A poster on the Facebook AD&D shared this comment from Frank Mentzer earlier in the week, in which he recalled a rules discussion with Gygax on the issue of magical light. Here it is:

Now, I'm not super thrilled about this (baroque?) ruling, and I wouldn't use it in my game. However, the way it was reasoned was considered a "real eye opener" for the person re-posting it on Facebook. Namely this part:

Probably for us here this is not very surprising; we've read enough of Gygax's writings, and his close associates, to at the very least interpolate that this was their standard thought process. Gygax would frequently write "O/AD&D" as a game system in the singular, and so forth. Many essential rules were not bothered to be copied forward from OD&D, even though the AD&D text in places just doesn't make sense without them, etc. But to some other gamers this may in fact be quite jarring, who want different editions clearly delineated and compartmentalized in a completely Cartesian fashion. So it's nice for Mentzer to clearly state this for once in exactly in the fashion here.

Actually, I wish that Gygax or some other editor had been willing to be more free about cutting out or overwriting certain bits when transitioning from OD&D to AD&D; treating the work as purely additive in all respects creates something... occasionally fossilized and burdensome. In software engineering we recognize the need to sometimes surgically cut out bad stuff as "refactoring". But nonetheless, it's useful to recognize the mind-state of the original designers in this regard, when interpreting the AD&D rules and associated writings.

## Saturday, March 30, 2019

## Monday, March 25, 2019

### Infravision Instances

I've spent a silly amount of time recently thinking about infravision. In particular: Should it be nullified when nearby to a torch or lantern? And what is the point of that rule in the first place?

See, in OD&D by the book there is no such rule for infravision being spoiled near artificial light. It seems like a rule that Gygax came to between the OD&D and AD&D publications. It is on AD&D DMG p. 59. It was missing from the Holmes Basic draft manuscript, but was added for publication, presumably by Gygax (p. 10; see Zenopus Archives). We're told by Ernie Gygax that what Gary had in mind was specifically this scene from the original

But it makes so little difference in-game that I've been ignoring that rule for a long time now. The DMG says, "It requires not less than two segments to accustom the eyes to infravision after use of normal vision" -- two segments being a small fraction of a round, so I'm not sure how that would ever make a difference in play. I suppose there's the range issue: infravision is 60', and torches as of AD&D light a 30' radius, so I suppose that cuts down the distance. But on the other hand, OD&D says that monsters are seen at 20-80' (Vol-3, p. 9), and AD&D likewise says "A light source limits the encounter distance to

So in these cases my approach is usually to research what would happen in real life as always the most solid, stable, and consistent way to play it. Previously I've used thermal-vision goggles as a serviceable model. Looking for real-world biological organisms with infravision is somewhat more elusive. Interestingly, there seems to be a lot of active progress in this area in recent years among biology researchers:

For all this, the

But I still can't quite see what the point of that rule was in the first place, actually. Perhaps it's just a totally academic issue.

See, in OD&D by the book there is no such rule for infravision being spoiled near artificial light. It seems like a rule that Gygax came to between the OD&D and AD&D publications. It is on AD&D DMG p. 59. It was missing from the Holmes Basic draft manuscript, but was added for publication, presumably by Gygax (p. 10; see Zenopus Archives). We're told by Ernie Gygax that what Gary had in mind was specifically this scene from the original

*Westworld*(1973).But it makes so little difference in-game that I've been ignoring that rule for a long time now. The DMG says, "It requires not less than two segments to accustom the eyes to infravision after use of normal vision" -- two segments being a small fraction of a round, so I'm not sure how that would ever make a difference in play. I suppose there's the range issue: infravision is 60', and torches as of AD&D light a 30' radius, so I suppose that cuts down the distance. But on the other hand, OD&D says that monsters are seen at 20-80' (Vol-3, p. 9), and AD&D likewise says "A light source limits the encounter distance to

**twice**the normal vision radius of the source" (DMG, p. 62) (as well as real-life experiments supporting the same thing, here; login required) -- so it seems like effective visibility is back to at least 60', equal to PC infravision. Moreover: I'm pretty sure none of us play like the monsters are blinded by PCs with a torch as in that*Westworld*clip, right?So in these cases my approach is usually to research what would happen in real life as always the most solid, stable, and consistent way to play it. Previously I've used thermal-vision goggles as a serviceable model. Looking for real-world biological organisms with infravision is somewhat more elusive. Interestingly, there seems to be a lot of active progress in this area in recent years among biology researchers:

- Snakes have specialized heat-sensing pits on their heads (2010).
- Mantis shrimp can see deep into ultraviolet and deep red wavelengths (2014).
- Zebrafish and bullfrogs produce special chemicals that can situationally shift their vision to near-infrared (2015).

For all this, the

*burning*question remains; if an infrared-sensitive creature stands near a hot torch or lantern, is its infrared visual capacity ruined? This does not seem to be a thing that anyone has tested to date. In fact, a few weeks ago I emailed one of the leading researchers in this area and asked, "If there is prey nearby and another very hot artificial source (say, fire), does the animal fail to detect the prey?". He kindly took the time to reply with: "Daniel, I don't know the answer to your question" (and also sent the inquiry to a second researcher). So: Still unknown as far as we can tell from current state-of-the-art science.But I still can't quite see what the point of that rule was in the first place, actually. Perhaps it's just a totally academic issue.

## Saturday, March 23, 2019

### Stats Saturday: Hydras

Generally we've found in our analyses of Equivalent Hit Dice (EHD) power-ratings for OD&D monsters that the danger level is linear in hit dice (for example, see here or other "Monster Metrics" posts). We've taken this to support a linear system of XP awards per Hit Die (e.g., the original Vol-1 system of 100 XP/HD over the later graduated table from Sup-I and all later editions).

Here's one notable exception: Hydras. As you can see in the chart below, Hydras are distinctly quadratic (parabolic) in their relation between HD and EHD. This isn't too surprising, because unlike other monsters Hydras are getting a double increase in attack potential per hit die (head): both adding to numbers of attacks, as well as increased chance to hit per attack (as by HD). In other words, they escape from the standard "action economy" limitation of most boss monsters, and wind up confronting PCs with staggering numbers of high-potential attacks per round.

Alternatively, one could approximate Hydra EHD in the allowed ranged with a linear regression of EHD = 2.26 Heads − 4.73 (R² = 0.99), or whatever level to which you want to round that off.

Here's one notable exception: Hydras. As you can see in the chart below, Hydras are distinctly quadratic (parabolic) in their relation between HD and EHD. This isn't too surprising, because unlike other monsters Hydras are getting a double increase in attack potential per hit die (head): both adding to numbers of attacks, as well as increased chance to hit per attack (as by HD). In other words, they escape from the standard "action economy" limitation of most boss monsters, and wind up confronting PCs with staggering numbers of high-potential attacks per round.

Alternatively, one could approximate Hydra EHD in the allowed ranged with a linear regression of EHD = 2.26 Heads − 4.73 (R² = 0.99), or whatever level to which you want to round that off.

**Tomorrow on the Wandering DMs Livecast:**We plan to discuss issues around adjusting your game for Conventions versus Campaign play. Tune in Sunday at 1 PM EDT (UTC -4) and add your comments and questions to the chat!## Monday, March 18, 2019

### More from Three Hearts and Three Lions

I recently re-read Poul Anderson's

However, with this read-through I was paying more attention to lesser-known, possibly overlooked tidbits, that either are or could be used as core parts of our game. My notes follow (page numbers from 1993 Baen printing):

*Three Hearts and Three Lions*. This is, of course, a critical piece of literature for D&D scholars (and other Appendix-N readers), granted how many ideas that were baked into the DNA of D&D originated here. Most of us are aware of the big-ticket items: Alignment (the cosmic system of Law-vs-Chaos; prior looks one, two), the paladin with a holy sword, a Scottish-speaking dwarf, and the swanmay (see also: mythological swan maidens).However, with this read-through I was paying more attention to lesser-known, possibly overlooked tidbits, that either are or could be used as core parts of our game. My notes follow (page numbers from 1993 Baen printing):

- General emphasis on
*smells*in sense-descriptions. - Ch. 5, p. 46: Land permanently in twilight for Chaos types (e.g., goblins).
- Ch. 6-9: Lots of interesting faerie magic, customs here.
- Ch. 7, p. 55:
**"All of them [elves] seemed to be warriors and sorcerers"**; menial work done by goblin/kobold slaves. (Meanwhile, elves cannot bear the presence of a cross or hearing holy words; see Ch. 3, p. 33; and Ch. 9. p. 76.) - Ch. 7: p. 57: Castle with magical moat, always circulating.
**Lord’s host sallies forth from castle**(compare to Vol-3 castle behavior?). - Ch. 8, p. 70:
**Doorknob turns to speaking mouth (**in castle of Faerie Duke, among many other enchantments.*magic mouth*?) - Ch. 10, p. 85: Supernatural enemies harmed by ultraviolet light.
- Ch. 12, p. 102:
**Law/chaos used interchangeably with good/evil**(last line of big paragraph). - Ch. 12, p. 107: Giants always carrying gold (and p. 113). Giants (“Great Folk”) sit in wintry halls for centuries practicing contests of skill, especially riddles.
- Ch. 12, p. 114: Giant (et. al.) turned to radioactive stone by sunlight.
- Ch. 13, p. 118:
**Iron passes through werewolf’s body without doing harm.** - Ch. 13, p. 119: Lycanthropy is generally inherited. May be bear,
**boar**, wolf, “or whate’er the animal may be for the person”.**Lycanthrope “Wounds knit upon instant”**(from non-silver). Possible recessive-trait werewolf who turns only when chaos magic ebbs over the land. - Ch. 14, p. 132:
**Iron hurts lycanthrope in human form.** - Ch. 15, p. 142: Description of enchanted Avalon, magical island drifting over sea.
- Ch. 15: p. 144:
**“Ever-filled purse” advertised on magician’s sign (as**See also Ch. 17, p. 162.*Bucknard’s everfull purse*). - Ch. 15, p. 145: Magician with diploma from magic university.
- Ch. 15, p. 146:
**Invisible (unseen) servant.** - Ch. 17, p. 160:
**Geas prevents spirits from assisting with divination.** - Ch. 19, p. 176: Nixie who tries to capture the protagonist.
- Ch. 19, p. 180: Undersea weed-house.
- Ch. 22, p. 217:
**Troll fights with dismembered hand, leg, jaw, ropy guts (!).** - Ch. 24, p. 230: Presence of Hell Horse (note 1941-1944 art journal in occupied Denmark called The Hell-Horse [
*Helhesten*]; "hell horse" is a synonym for the nightmare as per MM p. 74).

**Edit:**Mike Mornard helpfully confirmed on the OD&D Discussion board that the Vol-3 castle behaviors, as established by Gygax, were in fact inspired by the action in*Three Hearts and Three Lions*(as well as various Arthurian tales). See here.## Monday, March 11, 2019

### Magic Distribution in OD&D Dungeons

An observation: Boy, magic is really rare if using the dungeon treasure table in OD&D Vol-3 ("The Underworld & Wilderness Adventures", i.e., the DM's book). Prior to this table you get the dictate, "It is a good idea to thoughtfully place several of the most important treasures... Naturally, the more important treasures will consist of various magical items and large amounts of wealth in the form of gems and jewelry." Then for the rest of the dungeon, rules that 2 in 6 of rooms have monsters; with treasure present 3-in-6 with monsters and 1-in-6 without. Then this table is given for such treasure:

Now, I've sort of made my peace with this table for gems & jewelry; while rare (esp. at the lower-numbered levels), when present they come in batches and generally large values, so they sort of give some reasonably high value. (A bit like hockey or soccer: scoring is rare but each goal makes a big difference!) However, magic is very rare throughout dungeon levels 1-3, apparently only shows up one item at a time, and has the added limitation that when you turn to the magic tables, 25% of the time it turns into a treasure map, and not actually any magic.

Let's compute. Say on a standard piece of graph paper you get around 40 rooms per levels (that's what I tend to get), and randomly stock the whole dungeon with this method. Then on the first 3 levels combined you'd expect (adding terms for both with-and-without monster cases): 3 levels × 40 rooms × (2/6 with monsters × 3/6 treasure + 4/6 without monsters × 1/6 treasure) × 5% = 120 × (1/6 + 1/9) × 5% = 120 × 0.277 × 0.05 = 1.66 positive results for magic items. Adjusting for the chance of maps, we get 1.66 × 0.75 = 1.25 actual magic items. So on average we may likely get just one single potion on all of levels 1-3 of the dungeon, and no other magic whatsoever. That could make it a bit hard to fight 4th-level monsters like lycanthropes, gargoyles, or wraiths, that start showing up on the [checks notes]... um, 1st level of the dungeon.

Anyway, here's a complete table of expected number of magic items (discounted for chance of maps) per dungeon level under these assumptions:

As we've seen, on levels 1-3 we only expect about 1/3 of a magic item per level, so rounding to the nearest integer, this appears as zero (0) in the table above. At other levels you may get 1, 2, or 3 magic items on average from the Vol-3 random method. If you make a dungeon of 12 levels (~500 rooms?), then we'd expect a grand total of around 14 magic items in the whole complex.

So I think that most of us would agree that simply can't stand; we have to do something else to supply fighters with magic arms and armor, wizards with wands and spell-scrolls, non-renewable potions of healing et. al., and other stuff. The most obvious way is by DM fiat, thinking of the "thoughtfully place" dictum. But then we are left with no other guidelines for what kind of distribution is recommended in that advance process.

Perhaps one faint idea is to shift magic-positive results from one single item to 1d6 at a time (roughly tripling the numbers estimated above, on average). Any other ideas?

On the other hand, if we do permit maps in dungeon hoards, then perhaps we should account for the magic items to which they can lead you. Random maps have a 30% chance of leading to a "Magic Map" table, and 10% to a "Magic & Treasure Table". Those are both d8-based, and coincidentally, they each have an expected production of 19/8 = 2.375 magic items (individual results go as high as 5 items!). So together any random map expects to lead to 0.40 × 2.375 = 0.95 magic item. We might as well round that to "1", which tells us that we can effectively just ignore the map discount itself (each map leads to an average of one magic anyway). Either way, we're then back to an expectation of 1.66 items in the first three levels of the dungeon (not a big difference).

Or taking Daniel Wakefield's idea, maybe that gives us a clue for what a thoughtfully-placed "big magic cache" might be: 1-5 magic items or something like that. Other ideas are to scale it to the expected number of items PCs might have per level (almost embarrassed I didn't think of that earlier).

Consider: These days I roughly assume that pre-generated PCs might have a 1-in-6 chance for magic per level in each of 3 categories. (So: a 6th-level fighter with +1 sword, shield, and armor seems reasonable.) That implies about 3/6 = 1/2 item per level. If one PC level correlates with one dungeon level, and we have 4 PCs, then it suggests we want 4 × 1/2 = 2 items per level (permanent items?). Compare that if we say that random magic finds include 1d6 items (similar to the 1-5 range in the maps), then that multiplies our earlier per-level expectation and get 0.553 × 3.5 = 1.94 ~ 2 items per level. So those figures seem synchronous.

On the other hand, if you have big 8-person parties then you might consider the need to double that again? Geoffrey McKinney's stats for B2 indicate the per-area magic rate at about 80% × 50% × 46% = 0.184. So one of my 40-room levels would expect 40 × 0.184 = 7.36 items, or almost quadruple the figure in the prior paragraph. Hmmmm. At least that gives us a starting upper/lower bound for what we might choose.

Now, I've sort of made my peace with this table for gems & jewelry; while rare (esp. at the lower-numbered levels), when present they come in batches and generally large values, so they sort of give some reasonably high value. (A bit like hockey or soccer: scoring is rare but each goal makes a big difference!) However, magic is very rare throughout dungeon levels 1-3, apparently only shows up one item at a time, and has the added limitation that when you turn to the magic tables, 25% of the time it turns into a treasure map, and not actually any magic.

Let's compute. Say on a standard piece of graph paper you get around 40 rooms per levels (that's what I tend to get), and randomly stock the whole dungeon with this method. Then on the first 3 levels combined you'd expect (adding terms for both with-and-without monster cases): 3 levels × 40 rooms × (2/6 with monsters × 3/6 treasure + 4/6 without monsters × 1/6 treasure) × 5% = 120 × (1/6 + 1/9) × 5% = 120 × 0.277 × 0.05 = 1.66 positive results for magic items. Adjusting for the chance of maps, we get 1.66 × 0.75 = 1.25 actual magic items. So on average we may likely get just one single potion on all of levels 1-3 of the dungeon, and no other magic whatsoever. That could make it a bit hard to fight 4th-level monsters like lycanthropes, gargoyles, or wraiths, that start showing up on the [checks notes]... um, 1st level of the dungeon.

Anyway, here's a complete table of expected number of magic items (discounted for chance of maps) per dungeon level under these assumptions:

As we've seen, on levels 1-3 we only expect about 1/3 of a magic item per level, so rounding to the nearest integer, this appears as zero (0) in the table above. At other levels you may get 1, 2, or 3 magic items on average from the Vol-3 random method. If you make a dungeon of 12 levels (~500 rooms?), then we'd expect a grand total of around 14 magic items in the whole complex.

So I think that most of us would agree that simply can't stand; we have to do something else to supply fighters with magic arms and armor, wizards with wands and spell-scrolls, non-renewable potions of healing et. al., and other stuff. The most obvious way is by DM fiat, thinking of the "thoughtfully place" dictum. But then we are left with no other guidelines for what kind of distribution is recommended in that advance process.

Perhaps one faint idea is to shift magic-positive results from one single item to 1d6 at a time (roughly tripling the numbers estimated above, on average). Any other ideas?

**Edit:**Some folks in the comments take the interpretation that maps should not be generated from that dungeon-treasure table, actual magic only (maps for wilderness treasure only). There's definitely an intriguing case to be made there, but I'm not sure it's ironclad.On the other hand, if we do permit maps in dungeon hoards, then perhaps we should account for the magic items to which they can lead you. Random maps have a 30% chance of leading to a "Magic Map" table, and 10% to a "Magic & Treasure Table". Those are both d8-based, and coincidentally, they each have an expected production of 19/8 = 2.375 magic items (individual results go as high as 5 items!). So together any random map expects to lead to 0.40 × 2.375 = 0.95 magic item. We might as well round that to "1", which tells us that we can effectively just ignore the map discount itself (each map leads to an average of one magic anyway). Either way, we're then back to an expectation of 1.66 items in the first three levels of the dungeon (not a big difference).

Or taking Daniel Wakefield's idea, maybe that gives us a clue for what a thoughtfully-placed "big magic cache" might be: 1-5 magic items or something like that. Other ideas are to scale it to the expected number of items PCs might have per level (almost embarrassed I didn't think of that earlier).

Consider: These days I roughly assume that pre-generated PCs might have a 1-in-6 chance for magic per level in each of 3 categories. (So: a 6th-level fighter with +1 sword, shield, and armor seems reasonable.) That implies about 3/6 = 1/2 item per level. If one PC level correlates with one dungeon level, and we have 4 PCs, then it suggests we want 4 × 1/2 = 2 items per level (permanent items?). Compare that if we say that random magic finds include 1d6 items (similar to the 1-5 range in the maps), then that multiplies our earlier per-level expectation and get 0.553 × 3.5 = 1.94 ~ 2 items per level. So those figures seem synchronous.

