Monday, March 28, 2016

More Monster Metrics, Pt. 2

As mentioned last time, here are the results from v1.08 of the Monster-Metrics function of the Arena simulator. This shows how each monster from the OD&D dungeon tables match up with generic Fighters of levels 0 to 12. Equiv. HD (EHD) is what I would use to award XP now (using the original rule of 100 XP per HD). Or, for greater simplicity, you could look at the number of asterisks in the "Bonus" column and give that number of extra increments to XP (hat tip to the Cook/Moldvay notation).

Shading of the monster names indicates broadly where monsters were in the original six Monster Level Tables (OD&D Vol-3, p. 10-11). Although, if we were to revise those tables based on this data, it suggests that we could do something like this -- Level 1 would be all monsters shown with EHD 1; Level 2 would be EHD 2; Level 3 EHD 3-4; Level 4 EHD 5-6; Level 5 EHD 8-10; and Level 6 EHD 12+.

Thanks to the capacity for Party-on-Party combats in the current simulator, we can now gauge not just balance against one Fighter of similar level, but exactly what it would take to balance a given Monster against fighters of any level. (Recall: All fighter are assumed with 3d6 stats, chain and shield, +1 sword, no other special abilities). Where a fraction appears, it means that it takes several of the monsters to match a given fighter; e.g., in the top right, the "1/16" means that it takes about 16 Giant Rats to take on a solitary Lord of 12th level. Or where an integer appears, the reverse is true: the "17" in the bottom left, indicates that a Vampire is a fair match against 17 normal men-at-arms, under this model.

I think that the most important observation here is that not all monsters scale the same when you put them in groups, or face them off against higher or lower-level PCs. Two distinct categories of special abilities present themselves:
• Low-Level Killers: These are area-effect attacks, which can affect possibly an entire party at the start of a combat, but are limited in terms of times used (e.g., Breath Weapons) or being easily countered (e.g., avert eyes from Gaze Attacks). These abilities are likely to destroy large numbers of low-level fighters, but are of more limited use against higher-level PCs (who are likely to make the save and then keep fighting afterward). This is passingly similar to the magic-user "glass cannon" effect; a nuclear-type option that you can shoot off only once or twice.

• High-Level Killers: These are predominantly save-or-die-on-hit effects, that can only affect one target per round, but can be used round-after-round without limitation (e.g., Poison, Petrifying Touch, or Swallowing). These are unlikely to have any benefit whatsoever against low-level fighters; the target is likely to be dead anyway just from the 1d6 damage attack. But this effect is far more worrisome for high-level characters; if a Lord falls into a pit full of poisonous Spiders, the small amount of direct damage is of almost no concern, but the possibly multiple save-or-die rolls each round is a real problem (and the Lord's high hit points then become effectively irrelevant).

Granted that monsters scale differently in groups, I still wanted to generate a single value that we could use for experience awards (EHD), which incorporated all of the information across different levels of opponents (a more stable, consistent measure than before). So the metric I decided to use was to multiply the number in each cell times the level in question (that is: total raw hit dice value at that level), and then take the average of that across all levels 1-12. That is: $$EHD = (\sum_{n = 1}^N n \cdot f(n))/N$$, where $$f(n)$$ is the monster metric at level $$n$$ in the table above, and $$N$$ is the maximum level considered (in this case, $$N = 12$$). I think that overall this gives pretty good EHD figures.

Using these more robust EHD values, we see that some monsters have increased in value from our prior point estimates. For example: Spiders and Ghouls previously didn't seem to warrant any bonus to XP for their abilities; and we can see above that indeed, their special abilities do not put them at an advantage over a single 1st-level Fighter. But when appearing in groups their save-or-go-down abilities are much more dangerous; according to our data, it only takes 3 Ghouls, or 6 Spiders to threaten a solo 12th-level Lord with likely death in combat. Overall, then, we can confirm that they do warrant an XP boost across all the situations where they might appear.

