby the Immortal villain Alphaks decides to trick the PCs into releasing them, when they don't really need to. However, we never hear about Alphaks again and by the end of the module we indeed find the Carnifex about to invade the PC's home world, so they really do need to confront and stop them at just that time.

There's two overall parts to this module. Part I is called "The Seven Realms". In order to get to the prison realm, the PCs need seven key items, each held in a special pocket universe. These are reached through a Stonehenge-like edifice in the Broken Lands, each henge color-coded with a rainbow hue (and how many times have you seen that).

Although the realms can be entered in any order, each realm by itself is entirely linear, being a series of railroaded encounters -- between 7 and 3 encounters in each dimensional realm (dwindling in the amount of detail as you get further into the module). Most of the time, the encounters are just a single monster that jumps up out of nowhere and attacks. I counted at least 3 times where the setup was, quote, "[the monster] will immediately attack the characters, giving them just enough time to draw their weapons." (Yes, those exact same words are duplicated several times.)

Part II is "Carnifex Castle", which I basically described at the start of this review. The three encounter areas are a combat with a bunch of high-powered guardian undead, a corridor with a bunch of traps, and the Carnifex shrine. It's a bit unclear what order they're supposed to be in, because the text and maps for the areas come in different orders.

Again, the encounters a very programmed, with tiny little gestures attempting to act like the whole thing isn't totally linear. There's a horn at the gate that pulls one monster out of one encounter, which is then replaced by a different monster. In the combat zone there's a bunch of pillar/walls set up to little effect. The trap zone has traps like a pit of poison barbecue sauce (no, I'm serious -- because the Carnifex are hungry carnivores, get it?), a teleporter into a giant meat grinder, a "bakery walls" heat trap, etc., etc. Then at the end you fight 4 Carnifex wizard/clerics, and also the PCs get duplicated and have to fight themselves (ever seen that?). At that point they have to jump through the Carnifex's attack gate as overwhelming numbers run in from a nearby barracks living area or something, and of course the whole millenia-old castle collapses right at that moment.

This module is definitely the worst of the Master-level series, and it feels rushed and basically "phoned in". There's very little in the way of interesting, novel encounters to recommend it (there's one encounter with two black dragons trying to mate, so the male attempts to defeat the PCs in as showy a fashion as possible, the only scene that struck me as halfway clever). Areas are misnumbered and out-of-sync between map and text. The setup is poor, the planar realms are vague and uninspiring, the encounters are generally a railroaded sequence of solo monster attacks. There are no new spells or magic items, and even the arch-villain race of Carnifex are lacking a monster description entry! If you're looking for old D&D modules to pick up, this should be among the last of the publications that you consider.

Wednesday, February 4, 2009

Testing a Balanced Die

If you're like me, at this point you've seen a whole bunch of fellow players keeping track of their "lucky" dice, trying to "train" dice by storing them best-side-up, stuff like that. I got to wondering recently what it would take to practically test whether you've got a fairly balanced die or not, and fortunately I've read enough statistics at this point to finally track down what it would take. (I was a bit torn about whether to put this post here or in my math blog; I do think it belongs here a bit more, because it's a specific application of a pretty well-known "Pearson's chi-square" hypothesis testing procedure.)

Testing a d6: Let's say you've got a d6. Roll it 30 times. Keep a tally of how many times each face comes up, from 1 to 6. (Note that we expect the number of appearances from each face to be about 5; 30/6 = 5). At the end, go through the counts and subtract 5 from each, square them all, and then add them all up. For a fair die, the total at the end should be no more than 55.

Testing a d20: Alternatively, say you're looking at a d20. Roll it 100 times. Again, keep a tally of how many times each face comes up, from 1 to 20. For each of the counts, go through and subtract 5, then square them, then add them all up at the end. For a fair die, the total at the end of this process should be at most 150.

Comments: The process specified above uses the minimum number of rolls you can get away with (5 times the number of sides; more on that below). It has a significance level of 5%; that is, there's a 5% chance for a die that's actually perfectly balanced to fail this test (Type I error). There's also some chance for a crooked die to accidentally pass the test, but that probability is a sliding function of how crooked the die is (Type II error). A graph could be shown for that possibility, but I've omitted it here (usually referred to as the "power curve" for the test).

How does this work? As mentioned above, it's an application of the well-known "Pearson's chi-square test" in statistics. The process results in a random number whose probability follows a "chi-square" curve after some large number of rolls. Formally, the different faces of the die are called the potential "outcomes" of rolling the die; the count of how many times each one comes up is called the "frequency" of that outcome; and the number of times you expect to see each one is called the "expected frequency". For the "chi-square" function to be applicable, you've got to have an expected frequency of 5 or more for each possible outcome (hence the requirement for a number of rolls of at least 5 times the number of sides).

The numerical process after you're done rolling can be referred to as the (very, very common) method of finding the "sum squared error" (abbreviated SSE). The "error" is how far off each frequency was from the expected value of 5 (hence count - 5); the "squared error" is when you square that value (thus making everything positive, among other niceties); and the "sum squared error" (SSE) is when you add all of those up. If the die showed every face exactly 5 times, the SSE would be exactly zero; the more crooked it is, the more error, and hence the larger the SSE value at the end.

Normally in Pearson's procedure, you'd take each "squared error" and divide by the expected frequency, then add those, then check a table of chi-squared values to see how likely that result was (compared to your initial expectation). But since we expect every side of our dice to be equally likely, there's a simplification that I've done above. For example, for a d6 test at a 5% significance level (degrees of freedom one less than sides on the die, so df = 5), I go to a table of chi-square values and look up X^2(5, 0.05) and see 11.070. That means I would normally reject the fair-die hypothesis (i.e., the null-hypothesis H0, "there is no difference from a fair die") if X^2 > 11.070. Under Pearson's procedure this would imply the following:
Σ((O-E)^2/E) > 11.070
Σ((O-E)^2/5) > 11.070
Σ((O-E)^2)/5 > 11.070
Σ((O-E)^2) > 11.070 * 5
SSE > 55.35

(Above, the "O" stands for the actual observed frequency of each outcome, and "E" stands for the expected frequency of each side, which for us is 5 in every case). A similar simplification is done for the d20 process.

Now, if you wanted to improve the accuracy of the test you could obviously roll the die more times. You would then be able to reduce the chance for either a Type I or Type II error, but never totally avoid either possibility. (In practice, we normally keep the Type I error "significance level" fixed, and work to reduce the Type II error, thus improving the "power" of the test).

I'll end this with how you can form your own test, for any die, to whatever power level you desire. I'll assume you keep the significance level at 5%, and let E be the expected number of times each face should appear in your experiment (E = rolls/sides; must have E >= 5!). Then you will reject the H0 "balanced die hypothesis" if the SSE you get at the end is greater than X*E, where X is the table lookup for X^2(sides-1, 0.05). For a d4, X= 7.815; d6, X=11.070; d8, X=14.067; d10, X=16.919; d12, X=19.675; and for the d20, X=30.143. Have fun.

http://en.wikipedia.org/wiki/Pearson%27s_chi-square_test

Edit: Someone in the comments asked for a look at an actual working example -- included below in a photo of my scratch paper from the last time I tested a d20.


See also: