Man vs. Machine: Chainmail Jousting Live Tournament

“If ever Dan challenges you to a game of Chainmail Jousting, don’t do it. Just don’t do it! He has a system…”—Paul Siegel, Wandering DMs

I was properly warned. But when I got an email from Wandering DMs co-host Dan Collins earlier this week with the subject line: “Jousting Sunday?,” did I heed the warning? Of course not, I’m an adventurer after all.

This week on Wandering DMs, Dan and I tilt in the lists. My strategy is based on an analysis of Chainmail’s Jousting Matrix, outlined here. I rank each aiming point and defensive position using a simple point system.

Dan’s strategy is based on the Nash equilibrium. It’s a math thing. Essentially, as Dan explains, the goal of calculating the Nash equilibrium is to “optimize the possibly-infinite sequence of ‘if you know that I know that you know that I know’ decisions.” Or, as I understand it, Dan fed the Jousting Matrix to the machine, which coughed up the optimal strategy for winning a joust, and Dan turned the results into a weighted table.

It’s an age-old scenario: a human does a thing well until some other human builds a machine that does it better, faster, stronger… I’m not talking Steve Austin. I’m talking less fictional characters against automated opponents: John Henry vs. the steam drill, Garry Kasparov vs. Deep Blue, Jeopardy! champions vs. Watson.

In all these cases, the machine wins! Have I got a chance…?

Watch live, Sunday, July 24, at 1:00 p.m. EDT: “Jousting in Original D&D | Live Tournament,” on Wandering DMs. Root for the human.

The Phalanx and the Shield Wall

In three issues of The Strategic Review, Gary Gygax describes the spear and its kin: the javelin, lance, and pike, for fantasy adventure gamers (Vol. 1, No. 1, 1975) and details several other pole arms, giving use, length, and a drawing of each (Nos. 2, 4). He revisits the topic in a 1979 Dragon article, “The Nomenclature of Pole Arms” (#22), which is reprinted in Best of Dragon Volume II (1981) and, as an appendix, in AD&D Unearthed Arcana (1985). The corpus is often called Gygax’s “treatise” on pole arms.

The Pandemonium Society recognized the early rendition as a useful resource. Study of the historical use of pole arms, notably the pike, leads inevitably to the phalanx formation and house rules such as those Phenster describes in “Phalanx Fighting.” When we realize we can get two or more weapons in the same frontage normally reserved for one, the tactical advantage is clear.

Phalanx Formation [E]

Whatever its historical meaning, a phalanx, for the Pandemonium Society, is a combat formation in which a spearman—or any combatant armed with a long, thrusting weapon—fights from behind an ally. The phalanx formation is best achieved when the opponent is prevented from closing with the spearman by some means, multiple allies in the fore rank for example.

A pole arm at least eight feet long can attack through one rank; 12 feet or longer, through three ranks; 16 feet, four ranks; 20 feet, five.

Additional Weapons

Phenster’s article mentions two weapons not given in Holmes. For completeness sake, I give them here, each with its weapon class, qualities (see “Weapon Damage and Attack Priority”), and cost, the last of which I determined by my own fiat.

Short Sword [E]

Length: 1-1/2' to 2', weapon class: ordinary (damage: d6), weapon quality: short, cost: 7 g.p.

Long Spear [E]

Length: 10' to 15', weapon class: ordinary (damage: d6), weapon quality: long, two-handed, cost: 3 g.p.

Spear Against Charging Opponents [E]

Any spear versus a charging opponent adds 1 to damage. Set (as against the ground or a wall) versus a charging opponent, it adds 2 to damage.

Maximum Weapon Length in the Dungeon

Two options for limiting overlong weapons in the dungeon’s confines are given here. The latter gets tedious in execution, therefore, I put it in the [P] Pandemonium category (see “About the Reedition of Phenster’s”).

Maximum Length by Weapon Type [E]

In Greyhawk, OD&D Supplement I, Gygax prohibits weapons in the dungeon by type: pikes and pole arms “are not usable in dungeons as a general rule due to length” (15). This works well enough in a campaign with dungeons of usual dimensions.

Maximum Weapon Length Equals Ceiling Height [P]

Hazard’s ceiling height rule is more specific, and Phenster elaborates on the penalties. Carrying a weapon longer than the dungeon’s standard ceiling height is awkward and makes more noise. The DM should reduce the party’s movement to three-quarters normal rate and increase the frequency of wandering monster checks.

Close Order, Ranks, and Quarters

Phenster uses three terms that may be unfamiliar to adventure gamers without military or wargame experience. I define each as I interpret the text, though—still shy of the battle grid—I hesitate to give specific distances. Let’s assume that the normal distance in each case is five feet. The “close” modifier, then, implies some lesser distance.

Close Order: Refers to the distance between allies (left and right) in a rank.

Close Ranks: Refers to the distance between ranks (before and behind). A phalanx formation assumes close ranks.

Close Quarters: Refers to the distance between facing opponents—that is, they are standing toe-to-toe. The situation is achieved by move, when closing to melee, or maneuver, if already engaged (see Maneuver [E]). In close quarters, opponents may attack only with a short, thrusting weapon (a dagger meets these criteria). Otherwise, one or the other may step back by maneuvering (space allowing), or one may push the other (see Shield Wall, below).

Note: As “maneuver” occurs after the melee round in no particular order, when a combatant maneuvers into close quarters, the opponent, if space allows, may also maneuver to step away at the same time.

Shield Wall [E]

Multiple attacks against a single opponent may draw us into a phalanx, but in the shield wall the formation reaches its highest potential. After opposing forces have molested each other with pointy sticks, each side digs in to shove the other backward, employing the weight of its entire phalanx. The goal is to break the opponent’s formation. Once their formation is broken, troops tend to panic, leaving the field to the victors.

To form a shield wall, at least two shield-bearing combatants must be in close order, shields touching if not overlapping.

Shield Wall AC Bonus [E]

Each member of a shield wall, except the rightmost, benefits from the shield of the member on the right, gaining an additional +1 to AC.

Tortoise: Phenster mentions that rear ranks may hold their shields overhead against missile fire. The action is, however, cosmetic, as they would already count the shield’s bonus in their AC.

Shield Wall Push [P]

A shield wall, as described below, gets away with a lot of footwork. Previously eschewing Holmes’s static combat, Phenster allows a combatant, even while engaged in melee, to move but “just not very far” (L’avant garde #35, see also Maneuver [E]). A successful push can easily move both sides quite far. Perhaps the Pandemonium Society used other rules, unpublished, in addition to those in the article. As is, some extrapolation is required, which I do. For its ambiguity, I class Push in [P] Pandemonium. DMs should be prepared to adjudicate.

A shield-bearing rank, whether part of a phalanx or not, or an individual may, following the melee round (see Maneuver [E]), step into close quarters with an opponent and push the opponent backward.

  • Each side rolls dice equal in size and quantity to their total hit dice. All troops in the formation, no matter the number of ranks, are considered. In the case of large forces, the number of dice may be reduced proportionally. For example, 100d8 versus 80d6 becomes 10d8 versus 8d6.
  • The side with the higher result pushes the other side backward a number of feet equal to the positive difference. If a formation is pushed backward a distance equal to its combat move rate in a single round, or if it is pushed a total of three times its move rate, the formation is broken.
  • A broken formation must immediately make a morale check (NPCs only) with a -1 penalty. A failed check indicates a retreat: members run away (combat move ×3) and cannot defend (-2 AC). On a successful check, the force withdraws: members can defend (but not attack) while moving at combat speed to the rear. Once withdrawn, the phalanx is considered reformed at the beginning of the next round.

Missile Fire into Melee

In “Shooting into a Fight” (L’avant garde #39), Phenster addresses an age-old dilemma in D&D combat. I cover how the problem is handled in early editions and give my own solution in “Firing into Melee.”

On first reading, Phenster’s description of the “Friendly Fire Number” may seem complex. I attempt here to break down Hazard’s calculation into discreet steps. After some practice, I find it isn’t so difficult to do the quick mental math.

The task is less daunting if we remember the following points:

  1. Throughout the text, Phenster uses only even numbers up to the maximum of 8. So we only have four options: 2, 4, 6, or 8.
  2. We don’t have to take into account all figures in the melee. We only consider one or two friendly characters, between the shooter and the target, and only one creature as the target.
  3. Like Hazard says, “You don't have to get it exactly right.”

Missile Fire into Melee [E]

“Remember that spells and missiles fired into a melee should be considered to strike members of one’s own party as well as enemy” (Holmes, 20).

Holmes gives no further guidance on the matter. The following method may be used to calculate the chance of friendly fire.

Method

  1. Determine the friendly fire number (below).
  2. Subtract the friendly fire number from the attack roll and add bonuses and penalties as normal to determine a hit or miss on the target.
  3. If the natural dice result is the friendly fire number or less, a friendly is hit.

Short Range: According to Phenster, missile fire into melee is only permitted from the weapon’s short range. (But see below, Hail Mary [E].)

Target Has Precedence: In the case where the natural result would hit a friendly, but adding modifiers hits the target, then the target is hit.

Which Friendly is Hit: Phenster doesn’t say how to determine which of two friendly characters are hit. You could dice for it based on the relative sizes of the two characters, but that takes an extra dice roll. When two friendlies are in jeopardy, I call one odd, the other even, at the same time I call the friendly fire number. The attack roll then determines which is hit.

Sizes

Using Phenster’s examples, I extrapolate creature heights and add a couple other usual types.

Creature Sizes for Friendly Fire Number
Size Height Creature
Small Up to 3' Halfling
Man-Sized 4' to 6' Dwarf, elf, human
Big 7' to 9' Ogre, minotaur, troll
Giant 10' to 12' Hill giant*
Dragon Over 12' Dragon, other giants, purple worm
* Phenster says “giant,” but, depending on the type, a giant can be up to 24' in height. I think once the creature is more than twice man-size, the chance of friendly fire is null.

To Determine the “Friendly Fire Number”

I break down the scenarios into three cases by the shooter’s position in relation to the target and allies. The shooter is attacking either from behind allies in melee with the target, from the target’s flank, or attacking from the target’s rear while all friendlies are opposite.

From Behind Allies

In the standard case, where the missile fire attacker is trying to shoot into a melee from the same side as the friendly characters, consider one or two friendlies in melee with the target. We start with the base number, then add and subtract for friendlies and the target by size.

Base Number:

  • No matter who’s in the fight, the base number is 4.

Friendlies:

  • Subtract 2 for small friendly.
  • Add 2 for large friendly.
  • Add 2, 4, or 6 for second small, man-sized, or larger friendly.

Target:

  • Subtract 2 or 4 for larger target.
  • Add 2 for small target.

Other Considerations:

  • Assumes the friendly character is directly between the enemy and the would-be shooter. If the shooter is off to one side, but not flanking, subtract 2.
  • When two friendlies are before the target, the shooter may move to one side, space allowing, so only one friendly is in consequence.
  • The DM may add 2 or 4 for friendlies not in melee but masking fire in the middle field.
  • In no case should the friendly fire number be lower than 2.

From Flank

In the case where the attacker is flanking the melee without a friendly in the way, we consider all friendly figures on any side of the target as a group. The number is predetermined, no additions or subtractions.

  • Friendlies on one side: 2
  • Friendlies on two sides: 4
  • Friendlies on three sides: 6

Another way to think about this case, is simply 2 per side.

All Friendlies Opposite

When all friendlies are beyond the target, we calculate the number for one or two friendlies in the line of fire (as in From Behind Allies above), then subtract 2.

Modifiers to Mitigate Friendly Fire or No?

Hazard doesn’t count any modifiers in the chance to hit a friendly character. Should a shooter’s high Dexterity help to avoid friendly fire? How about a magic bow? Should a high-level shooter have a better chance?

The answers to these questions may depend on how high a Dexterity bonus can be in your game and how much magic treasure comes into the campaign. Another consideration may be the calculation required. It’s easy math for sure, but that extra step in the players’ minds takes a little of the immediate impact out of the dice roll. I like for everyone to know immediately when the dice stops whether the aim was true or somebody needs to make an apology.

The following rules take modifiers and high levels into account. You can use either or both with Missile Fire into Melee [E].

Add Modifiers to Mitigate Friendly Fire [E]

Use the modified dice result to determine friendly fire.

Add Level “Steps” to Mitigate Friendly Fire [E]

Add 1 to the attack roll for each “step” above the first on the attack matrix, one step being three levels for fighters, four for clerics and thieves, and five for magic-users.

Caveat: Using both these rules, a 4th-level fighter or a 5th-level thief (second step, +1) with a +1 Dexterity bonus for a score of 13 or more and a +1 magic bow at short range (+1), has a +4 on the attack roll, which allows them, in the standard scenario, to shoot into a melee with impunity. Careful.