Anything surprising you can see in that output data? Next time we'll focus specifically about the monsters with the uppermost EHD values in that table.

Friday, March 25, 2016

One thing that sometimes bothers me in placing decrepit dungeons/ ancient tombs/ lairs of chaos – each containing fabulous treasure and magical prizes – is: Why hasn't anyone looted it out before? Here are 20 reasons from a brainstorming session, feel free to add your own:

One of the things to look out for is that some of these work better to restrict dungeon access to high-level, and some to low-level, PCs. For example: Having to navigate through a long and dangerous wilderness, travel undersea, cast a high-level spell, conquer a foreign country, etc., set a bar for high-level PCs only. On the other hand: A dungeon magically restricted to low-level characters, cursed to foreigners but not locals, or possibly secreted or appearing only intermittently, could be used to explain why your 1st-level PCs are the first people to stumble into a particular place and thereby start their careers as freebooting adventurers.

Monday, March 21, 2016

More Monster Metrics, Pt. 1

The previous series of articles on assessing monster threat levels by way of simulation had, arguably, one weakness -- the simulator was only able to model a One-on-One battle between one fighter and one monster. It then found the level of fighter that gave as close to a 50/50 chance of beating the monster in question, which was correct insofar as it went. However: in statistical inference we might semi-disparagingly call this a mere "point estimate"; it tells us what happens at that one level, but little to nothing about the variance found for matchups at other levels.

So here's the solution to that, and, I think, a much more stable and trustworthy measurement of what I'm calling Equivalent Hit Dice (EHD), which can be used to gauge threat levels and apply simple XP awards without complicated table look-ups (e.g., as in the original system awarding 100 XP per HD). As of version 1.08 of the Arena simulator, I implemented Party-on-Party battles, so that arbitrary numbers of men or monsters could face off against each other and get a view of data on those scenarios.

Before I present the output from that Party-based simulator (that will be next time), a few thoughts on the model of combat that went into it. First: exactly who attacks who? In the AD&D DMG, Gygax gives recommendations which fully abstract the selection, reflecting the fact that his games had removed miniatures from the table by that point:
As with missile fire, it is generally not possible to select a specific opponent in a mass melee. If this is the case, simply use some random number generation to find out which attacks are upon which opponents, remembering that only a certain number of attacks can usually be made upon one opponent... (AD&D DMG, p. 70)

That seemed like a pretty handy model to implement, because it meant that I didn't have to track actual position or movement of any of our simulated combatants. In fact, I went so far as to randomize the target of every separate blow, which was a pretty simple way to parse out multiple attacks (e.g., as for those creatures who can battle multiple opponents). Furthermore, I set a maximum number of attacks against any creature in a round at 6 (again, taking a cue from the hex-based rule in DMG p. 69; and didn't bother to adjust for relative sizes). Note that a side-effect of this is to come close to coincidentally implementing the AD&D rule for Hydras that "up to 4 heads are able to attack the same target simultaneously" (while not going quite so far with the safety bumper, this will reduce the immense number of attacks that made Hydras look so deadly in the prior model against just one target at a time). Another side-effect is that there will possibly be some small bit of bias for mobs of many creatures, as one can possibly take a hit and keeping fighting as other targets take their place (as opposed to a fighter picking one target and attacking until that one goes down from the fight).

The Party-based combat also made it worthwhile to implement a number of special abilities that had to be left out previously, notably area-attacks such as: Breath Weapons, Petrifying Gaze, Manticore Tail Spikes (shot among many opponents), Vampire Summoning, and Dragon Fear. With each of these special attacks I assume that the monster in question automatically gets one "entry attack" usage before melee begins. For example: With Petrifying Gaze (e.g., Basilisks and Medusae), the entire opposing party all have to make initial saves or turn to stone at the outset; thereafter, I assume everyone knows to "avert their gaze", and the melee proceeds with everyone getting −4 to hit the monster, but no further gaze attacks are assessed. Breath Weapons are abstractedly assumed to hit about half the opposing party (meleers on one side?), or a maximum number scaled to the breath area (4 for Chimera, 7 for Gorgons, 14 for Dragons). Fear from Dragons (included as part of the description from Chainmail) was assessed as a D&D morale/loyalty roll; all opponents must roll 2d6 + HD ≥ 6 or else flee before the fight starts.