On the other hand, if you have big 8-person parties then you might consider the need to double that again? Geoffrey McKinney's stats for B2 indicate the per-area magic rate at about 80% × 50% × 46% = 0.184. So one of my 40-room levels would expect 40 × 0.184 = 7.36 items, or almost quadruple the figure in the prior paragraph. Hmmmm. At least that gives us a starting upper/lower bound for what we might choose.

## Monday, March 4, 2019

### More Missile Modeling

I've written so many letters on the physics and statistics of missiles, archery, and ballistics that it could sink a warship (search the blog, you'll see). So much so, sometimes it's easy to lose the plot at this time. I figured I'd summarize some of our findings to date.

We have two primary sources of data. One is from Longman and Walrond,

A second source of data is from more recent UK "clout" long-distance longbow competitions. Results from a competition in 2016 show that at a range of 180 yards, competitors hit a 12-foot radius target 42% of the time, and an 18-inch radius central target only 1% of the time. (Full data and spreadsheets on the blog here.)

Using that as a guideline, we've developed a simple physical simulation to model archery shots, using an idea I first saw in Conway-Jones, "Analysis of Small-Bore Shooting Scores", Journal of the Royal Statistical Society (1972). The idea is fairly simple: model shooting error in both the x- and y-axes directions as two independent normal curves, which we call the "bivariate normal distribution". (First noted on the blog here.)

The simulation of that is written as a Java program and posted to a public code repository at GitHub (here). If we run that program with settings of precision = 6.7 (extremely high skill!), target radius = 2 feet, and long output form (that is: parameters

Likewise, if we run the program with precision = 1.6, target radius = 12 feet (parameters

Also, if we set the target radius of this latter experiment to 1.5 feet (that is, 18 inches), then the hit rate at 180 yards becomes 1%, exactly as seen in the real-world data. Comparing these two data sources, we might be led to think that English archery skill has dropped off precipitously between 1894 and 2016 (precision 6.7 in the former and 1.6 in the latter). But based on the short quote regarding the first data source, we might say that it was cherry-picking its data; the best dozen results across all tournaments in England in a year. Contrast that with the second data source which includes all 30 competitors in one single tournament, whether they performed well or not. So the jury is still out on that issue.

That ends the recap. Now for a new thought: What is the best statistical model for these numbers? Clearly it's capped above and below: the chance to hit (or miss) cannot possibly be more than 100%, or less than 0%. Presumably we want a smooth, continuous curve, and one that can theoretically handle any arbitrary distance. Effectively we have just given the definition for a sigmoid curve, that is, an S-shaped curve seen in many probability cumulative distribution functions. The simplest model for this is the logistic function, as applied in logistic regression analysis.

One problem with this observation is that logistic regression of this sort is not built into standard spreadsheet programs (Libre Office, Excel) like many other types are (linear, polynomial, exponential, etc.) So what I've done below is this: Used the model derived from 2016 clout shooters (second experiment above; precision = 1.6, set target radius = 2 feet); increased granularity of the output to increments of 2 yards (for added detail); converted hit chances to miss chances (because the logistic curve expects numbers to be increasing from left-to-right), and used the online Desmos graphing calculator site (here; thanks immensely, guys!) to regress it to a logistic function. We get the best possible fit as follows:

Note that our regression (orange curve) has an

Let's find an approximating line for that "critical" part of the curve. Our regression formula generates the points (20, 0.28) and (40, 0.73) -- so, this is the region where hit-or-miss chances vary from about 25% to about 75%. Solving for an equation of a line through those points (using Wolfram Alpha or good ol' college algebra) gives: y = 0.0225x − 0.17. Note the slope m = 0.0225, which means the chance to hit drops by 2.25% per yard on that region. Converting to feet we get 0.0225/3 = 0.0075, so: 0.75% per foot, or

In conclusion: It seems like our data and multiple models are telling us that there's a consistent dropoff in hit rates of around 7.5% per 10 feet, in the part of the range where it matters (neither a near-automatic hit or miss). This is why in the last few months in my D&D game I've shaved this number off to 5% and simply said there's a

(P.S. Keep in mind that the exact hit-or-miss numbers shown above assume a single unmoving, undefended, man-size target of radius 2 feet or so. In practice, we need all kinds of extra modifiers to account for aware, defensive men in the field; shooting at a clustered army of bodies; and so forth. But from what we can tell the specific range modifiers increments would be generally consistent regardless of other considerations.)