Hail Mary [E]

It’s risky, but in a game where anything is permitted, players may want fire into melee at medium or long range. In this case add 4 to the friendly fire number for each longer range: medium +4, long +8.

No Shooting Into Melee [H]

“Once the party is engaged in melee, arrows cannot be fired into the fight because of the probability of hitting friendly characters” (Holmes 20).

“…and then melee is joined, after which no missile fire is permitted because of the danger of hitting friendly forces” (21).

In two of three mentions, Holmes interdicts missile fire into melee. Unsatisfactory as this may be, the easiest way to handle the situation is to disallow it. I include this option to remove any ambiguity, should the DM provide a list of house rules to players.

Maneuver, Initiative Order, and Multiple Attacks

By neglecting some rules in “Rules the Pandemonium Society Doesn’t Use,” Phenster obliges us to clarify. He also adds a house rule for acting later in a melee round. To all that, I add multiple attacks per round, suggested by a L’avant garde reader in response to Phenster’s earlier article.

Each house rule is followed by a category designator in parenthesis. See “About the Reedition of Phenster’s” for category descriptions. Text under headers sans designator is just me talking.

A Word About Figurines

Holmes mentions the use of figurines, claiming, “The game is more exciting and spectacular using the lead miniature figures” (5). Many gaming groups of the ’70s and ’80s had and used figurines at the table. I suspect the majority did not employ them in combat in the meticulous manner assumed by some later D&D editions.

My experience with figurines in the ’80s and ’90s, other than admiring the paintwork, was limited to their use to designate order of march and the occasional table arrangement to show the more complicated battle arrays. Even in this later case, we didn’t often use figures to represent monsters. Too many monsters, not enough cash. We just said, “The [monster] is over here,” sometimes placing dice or a soda can.

Phenster never mentions figurines in relation to combat. In the reedition, I intend to keep the rules light enough that one is not forced to break out the miniatures.

For example, in the following rule for maneuver, I avoid delimiting a certain distance a character can move in combat and eschew terms like “square.” I prefer the Holmes term “space,” which leaves the theater free of any grid.

You may, of course, employ figurines, or not and to any degree, as you please.

Maneuver [E]

While engaged in melee, a combatant may move into any open space behind or beside. If the combatant turns their back on the opponent or is otherwise distracted, the opponent gets a free attack, as if the combatant were fleeing. (See Holmes, 21.) This movement occurs after the melee round with any other movement (see Holmes, 20).

Disengagement

When using the maneuver rule, ignore the suggestion, under Caveat in “Weapon Damage and Attack Priority,” to use the parry rule (Holmes, 21) to disengage. With the maneuver rule, disengagement is simpler. If an engaged combatant steps away from the opponent, the opponent may follow at the same time. In this case, disengagement does not occur. If the opponent does not follow, disengagement occurs.

According to Attack Priority by Weapon Quality [E], without the necessity to parry, a character armed with a long weapon may step back from an opponent, and, assuming the opponent does not follow—to avoid being flanked by another enemy, for instance—the character may strike the first blow in the next round, while the opponent cannot return the blow, unless it is also armed with a long weapon. The character may remain in position on subsequent rounds, getting the first blow without the chance to be attacked. This is essentially what Phenster calls a “phalanx,” which he covers later.

“Withdraw” or Retreat

Holmes’s description of withdrawing from melee (21) implies the character’s back is to the enemy. Suffering a “free swing” at +2 and not counting shield seems more like a retreat. Using maneuver, if a combatant moves back a space, and the opponent does not follow, the combatant may, in the next round, turn tail and run without consequence.

Drop Items on Surprise [E]

Ignore the rule that says a surprised character drops any items in hand (Holmes, 10).

One-sixth of ⅓ is about 5%. Before arguing about whether the chance is too high or too low, we drop the rule because an extra dice roll per character, including entourage, every time the party is surprised (2 out of 6) is too much for what it gives the game.

I put this one in the [E] Extra category. Holmes straight up, I use the rule as written. It’s unique to Bluebook D&D.

Initiative Order

The following rules assume the use of the initiative-by-Dexterity system in Holmes. In that system, a normal round starts at 18 and counts down to 3, where combatants act on the “count” equal to their Dexterity score.

Simultaneous Combat [E]

Ignore the instruction to dice for first blow when two opponents have similar Dexterity scores (Holmes, 21). If opponents act on the same count, the actions are simultaneous. The success or failure of all simultaneous actions are determined before results, usually damage, are applied.

Hold Action [E]

A character may wait to take their action on a later count in the initiative order. An action might be held so as to work in conjunction with another’s action or to interrupt it. Holding one’s action changes the character’s initiative count in subsequent rounds to that on which they act in the current round.

To hold an action, the player states their intention on their normal initiative count. The character then takes the action when the situation matches the intention. The player may, at any time, change their mind and take some other action. An action not used may be executed at the end of the round or held until the next round.

Multiple Attacks per Round [E]

Referring to weapon classes in “Weapon Damage and Attack Priority,” a combatant wielding a weapon two classes lighter than that of the opponent gets two attacks per round. Three classes lighter, three attacks.

No matter the difference in weapon class, combatants are still limited to one attack per phase: beginning, middle, end. (See heading Go First, Go Last in “Weapon Damage and Attack Priority.”) Usually, the last of two or three attacks is taken at the end of the round; the second of three is taken in the middle. The usual case may change, for instance, when the combatant holds an attack. If not using those phases, the DM may adjudicate whether the combatant can get in all attacks.

A combatant gets only one attack per round when closing to melee or in any round after moving, not including maneuvering.

With a weapon of a heavier, same, or one class lighter than the opponent’s weapon, a combatant gets just one attack per round.

Weapon Damage and Attack Priority

With his first house-rules article (see “Pandemonium Society House Rules”), Phenster attacks the most salient problem in the Holmes edition. In a world where all weapons do the same damage and light weapons attack twice per round, daggers get a lot of use, and we wonder why swords—or indeed any other weapons—ever came under the blacksmith’s hammer.

Charming Solution

By far the simplest solution is to ignore Holmes’s varying number of attacks per round by weapon. Thus, every weapon strikes once per round and does d6 damage. Weapon choice then becomes purely aesthetic. This solution has its charm.

Using the following weapon damage and attack priority rules together disarms the dagger-wielding fighter and gives the adventurer meaningful choices when considering arms.

Damage Dice by Weapon Class [H]

Phenster notes a d6 modified by -1, +1, and +2 for three weapon classes. I assume he intends a fourth class for medium weapons—Holmes uses “ordinary”—which inflict damage equal to an unmodified dice.

In the table below, I note the weapons in each class. Where neither Phenster nor Holmes (20) specifies, I use the weapon damage versus man-sized opponents from Greyhawk (OD&D Supplement I, 15) as a guide.

Damage Dice by Weapon Class Table
Class Weapons Damage Alternative
Light Dagger, sling (stone) d6-1 d4
Ordinary Bow (arrow), hand axe, javelin, light crossbow (bolt), mace, spear d6 d6
Heavy Battle axe, flail, heavy crossbow (bolt), lance, morning star, pike, pole arms, sword d6+1 d8
Extra-Heavy Halberd, two-handed sword d6+2 d10

I show an alternative method, not considered in Phenster’s article, that is more familiar to us from B/X. Comparing it to the d6 method, the average damage is the same or, in the case of d6-1, comes close: 2.5 for a d4 versus 2.67 for d6-1, but the ranges of possible results differ.

d-6 Based Damage vs. Alternative Method Comparison Table
d6-Based Average Range Alternative Average Range
d6-1 2.67 1-5 d4 2.5 1-4
d6 3.5 1-6 d6 3.5 1-6
d6+1 4.5 2-7 d8 4.5 1-8
d6+2 5.5 3-8 d10 5.5 1-10

Heavy and extra-heavy weapons have a higher minimum and lower maximum possible result than the alternative method. Light weapons, while they have two chances in six to do 1 point of damage, might do up to 5.

I like the alternative method, because it makes use of more of the “crazy dice.” But it tilts the rules terrain toward B/X, and that’s a slippery slope. Moreover, it veers from the Holmes spirit. Rolling a d6 for damage feels more like Bluebook D&D.

Attack Priority by Weapon Quality [H]

To determine who gets the first blow, Phenster gives priority to certain weapons, which I separate by melee and missile and sort into three qualities each: (melee) Short, Long, and Two-Handed—if my interpretation of “Two-handed swords, et. al.” is correct—and (missile) Slow, Fast, and Loaded.

Engagement

From Phenster’s “when you’re fighting something” and from the example of Beowulf versus orcs, I derive the term “engagement.” An engagement occurs between individual combatants. A melee comprises one or more engagements.

Phenster’s example:

So, when Beowulf is charging into a horde of orcs with his two-handed sword, he gets the first blow against the first orc that's fighting with an axe. But after that he goes last, until he wins the fight and goes to fight another orc.

L’avant garde #32 (August 1980)

Attack Priority by Weapon Quality Table
Melee Weapons
Quality Weapons* Attack Priority
Short Dagger, hand axe
  • Last blow in first round of an engagement.
  • First blow in subsequent rounds of an engagement.
Long Halberd, lance, pike, pole arms, spear, two-handed sword
  • First blow in first round of an engagement.
  • Last blow in subsequent rounds of an engagement.
Two-Handed Battle axe, halberd, pike, pole arms, two-handed sword
  • Last blow in a round unless also Long.
Missile Weapons
Quality Weapons* Attack Priority
Slow Heavy crossbow†
  • Shoot every other round.
Fast All bows†, dagger
  • If otherwise inactive, make second attack at end of round.
Loaded‡ All bows†, all crossbows†
  • Shoot first in first round.

* Unless otherwise specified, a weapon’s priority is normal. That is, the wielder attacks in initiative order.
† Though it does not effect attack priority, bows and crossbows require two hands.
‡ To gain the Loaded quality, a bow must be readied (arrow knocked) and a crossbow must be loaded before combat. During combat, if the bow- or crossbowman does not shoot in the normal initiative order, the weapon may gain the Loaded quality.

Go First, Go Last

Phenster gives no indication as to how we should integrate first and last strikes into the initiative order. Assuming the Pandemonium Society uses Holmes’s initiative-by-Dexterity system, we might do it the same way we integrate the Editor’s directions about magic spells and missile fire:

“When there is time, or when a magic-user says he is getting a spell ready, magic spells go off first. This is followed by any missile fire…” (Holmes, 21)

In play tests, I divide a round into beginning, middle, and end phases, handling all actions (missile, magic, melee) within each phase in Dexterity order.

I add the Loaded quality to bows, in the case where a bowman “knocks an arrow” just prior to impending combat. Note that, unlike a crossbowman, the bowman’s arm tires quickly, so the knocked state cannot last long.

Caveat to Short and Long Weapons

When an attacker with a short weapon gets inside a longer weapon’s reach before the defender can react, the DM might rule that the short weapon gets the first blow.

For examples, when striking from behind, of course, and when closing on an opponent already engaged in melee with another.

Similarly, a combatant with a longer weapon (e.g., normal vs. short or long vs. normal) may use the parry action (Holmes, 21) to step back, thus disengaging. If the parry is successful, i.e., the parrying combatant is not hit, and if the two opponents come together in the next round, it is considered a new engagement, where the longer weapon again gets the first blow.

Source

Though Phenster does not mention a source in the 1980 article, the attack priority system for melee weapons yields results similar to the man-to-man initiative system given in Chainmail (25-26), and the missile weapon attack priorities are not dissimilar to its mass combat rates of fire (11).

Example: Attack Priority

In this example, I ignore movement rates as well as hits and misses. I also ignore Holmes’s instruction to dice for first blow when “dexterities are within 1 or 2 points of each other” (21), as does Phenster [covered later]. The first two combat rounds are shown, divided into beginning, middle, and end phases. Any movement, which usually takes place after the melee round, is included with the character’s action.

The order of march gives the character class of each party member and their weapons with any notes, including weapon qualities (in parentheses). Dexterity scores are shown [in brackets].

Player Party Order of March:

  • Fighter [12], sword
  • Fighter [6], spear (Long)
  • Magic-User [11], dagger (Short)
  • Thief [15], bow (Fast) and dagger (Short)
  • Elf [10], light crossbow (Loaded) and sword

While the party traverses an intersection of two 20'-wide corridors, three gnolls, approaching from the corridor on their right, see the light and charge. The 1st and 2nd Gnolls [14, 9], armed with maces, lead the charge. The 3rd Gnoll [7], wielding a halberd (Long, Two-Handed), trails, so, closes to melee in the second round.