Discuss these rulings for mass melee: reasonable or unreasonable? Which of the special abilities mentioned above would you guess are the most dangerous?

Friday, March 18, 2016

Other Monster Languages

Let's say that you're determining languages known for a pregenerated PC, an NPC, or a magic sword. The table on AD&D DMG p. 102 is a good resource; it includes most of the well-known monster types for the game, with preference given for the most common types (e.g., 5% each for elves, dwarves, and orcs; combined 8% for giants and 10% for dragons; 4% for ogres, 3% for lizard men, etc.). One limitation: the largest category of all, the 86-100 roll, is devoted to "Human foreign or other (Select a foreign tongue, choose an unlisted creature language...)". What to do in that case?

Here's a supplemental tables of 100 other monster types that you can roll on in that case. These include only creatures from the Monster Manual; and Intelligence must be "Low" (as for ogres, trolls, hill giants) or better. I've deleted unique types, most undead, and some other subjective choices. I've left in a few types of exotic Men, assuming that they have their own dialects. (Here's an ODS spreadsheet if you want a closer look or to edit your own selections.)

Monday, March 14, 2016

Monte-Carlo Measures of Monster Levels, Pt. 7

Finally, some squirmy-wormy creatures at the end of our list who may have evidence of being possibly broken in our Arena monster combat simulator:

Monsters vs. Spells: The Arena OD&D simulator is purely a 1-on-1 man-vs-monster combat (as of this writing); monster of high Hit Dice value need to be cut down one d8+1 sword blow at a time. Of course, an associate wizard with proper spell selection could radically simplify the process, but this depends on the monster involved. For low-level mobs, then area-effect spells could clear the room, as with sleep, fireball, or cloudkill. But for single high-powered monsters, then a targeted save-or-submit spell like charm or hold monster or disintegrate could end the fight immediately. While I'm happy to use the fighting EHD as a baseline for lesser monsters, at the upper end things get a lot more hazy about whether spell options should really be the major consideration.

Hydras: Hydras with 6-8 heads are listed in OD&D monster table Level 5; and Hydras with 9-12 heads are on the Level 6 table. As usual, in each case I picked one representative type for my Arena input. The 6-headed Hydra is reasonably given EHD 11, about double the HD 6. But the 10-headed Hydra came out at EHD 44, unprecedentedly more than 4 times the HD of 10, and at least twice the EHD of the next-lower monster on the list.

So: Hydras are dangerous, really dangerous. The fact that it can make so many attacks, possibly up to 10 dice of damage per turn, and those attacks are at a high hit-level of 10 HD or more (even after losing some heads), really makes it a whirlwind of death to a lone fighter. A high-level fighter with higher abilities, with magic plate, shield, and sword of multiple bonuses, will have a much better time against it (both in avoiding more of the bite attacks, and getting average damage above 6 points means you can probably reduce the head count by at least one each round). The AD&D Monster Manual made a major rule adjustment: no more than 4 heads can attack any one creature; but that limitation does not appear in OD&D. Perhaps that restriction is both reasonably balanced and also intuitively realistic (in that some limitation should prevent innumerable heads from all getting tangled up in one space, just like any other monster is limited in number of attacking bodies they can throw against a PC).

If we think about including spell-casters in our game, then subjectively this could be altered further downward, but the exact level depends keenly on adjudications regarding Hydras, not explicated in the rules text for any edition. For example: I've seen many D&D games get quagmired when the DM drags out a Hydra, a PC hits it with a spell, and then no one has any idea how rule on it. For example: Say a fireball is rolled for 20 points of damage; does this apply one time to the whole Hydra (killing at most 3 heads), or area-effecting all the heads separately (killing them all)? Assume that a charm monster spell is cast at a Hydra: Does it affect a single head (causing it to attack other heads), or does the entire Hydra become subdued with a single saving throw? What about disintegration? Would a cloudkill or death spell consider each head as separate 1-HD creature (if "yes", then the hydra is defeated; if "no", then the hydra is immune).

Now, personally I would rule that in the spirit of the thing, heads are never treated as separate creatures for the purpose of spell effects, and I hope that most of us would agree (so: average fireball gets maybe 3 heads; sleep, cloudkill, and death spell have no effect; but charm, hold, and disintegrate stop the whole monster). On the other hand, I have seen some DM's go the other way with that; and more worrisome, it's a little unpredictable because it seems that many of us haven't thought it through in advance. The exact risk level, and hence XP value, of Hydras depends a lot on this off-book decision-making. But clearly they are dangerous top-of-the-food-chain monsters.

Purple Worms: I set my simulator to do a binary search of fighters between levels 0 and 50 to find the best match for any given monster (with 10,000 fights between random monsters and fighters at each level). Initially it had a maximum level of 20, but I found a number of monsters bumping up against that ceiling. We can find one monster that will bump up against any ceiling we specify, that is, it has undefinable (effectively infinite) EHD in this model; the Purple Worm (HD 15).

The save-or-die poison tail sting is bad enough, but as Vol-2 rightly asserts, "its mouth is the more fearsome weapon". With a roll of more than 4 points over the to-hit number, the purple worm swallows its prey into its gullet, where it dies automatically 6 rounds later (and is irrecoverably disintegrated 6 rounds after that). Note that for our baseline fighter in chain & shield, AC 4, the worm effectively always hits (20 − 15 − 4 = 1 to hit!), and thus swallows on any roll of 6 or more -- 75% chance on any attack to instantly take the fighter out of the fight, of any level (for a total probability of 99.99% to win such a fight). Or let's compare to the case of a fighter fully equipped with magic +5 plate, +5 shield, and +5 sword (assume average Dexterity), AC −8; so the worm still hits almost half the time (20 − 15 − (−8) = 13), and still swallows on an 18+ roll -- 15% chance to end the fight on any attack, regardless of opponent level (overall, the worm is favored at 57% to win a fight against any fighter of any arbitrarily high level in this model).

As an aside, we see that the Vol-2 rule clause of "or 100% [natural 20] in any case" is effectively redundant, as any character of any possible AC value is swallowed on some roll of less than 20 anyway.

In OD&D, there is simply no mechanism mentioned such that a swallowed character can escape on their own (excluding friends presumably killing the beast and cutting them out in time), so we know that the solo fighters in the Arena will definitely be destroyed as soon as a Purple Worm swallows them. The later AD&D Monster Manual added a remedy to this situation, by stating, "Note, however, that a creature swallowed can try to cut its way out of the purple worm's stomach. The inner armor class of the 'worm is 9, but each round the creature is in the worm it subtracts 1 from the damage each of its  attacks does." Now, at first glance this rule might seem reasonable to us (a nod to Jonah), but personally it always gives me indigestion. Notice that the character swallowed is under no threat of attack or damage while inside, and the inner AC of 9 is the easiest possible target in the game (a high-level fighter should automatically hit it, much like the Purple Worm automatically hit him in the first place). So in practice it turns out that, under this rule, the safest place to fight a Purple Worm is inside it. I've seen fights degenerate into a comical sequence of: fighter gets swallowed; fighter cuts way out; fighter gets swallowed again; rinse and repeat, etc. With this rule, the Swallow attack is not so much a feared ability, it's actually the best defense against the Purple Worm itself. For this reason, I am very much opposed to the AD&D rule, and keep the special ability as a Doom-That-Came-To-Swallow kind of menace. Bring lots of friends to cast missiles and cut you out, or don't face off against a Purple Worm in the first place.