Neither side is surprised. During the gnolls’ charge, the two fighters (Swordsman and Spearman) step in front of the magic-user, who prepares to cast a spell. The thief could knock an arrow, thus adding the Loaded quality to the bow, but the gnolls’ charge catches the player flat-footed.

First Round

Beginning:

  • Magic-user [11] casts shield.
  • Elf [10] shoots crossbow (Loaded) at 1st Gnoll.
  • Spearman [6] (Long) attacks 2nd Gnoll.

Middle:

  • Thief [15] shoots at 1st Gnoll—who is not yet engaged in melee; see next.
  • 1st Gnoll [14] attacks Swordsman—now it’s engaged.
  • Swordsman [12] attacks 1st Gnoll.
  • 2nd Gnoll [9] attacks Spearman.

End:

  • Thief [15] shoots bow (Fast) at 3rd Gnoll.
  • Magic-user [11] draws dagger, steps up to flank 2nd Gnoll.
  • Elf [10] drops crossbow, draws sword, steps forward.

Second Round

Beginning:

  • Magic-user [11] with dagger (Short) attacks 2nd Gnoll—because the gnoll is already engaged (see Caveat above).
  • 3rd Gnoll [7] with halberd (Long, Two-Handed) attacks Elf.

Middle:

  • Thief [15] drops bow, draws dagger, moves to attack from behind (next round).
  • 1st Gnoll [14] attacks Swordsman.
  • Swordsman [12] attacks 1st Gnoll.
  • Elf [10] attacks 3rd Gnoll with sword.
  • 2nd Gnoll [9] attacks Magic-User.

End:

  • Spearman [6] attacks 2nd Gnoll.

Should the Spearman fell the 2nd Gnoll at the end of this round, he may then engage the 3rd Gnoll, attacking in the beginning of the next round.

About the Reedition of Phenster’s

In “Pandemonium Society House Rules,” L’avant garde #32, Phenster takes offense at the accusation that the Pandemonium Society plays an unsophisticated children’s game. As proof their D&D game is “plenty sophisticated,” he gives examples of the group’s house rules. Many more house rules follow in later issues of L’avant garde.

Phenster’s style, while explanative, is verbose and haphazard. My purpose in the reedition is only to make the rules more concise. Where I make assumptions or attempt to patch any holes Phenster leaves, I make it clear in the text.

Sources

In the August 1980 article, Phenster hints the starting point for the house rules is the Holmes edition of Basic D&D, which, once the perforated leaf is removed, has 46 pages. Later, he also writes that he started playing D&D when he got “a strange game that came in a box with a dragon on the top with 5 crazy dice” as a gift for Christmas in 1979.

He doesn’t specify the edition of Holmes. From the late date, we might assume 2nd or 3rd Edition. Other clues bear out the hunch. One article in the series adds a curious assortment of rules. Three entries among them come from the earliest printings of the D&D Basic Set:

Phenster adds an “enormous type to the spider list,” which has 6+6 hit dice. A new magic item is a “ring of plate mail,” and “Throwing salt on zombies makes them dry up and wither.”1

…[Phenster] started playing D&D when he got “a strange game that came in a box with a dragon on the top with 5 crazy dice” as a gift for Christmas in 1979.

That these are added as house rules further supports the idea that Phenster’s base rule set is not the 1st Edition (first three printings) of Holmes. The presence of only five dice in the box narrows the field to the fourth or fourth+ printings—both 2nd Edition, or the seventh, which is the last printing of Third Edition Holmes.2

The dungeon Phenster mentions in his “Welcome” message in Paradigm Lost #1 (April 1980) is titled “The Great Halls of Pandemonium.” The name suggests there are multiple Great Halls, and the adventure locale described, “Great Crone Hall,” bears cosmetic resemblance to the Caves of Chaos from Dungeon Module B2. The dungeon consists of several rooms, densely populated by monsters. The accompanying narrative “dungeon report” indicates Great Crone Hall is higher up a cliff face from the previous dungeon and makes the conclusion “so it’s more dangerous.”

Similarities to the Caves end there. The map of Great Crone Hall shows the straight lines of stone-block construction, not natural or rough-hewn cavern walls. In the narrative, player characters 3rd-level and higher encounter monsters of up to 10 hit dice. Though when two player characters are felled and others wounded and short on spells, the party goes “back to the fortress to heal and get the gemstones appraised,” activities typically conducted in a base town.

But this only hints that Hazard, the DM who we presume made the dungeon, had some experience with The Keep on the Borderlands, which circulated by then in the sixth and seventh printings (November-December 1979) of the Holmes boxed set. I don’t know when the module became available separately in stores. In any case, the similarity does not mean B2 came in Phenster’s Christmas set.

…this only hints that Hazard, the DM who we presume made the dungeon, had some experience with The Keep on the Borderlands

Although Phenster claims the single book in 1980, some additions and rules changes in articles from later years appear to be inspired from other sources. For examples: He makes specific reference to pole arm articles in the Strategic Review. He notes, in separate articles, that the Pandemonium Society uses the experience point tables for high level characters from Men & Magic and Greyhawk, and psionics from Eldritch Wizardry.

Categories

Phentster’s are a diverse array of house rules. He covers topics from the mundane cost of a sling, through the fantastic summoning of infernal beings, to the gonzo “rod of ICBMs.”

I class the house rules into the following four categories:

[H] Holmes

These house rules patch the Holmes edition to make it a more coherent game, while maintaining its simplicity. In this regard, Phenster does a decent darning job without sewing a whole new garment.

Examples: Damage and attacks per round by weapon class.

[E] Extra

Not strictly necessary, the rules in this category extend the game beyond 3rd character level and add elements that most gamers were used to even in the early ’80s. While these come with more complexity, they also add real value to the game without interfering with play.

Examples: Additional adjustments to ability scores, advancement to higher levels, wilderness exploration.

[C] Campaign

This category adds elements as suggestions or models to apply to a campaign. These ideas might be applied to a bare-bones campaign to give it some ambiance without much effort.

Examples: Coins of the realm, human languages, birds of war.

[P] Pandemonium

These are additional house rules employed by the Pandemonium Society. They often go a step—sometimes leagues—beyond fair or useful. But they can be fun.

Examples: Magic-use without spell books, combat complications, a number of improbable and horrific monsters, and “the Boomtown Rule.”

While Phenster’s start point is Holmes, many of the house rules are applicable to other old-school editions of the world’s most superlative role-playing game.


1 Again, Zach Howard’s Zenopus Archives cast light: in the Holmes first printing, an “enormous” 6-HD spider lurks in Room J (44), a ring of protection “serves as plate armor +1” (38), and zombies are vulnerable to salt (14).