The Arena simulator reflects this pure-OD&D adjudication, and so the Purple Worms are favored to eat up any single fighter regardless of level; and thus the EHD is effectively undefined for this type. Perhaps we should use other monsters as a guidepost and say that it likely tops out at about triple HD/XP?

Conclusions: Taken as printed in our Arena output chart, we could take the listed EHD's as the exact number to use in place of HD when we look up awards in the XP table; or more simply using Vol-1 rules, compute EHD × 100 in each case. But granted that even this is a rough model, and that some subjectivity exists for consideration of multiple party members and spell-casting, perhaps the simplest and most honest thing to say that at best we may have the right magnitude of multiplier by which to adjust XP. That is: either double or triple XP, as shown under the "Multiplier" column in the table, and not pretend to more precision than that.

Even this has some interesting discoveries, I think, like: Wights, wraiths, mummies, lycanthropes, and even giant constrictor snakes probably don't deserve any XP bonuses. Basilisks and Medusae are probably reversed in the tables in terms of danger level. Most monsters likely don't deserve more than a single doubling of XP, with only the rarest of exceptions (even Vampires with their half-dozen abilities may only deserve one doubling, according to this).

Now, while OD&D Vol-1 didn't make any mention or awareness of the need to adjust monster XP for special abilities (actually, the Troll example explicitly avoids making any such adjustment), the need must have been rather quickly noticed, because a D&D FAQ article in Strategic Review #2 (Summer 1975) did bring up the topic:
Experience: ... For purposes of experience determination the level of the monster is equivalent to its hit dice, and additional abilities add to the level in this case. A gorgon is certainly worth about 10 level factors, a balrog not less than 12, the largest red dragon not less than 16 or 17, and so on. The referee's judgement must be used to determine such matters, but with the foregoing examples it should prove to be no difficulty.

In broad strokes, this is the same method we recommend here (add some integer number to the Hit Dice to adjust for special abilities, before computing XP, without any extra table or more complicated calculations being made). How much do we agree with those suggestions? The Strategic Review article probably undervalues those top-level monsters by quite a bit. For Gorgons it suggests "about 10 level factors", but here we think 15 is better. For a Balrog it says "no less than 12", and strictly speaking we agree with the inequality statement, a value of 22 being reasonable. For the largest Red Dragon it says "not less than 16 or 17", and the Arena simulator would suggest that a value of even 45 would not be out of the question for eldest type.

Hopefully this gives some additional perspective and confidence when assigning XP under the Original D&D (Vol-1) system. Perhaps equally important, it was a great motivation to crawl into the guts of the OD&D monster special abilities, and be forced to think about the specifics of adjudicating each, given the level of complete discipline and specificity that our computer programming framework forces upon us.

Eaten any thoughts? Devoured any intellects?

Friday, March 11, 2016

Jon Peterson Interviews Len Patt

Jon Peterson interviews Len Patt, who was recently discovered to have initiated many of the rules underlying Chainmail Fantasy, and hence D&D (and hence all of fantasy adventure gaming). Somewhat amusing that while we now have documentation of that fact, since Patt left gaming entirely to pursue chemistry in college, he himself has only the faintest recollection of that time. Jon draws a moving lesson from this at the end.

Monday, March 7, 2016

Monte-Carlo Measures of Monster Levels, Pt. 6

Six levels of monster tables in OD&D, and six posts discussing the details of monster special abilities (so far). Let's see what happens at the top end of the tables (data here):

High Level Fighters: One possible wobble in our simulator is that Fighters stop getting full Hit Dice after level 9, so the baseline of pure Hit Dice comparisons (Equivalent Hit Dice, EHD) might be a bit biased after this point. But the effect should be fairly minor; fighter Hit Dice here are d8's by Sup-I rules (expectation 4.5), abilities are straight 3d6 (so no Constitution bonus on average), and I award 3 hit points per level after 9th (end result: only a 1.5 pip difference between lower levels and higher levels). Plus, fighters continue to get normal improvements to attack chances at higher levels. Perhaps if we were completely precise we should reduce EHD by 1/3 for each level after 9th (so: Medusae would by EHD 11, Balrogs would be EHD 18, a slight modification from our table).

Chimeras and Gorgons: Both of these have a deadly breath weapon that places them solidly in the Level 6 table (the former, 3d6 fire damage; the latter, petrification). Each is an area attack, of course, so subjectively we could boost their EHD even higher for having the capacity to take out multiple opponents at once. Note that in OD&D, no specification was given for how often they can use these attack forms, or at what frequency. It seemed elegant to me when coding them to use the exact same method as for Dragons; up to three times per day, with a roll of 2d6 ≥ 7 (58%) indicating breath instead of standard attacks. The range of the breath is stated, 5" or 6", that is, about half to two-thirds that of a Red Dragon.

Vampires: The blood-drinking master undead have a veritable Fibber McGee's closet of special abilities, more than twice as many as any other monster type -- flying, magic-to-hit, energy drain, regeneration, charm person, summoning, and polymorph (as well as the well-known special vulnerabilities to keep in mind). The Arena simulator makes use of the energy drain, regeneration, and even charm person (the vampire gets an initial entry gaze attack with −2 save modifier, but I assume that if saved that the fighter is then immune and doesn't have to avert gaze thereafter). The magic-to-hit defense is a non-issue, since our fighters (surely of this level) are all assumed to have a +1 sword. The flying is not simulated (nor is any specific movement). Neither is the polymorph or summoning; the former is mostly just a retreat mechanism, while the latter (10-100 rats or 3-18 wolves) would break the current program architecture in expecting just 1-on-1 fights in the Arena. This could have been a notable factor, but of course these are very low-powered helper monsters in any event. (What's your guess? We'll see in Arena v.108.)

All told, Vampires are assessed at EHD 15, just about double their actual HD 8, so doubling the XP for their special abilities (at least) is definitely justified.

Dragons: As with Lycanthropes and Giants, I tried to pick the one most representative type for my data input; in this case, a smaller Red Dragon of 9 HD, which puts it at about the median of available Hit Dice points for Dragons. Hit points are rolled as per the book: a single d6 indicates age level and hence uniform hit points for each HD (so either 9 hp or 54 hp are as likely as anything in between); note that this makes the variance in hit points tremendously greater than for other monsters (who otherwise have bell-shaped distribution with the most likely hit points around the median). The functioning of the fire breath is explicit in terms of frequency and maximum usage (see Chimeras/Gorgons above).

I interpret the damage from that breath in the harshest terms, that is, always equal to Dragon's maximum hit points no matter how much damage it has taken (as opposed to Moldvay Basic D&D where the current hit points is the damage output, so much less dangerous after a few rounds of hacking it down). I don't give their claw/bite routine the many dice of damage suggested in Sup-I, but I do give 2 attacks for 2 dice each here (or in practice, 1 bite for 2d6 and 2 claws for 1d6 each, mathematically the same). Of course, it was always a bit wonky that the youngest 9 hp Dragon, and the oldest 54 hp dragon, both have the same attacks, damage, saves, and to-hit under these rules (the AD&D Monster Manual slightly adjusted this for the saves, not the other factors), but this is maintained as-written in the Arena simulator.

So, for the case of Dragons it makes sense to do a run for each of the separate Age categories and see how our model measures them (again, mid-level 9 HD dragon in each case):

So, we see that even a Very Young dragon with 9 hit points is an even match for a 6th level fighter; if it gets initiative it can tear into a man and do several dice of damage (4d6 in this interpretation; and it also has the best armor in the game to this point, as plate & shield, AC 2). A Very Old dragon with 54 hit points is assessed the same as a 45th level fighter (!), although with pro-rated-hit-dice adjustment, we might call that only around 33rd level to be nice. And recall that our baseline fighters still only have straight-3d6 stats, chain & shield, and a single +1 sword, no matter their level; more well-equipped fighters will be better off against this challenge. But still, getting hit by 54-damage fire breath a few times in a row will pretty much ruin your day no matter who you are.