2 See the Acaeum for an exhaustive list of D&D Basic Set contents by printings.

Recalculating a Coin’s Weight

In “Holmes on a Coin’s Weight,” we take the Editor’s proposed weight of a standard coin—twice that of a quarter—and calculate that 40 coins make a pound. This was prompted by questioning the validity of old-school D&D’s standard, ten coins to a pound, to measure encumbrance.  Now I’m curious about the real weight of coins made from precious metals.

Source of Incongruence

In his review of the TREASURE chapter of the Holmes manuscript, Zach Howard notes that the section with heading BASE TREASURE VALUES (Holmes, 34), in which the weight of a coin is specified as twice that of a quarter, is not present.1 We deduce, then, that neither the 110 nor the 140 pound coin is proposed by Holmes. Rather, the incongruous weights entered the publication during editing.

I added a brief mention in an update to the earlier article.

Precious Metal Coin Weights

A US quarter-dollar piece, 1.75 mm thick and 24.26 mm in diameter, has a volume of 808.93 mm3 or 0.81 cm3. By the weight of the precious metals from which D&D realms mint coins, we can calculate the number of coins in a pound by metal. We ignore electrum as the alloy varies in weight depending on its composition.

Precious Metal Pieces
Piece Copper Silver Gold Platinum Average
Volume (cm3) 0.81 0.81 0.81 0.81 0.81
1 cm3 Weight (lbs.) 0.0197 0.0231 0.0426 0.0473 0.0332
Piece Weight (lbs.) 0.0160 0.0187 0.0345 0.0383 0.0269
Pieces in 1 lb. 62.64 53.38 28.99 26.11 42.78

More precious metals are heavier. A pound of copper counts 64 pieces, while less than half that number make a pound of gold or platinum, 29 or 26 pieces respectively.

Forty Coins to a Pound

We could justify a pound of 40 coins by assuming most treasure hauls will have a mix of silver, gold, and platinum, with silver making up a half. We leave the copper pieces in a trail behind us, so we can find our way back to the hoard for a second load.2 The average of 53, 29, and 26 is 36 coins, which rounds up to an even 40.

And let’s take another look at the Holmes quarter-sized coin. Its weight, 0.025 pounds, is practically the average of the ensemble of precious metal coins: 0.027 pounds.

Precious Metal Pieces Compared to the Holmes Quarter
Piece Average of Precious Metal Pieces Holmes Quarter
Volume (cm3) 0.81 0.81
1 cm3 Weight (lbs.) 0.0332 0.0309
Piece Weight (lbs.) 0.0269 0.0250
Pieces in 1 lb. 42.78 40.00

The average number of pieces per pound is 42.78. Adding electrum (not shown) with equal parts gold and silver brings the average down to 41.74.

Aside: Early Calculations

That the average weight of precious metal coins comes so close to Holmes’s twice-a-quarter’s-weight makes me wonder whether some editor might have done the research and made the calculations.

In the Internet Age, out of sheer curiosity, I looked up the precise dimensions of a quarter and plugged them into a volume calculator, found a web page that gives weights of metals by volume, and entered a few simple formulas into an electronic spreadsheet.

Certainly, the average 1970s high school student could accomplish the same,3 though by other means. All the calculations—the coin’s volume and each formula for each metal—must be done by hand, possibly with the assistance of a handheld calculator. Before doing the numbers, the research to find the weights of precious metals—unless one had a set of encyclopedias on the home shelf or a reference work noting specific gravities of metals—required a library trip.

Again, it was doable without the web, but it took more time and effort. Whoever did it, if it was done prior to 1977, had to be motivated.

Ten Coins to a Pound

To weigh one-tenth of a pound, how big would a coin have to be?

The average weight of 1 cm3 of the given precious metals is 0.033178 lbs. One-tenth pound divided by 0.033178 is 3.014. So we need about 3 cm3 of metal. A coin of that volume and, let’s say, twice a quarter’s thickness, 3.5 mm, must have a diameter of 33.1 mm, which is 1.30 inches or just shy of 1516.

Coincidentally, the Eisenhower dollar coin, with a 1½-inch diameter and 110-inch thickness, has a volume of 2.8958 cm3. It weighs 24.624 grams or 120 of a pound. So, instead of a quarter dollar, we might say coins in D&D are the size of an Eisenhower dollar and twice the weight.

In a world of fantasy adventure, I could go with a coin of such an important size. It’s treasure, after all. It ought to look like treasure!

Still, even at quarter-size, we could argue for the ten-coin pound. As Moldvay suggests, when measuring encumbrance, we mustn’t neglect bulk. A coin seems to be the antithesis of bulk. It’s small, stackable with others, creates minimal lost space between pieces, and fistfuls of them fill voids between silver goblets and gold statuettes.

But a sack of coins isn’t rigid. I’m guessing that the only difference between a sack of 1,750 metal pieces and a party member’s corpse carried over your shoulder is that one of them will pay for a round at the base town tavern.

Euro Equivalents
Euro Equivalents.
The 2- and 1-euro coins are just larger and just smaller than a quarter: 25.25 and 23.25 mm in diameter, respectively. The 50-cent piece is the closest match at 24.25 mm, though its thickness, 2.38 mm, is a third again that of a quarter.

Notes

1 Zenopus Archives blog, “Part 34: ‘Many Monsters Carry Treasure.’

2 In adventurer jargon, copper pieces are called “dungeon marks.”

3 In a December 1983 Dragon article, David F. Godwin makes such calculations. “How many coins in a coffer?” (Dragon #80, 9) doesn’t question the tenth-pound coin but addresses the related problem of a coin hoard’s volume. One point Godwin makes is that, due to the heavier weights of metals, the volume of coins in a “full” sack is much less than the sack’s volume. Imagine a stack of ten quarters. It weighs one pound. Make six rows of stacks by ten columns. Rounding to convenient dimensions, a stack of ten quarters takes up a volume 1" × 1" × ¾". Stacked, the 600 coins take up 6" × 10" × ¾". Dump them into a large sack. Any more weight would burst the seams, but there’s still a lot of air in the volume. So much that even four times as many coins doesn’t begin to take up the space.

For an example of a large sack overfull, see the Erol Otus illustration in Moldvay’s D&D Basic Rulebook, B20.

Holmes on a Coin’s Weight

“…for 300 gold pieces are assumed to weigh about 30 pounds” (Holmes, 9).