And keep in mind again that the breath weapon is also an area attack that can probably torch a whole party if the lone fighter brought any friends, so feel free to discuss how many more EHD's should be added to account for that. Clearly the XP could be doubled, tripled, quadrupled, or more over the base HD, depending on age category.

Balrogs: Yes, Virginia, OD&D originally had Balrogs in the core rules (and in fact, they're probably referenced as an example throughout the little brown books more times than any other monster), before they were taken out from printings after the 1st. See the Balrog Reference Sheet generously provided by Zenopus Archives if you want to fill in that gap for your game. The Balrog has flight, top-level armor (AC 2), a magic sword and whip, among the most Hit Dice in the game (HD 10), a special immolation attack, and also the first instance of Magic Resistance in the game of D&D (though not simulated here, since it makes no difference against our non-spell-casting fighter opponents). They get two attacks, which here I assume get 2d6 damage each (Sup-I says 1-12 points for the sword); my ruling on the whip/immolation is that the whip can strike for 2d6 damage normally, and on a hit then we also check 2d6 ≥ 7 to see if the target is pulled into the Balrog's immolating body for an extra 3d6 fire damage (the average size), save for half. The Arena simulator says this combination is worth EHD 22, again about double the base HD of 10 (and somewhat more dangerous than the average mid-level Dragon). Note that by the book's Wilderness random encounter tables, you might run into 1d6 Balrogs just wandering around together on any given day; giants truly walked the earth then, and lifespans were indeed very much shorter.

Any interpretations here that burn your britches? Still worse things to come.

Friday, March 4, 2016

Area of Effect Images

When I was working on the 1st Edition of Book of War, at some point I drew out models of some of major area-of-effects (namely, fireballs and breath weapons) to see how many character figures would fit inside them (i.e., be normally subject to hits). I find that I keep wanting to refer back to them for many different types of work, but I always forget where I have them filed. So for personal reference here there are on the blog.

Now: Keep in mind, as usual, that I interpret the distances in inches literally on the tabletop, as given in Original D&D Vol-1 and Vol-2 (Men and Monsters); and assuming the same scale as the figures used, so about 1" = 5 feet. The prescription that appeared in Vol-3, that 1" = 10 feet, I ignore. (See the sidebar on house rules for more discussion.) The figures and counters that I used are at the standard 3/4" (or 20mm) square base for men; or 1×2" (25×50mm) for horses.

Also, my general assumption is that there is some spread between figures; that is, a few feet between ranks of soldiers, or some amount of scattering in the face of melee/ missiles/ magic fire. Here's what that "sparse" crowding looks like:

 Areas of Effect: Sparse Crowding

As you can see, a fireball catches about 7 men; the red dragon breath cone catches about 14 men, the green dragon cloud a similar 15, but the blue dragon's line is smaller with only 7 men encompassed (further reduced for lesser types like black and white dragons). For cavalry, the numbers would technically be reduced (3, 5, or 8 figures hit by breath). But for simplicity in Book of War, I rounded off and said that all of these attack forms hit about 10 men, that is, 1 figure at mass scale.

However, perhaps you don't share my assumption about men being spread out a bit in most cases. If the figures are packed maximally "dense", then here's what you get instead:

 Areas of Effect: Dense Crowding

In this case, a fireball hits about 13 men; and the breath from red dragon hits 26, green dragon 30, blue dragon 13. Here we find 5, 8, or 12 cavalry figures in a dragon's breath area. In principle, this might argue for the red and green dragon breath types to hit up to 3 figures of men at mass scale; but all of the others (like fireball) still round off to 10 men, so I think that gives extra evidence for the simplicity of the Book of War rule that only 1 mass figure can be affected.

More generally, I find that keeping this in mind helps to adjudicate at any time we're abstracting the action away from actual figures on a tabletop (for example, when using random/abstract opponent selection as in the AD&D DMG).