Melqart raised the torch over his head. Flickering light glinted off gold and silver. An alabaster frieze decorated the far wall. Before it, coins spilled from coffers, chests, and brass urns. A gold chain adorned with precious stones sparkled red and green.

Melqart drew a breath. “How many rounds1 you reckon, Hathor?”

Hathor-Ra stood, shield lowered, mace pointing down, mouth agape.

“Hathor?”

She blinked at the dazzling mound. “Thousands and thousands!”

“How many sacks do we have?”

“Three large, one small… and I’ve got room in my backpack.”

In early D&D editions, the base unit to measure encumbrance is the “coin,” and ten of them weigh one pound. I struggled with that idea for a long time. Even if we assume that encumbrance is “a combination of weight and bulk,” as Tom Moldvay puts it (B20), a one-tenth-pound coin seems hardly credible. Eventually, I came around to accept the absurdity in favor of playability.

Ten coins to a pound started as early as OD&D, in which the average man weighs 1,750 coins (Vol. 1, 15). That the entry tops the encumbrance list is either to set a benchmark—175 lbs. was average for a 1970s American male—or to remind us it’s a dangerous world: there are rules for carrying a comrade’s corpse.

The ten-coin standard continued through AD&D and the “Basic” line (B/X, BECM/I, and the Rules Cyclopedia). It was abandoned in 2nd Edition, which uses pounds to measure encumbrance.

The quote at top from the section on encumbrance in Holmes Basic D&D pulls the heavy coin forward from OD&D. But Zach Howard’s reading of the Holmes manuscript implies that it wasn’t the Editor who wrote the encumbrance section,2 but rather a subsequent editor.3

Elsewhere in Holmes we read:

“All coins are roughly equal in size and weight, being approximately the circumference and thickness of a quarter and weighing about twice as much” (34).

Reading Zach Howard’s discussion of the Treasure section in the Holmes manuscript, I see that Holmes didn’t write about the size and weight of coins either. [22:30 13 February 2022 GMT]

This gives us the idea that Holmes used, at least in his own game, a smaller value for the weight of a coin.4 A US quarter-dollar piece weighs 5.67 grams. Twice that, 11.34, is 0.025 pounds. Using this as the standard, there are 40 coins in a pound.

Do you know what that means? You can carry four times more treasure out of the dungeon. That’s four times more treasure! More treasure for you, more treasure for me—more treasure for everyone!

Laden Thieves
Laden Thieves.
Adventurers carry 9,600 rounds in four large sacks.

Notes

1 A round, in adventurer jargon, is a precious-metal coin of any realm, past or present.

2 Zenopus Archives blog, “Part 6: ‘Fully Armored and Heavily Loaded’

3 Howard suggests, with compelling evidence, Gary Gygax for the Editor’s editor: “Interlude: Who Edited the Editor?

4 Written accounts from the Editor himself indicate that Holmes knew and used some rules from an early third-party OD&D supplement called Warlock. I wonder if a coin’s weight is addressed in those rules. Zenopus Archives blog, “Warlock or how to play D&D without playing D&D?

Having now had the opportunity to read Warlock as printed in The Spartan #9 (August 1975), I can report that, other than that it weighs one unit, no mention of a coin’s weight is contained therein. Nor is any other of Holmes’s unique rules. [18:34 19 May 2022 GMT]

Firing into Melee

“Once the party is engaged in melee, arrows cannot be fired into the fight because of the probability of hitting friendly characters.”

—Holmes, 20

“Kaytar attacks it with his dagger.”

“And I shoot an arrow at it.”

“You can’t fire into melee.”

“Why not?”

“You might hit Kaytar.”

“He’ll live!”

I was never big on tinkering with the game rules. But the first rule I ever questioned was no missile fire into melee.

For me, two big attractions to the game are that it simulates a reality of heroic fantasy and that, through a character, you can do anything within the realm of possibility. A well aimed arrow between comrades to save the day seems to fall within its purview.

In the scene above, Garth relented and let me roll the missile attack.

“A 9, you miss,” he said. “There’s a fifty-fifty chance you hit Kaytar.”

It was an embarrassing moment. All at once, I fumbled an opportunity to make a difference, put Kaytar in jeopardy, and interrupted play with the argument—however brief—plus an extra dice roll.

I don’t find in the OD&D booklets any reference to firing into melee. I turn, therefore, to Chainmail, which reads: “Missiles cannot be fired into a melee” (16). Terse and unambiguous for once—we count ourselves lucky this day.

Holmes, quoted at top, carried the Chainmail rule forward. But later, on the same page, the text is less definitive:

“Remember that spells and missiles fired into a melee should be considered to strike members of one’s own party as well as the enemy” (20).

The Editor doesn’t give us any kind of rule to go with the permission. We might take “as well as” to mean the chance to hit an ally is equal to the chance to hit the enemy.

I could have missed it, but I don’t see a reference to firing into melee in AD&D or in B/X. Firing into melee is allowed in 2nd Edition AD&D, but it’s a “risky proposition.”

It’s also awkward. Before the missile is fired, allies and enemies are counted and weighted by size to calculate the chance that the impending attack roll will be against a party member’s AC (PHB, Chapter 9: Combat, Firing into a Melee). Depending on the result, we may have a distasteful situation where a player must roll an attack on a friend’s character.

A reader points me to a page in the AD&D DMG, where Gygax tells us how to handle “the discharge of missiles into an existing melee” (63). The system is similar to that of 2nd Edition, though more wordy. [05:54 29 January 2022 GMT]

From memory, 3E (in which I include 3.5) allows firing into melee with a simple −4 to the attack roll and no possibility to hit an ally. An optional rule allows for friendly fire—if the dice result would have hit the opponent without the penalty.

This lacuna in early editions has been the target of countless house rules. I’ve tried a few approaches in my own games, none satisfactory. Most require an extra dice roll, like 2nd Edition, or some additional calculation, like 2nd Edition again and 3E.

I recently made another attempt. No extra dice rolls in this one, no complex calculations, and it’s easy to remember. It assumes that allies give the opponent some cover from the shot. As soon as the dice comes up, you know whether you save the day or cripple a comrade.

One caveat: when playing with young children or sensitive adults, consider applying the attack roll penalty but ignoring the chance to hit an ally.

Odd Miss Hits Friendly

When targeting an opponent engaged in melee with friendly figures, subtract 4 from the attack roll. If the shot misses and the natural dice result is odd, the missile hits the friendly figure nearest its flight path.

Firing Into Melee
Firing Into Melee.
A reconstruction from a vague memory. The monster—I don’t remember which type—surprises the party and closes to melee with Kaytar. The neutral human fighter takes careful aim.

Progressive Dice, a Misnomer

This is a follow-up article to “Progressive Dice for Effects Durations,” in which I propose a method to roll each turn for the chance for an effect to end. This, in order to maintain the secrecy—and suspense—of an effect’s duration when playing solo or otherwise without a DM.

So-Called “Progressive” Dice

“Instead of rolling for the effect duration at the trigger to know on what turn the effect ends, we roll the same dice at the beginning of each subsequent turn to see if the effect ends in that turn.

“A dice result equal to or greater than the current subsequent turn indicates the effect continues. Roll again at the beginning of the next turn. A lower result means the effect ends.”

—“Progressive Dice for Effects Durations

I’ve used progressive dice for effects durations and a number of other things for years. My assumption was that the chance for the effect to end each turn stands alone turn by turn, increasing as turns go by, therefore “progressive.” I also assumed that the overall probabilities, compared to the traditional method, were somehow the same.

Progressive Dice - Assumption

Writing the previous article forced me to think a little deeper on the method. I wondered if I’d got it right. Does it really yield a progressive chance, turn by turn, for the effect to end?

The smartest D&D mathematician I know is Dan Collins of Wandering DMs and Delta’s D&D Hotspot. A 40+ year D&D veteran, Dan is also a university lecturer in mathematics and computer science.

So, I sent him a query outlining the problem. Dan’s response, a few lines and a table of probabilities, shows how it is that progressive dice are not so progressive after all. For, using the proposed method, the chance of the effect ending is much higher in the initial periods than the later, so, not at all statistically equivalent to the traditional method.

In a traditional game, the DM rolls a single dice (or combination thereof) when an effect is triggered to determine its duration. A duration of 1 to 6 turns, say, is rolled on a d6. The probability that the effect ends on any turn is ⅙ or 16.67%.

Single Dice Roll [Traditional]

Using so-called progressive dice, “It stacks up differently,” Dan writes. “It’s very unlikely that you’ll get to turn 5 or 6, because you have to survive all the prior rolls to get there. Over half the time you’ll have the effect stop after two or three rounds.”

Here I had to make a saving throw vs. Death Ray. Reading the email, I was talking to Dan through the screen: The progressive dice method is so elegant, man—it has to be right!

Dan goes on to explain: “Computing a compound probability like this is a series of multiplications…” He also includes a table with a note that, if the calculations are correct, the sum of all chances should be 100%. I reproduce the table here.

Progressive Dice, Chance to End After Turn n
Turn Multipliers Product
1 16 16.67%
2 56 × 26 27.78%
3 56 × 46 × 36 27.78%
4 56 × 46 × 36 × 46 18.52%
5 56 × 46 × 36 × 26 × 56 7.72%
6 56 × 46 × 36 × 26 × 16 × 66 1.54%
Total   100%

So, it isn’t just the simple chance (bold) each turn that the effect will end. We have also to factor in the cumulative chance (italics), which is each previous roll inverted, that the effect hasn’t already ended.

Note that, in the previous article, we roll to see if the effect ends at the beginning of the next turn. “Ends after turn n” is a different way to say the same thing.

Progressive Dice - Correction

Therefore, at best, I misnamed “progressive dice.” Though the number to roll increases turn by turn, the chance to make that number is not at all progressive. The chance to end the effect after the second or third turn is much higher than the first or later turns.

Alternatives

So, what is a DM-less player to do? We might accept the statistical difference and use the so-called “progressive” dice in play. Or we might seek out other solutions. We look here at two—one of them works.

Single Dice, Effect Ends on a 1 (Not a Solution)

I thought of an alternative method. Roll the same dice every turn, with a result of 1 signaling the effect’s end. The effect ends automatically at the end of the  maximum duration.

It’s more simple than counting turns. But, if I’m following Dan’s lesson well, we still have to factor in the chance that the effect ended with the previous roll(s).

Effect Ends on 1, Chance to End After Turn n
Turn Multipliers Product
1 16 16.67%
2 56 × 16 13.80%
3 56 × 56 × 16 11.57%
4 56 × 56 × 56 × 16 9.65%
5 56 × 56 × 56 × 56 × 16 8.04%
6 56 × 56 × 56 × 56 × 56 × 16 6.70%
Total   66.51%

Ends on 1

Furthermore, I note that the total percentage is only 66.51, which is 33.49 short of 100. I’m guessing that’s because the effect automatically ends at the duration’s upper limit. The chance that it will end after the 6th turn is, in fact, 6.70 plus the remaining 33.49, or 40.19%.

Ends on 1 - Corrected

1 to n Cards

Dan suggests a card solution: a number of playing cards n equal to the upper limit of the range, 1 to n, one of which is an ace—or, if you have a deck of many things on hand, the Donjon (ace of spades) or the Fates (ace of hearts).

Shuffle the deck once when the effect is triggered. Draw one card from the top of the deck at the beginning of each turn. When the ace comes up, the effect ends.

Here, the shuffling is the dice roll, which determines on which turn the effect ends (on the ace). The chance that it will end in any particular round is 1n, just like a single dice roll. The only practical difference from the dice roll is that the ending turn, while predetermined, is hidden within the deck. Also elegant.

A disadvantage is that the card method cannot duplicate dice combinations. Melqart’s stun duration, 2d4 turns, for example, cannot be reproduced using this method. In this case, it was the first effect duration of the campaign, but dice combinations might be infrequent.

Another disadvantage is that you have to manage an additional tool at the table. The suspense about when the effect ends, though, may well be worth the trouble.

For myself, I love to incorporate playing cards into my D&D, and if there’s an opportunity to get more use out of a deck of many things, I’ll take it.

Other Solutions?

I’m interested to hear your suggestions for maintaining the secrecy of effects durations in a DM-less game. I would also entertain a counterargument showing that progressive dice do in fact produce progressive results. Because it’s elegant, man, it has to be right!

My thanks to Dan Collins for his statistical analysis of the problem as well as an alternative solution. For interested readers, Dan offers several venues to learn more about dice and probability. In an episode of Wandering DMs, Dan gives a course in Basic Dice Math, and in another episode with cohort Paul Siegel, he talks Dice Mechanics. In addition, you’ll find a plethora of articles about dice statistics on Dan’s blog.