We have two primary sources of data. One is from Longman and Walrond,

*Archery*(1894) -- as reported by Barrow in*Dragon*#58, "Aiming for realism in archery" (Feb. 1982). He writes (first noted on the blog here):English archers use a 48-inch-diameter target in tournament competition... A compilation of the twelve highest tournament results during a one-year period shows that the “hit” percentages of England’s finest archers at three ranges were: 92% hits at 60 yards, 81% at 80 yards, and 54% hits at 100 yards distance.

A second source of data is from more recent UK "clout" long-distance longbow competitions. Results from a competition in 2016 show that at a range of 180 yards, competitors hit a 12-foot radius target 42% of the time, and an 18-inch radius central target only 1% of the time. (Full data and spreadsheets on the blog here.)

Using that as a guideline, we've developed a simple physical simulation to model archery shots, using an idea I first saw in Conway-Jones, "Analysis of Small-Bore Shooting Scores", Journal of the Royal Statistical Society (1972). The idea is fairly simple: model shooting error in both the x- and y-axes directions as two independent normal curves, which we call the "bivariate normal distribution". (First noted on the blog here.)

The simulation of that is written as a Java program and posted to a public code repository at GitHub (here). If we run that program with settings of precision = 6.7 (extremely high skill!), target radius = 2 feet, and long output form (that is: parameters

**6.7 2 -L**), then we get to-hit results very close to the 1894*Archery*figures (compare highlights to quote above):Likewise, if we run the program with precision = 1.6, target radius = 12 feet (parameters

**1.6 12 -L**), then we get results very close to the recent UK clout tournaments:Also, if we set the target radius of this latter experiment to 1.5 feet (that is, 18 inches), then the hit rate at 180 yards becomes 1%, exactly as seen in the real-world data. Comparing these two data sources, we might be led to think that English archery skill has dropped off precipitously between 1894 and 2016 (precision 6.7 in the former and 1.6 in the latter). But based on the short quote regarding the first data source, we might say that it was cherry-picking its data; the best dozen results across all tournaments in England in a year. Contrast that with the second data source which includes all 30 competitors in one single tournament, whether they performed well or not. So the jury is still out on that issue.

That ends the recap. Now for a new thought: What is the best statistical model for these numbers? Clearly it's capped above and below: the chance to hit (or miss) cannot possibly be more than 100%, or less than 0%. Presumably we want a smooth, continuous curve, and one that can theoretically handle any arbitrary distance. Effectively we have just given the definition for a sigmoid curve, that is, an S-shaped curve seen in many probability cumulative distribution functions. The simplest model for this is the logistic function, as applied in logistic regression analysis.

One problem with this observation is that logistic regression of this sort is not built into standard spreadsheet programs (Libre Office, Excel) like many other types are (linear, polynomial, exponential, etc.) So what I've done below is this: Used the model derived from 2016 clout shooters (second experiment above; precision = 1.6, set target radius = 2 feet); increased granularity of the output to increments of 2 yards (for added detail); converted hit chances to miss chances (because the logistic curve expects numbers to be increasing from left-to-right), and used the online Desmos graphing calculator site (here; thanks immensely, guys!) to regress it to a logistic function. We get the best possible fit as follows:

Note that our regression (orange curve) has an

**R² = 95.87%**match with the numbers from our simulated physical model of UK long-distance clout shooters (black dots). One possible downside: the logistic formula shown in the bottom-left is probably too complicated to use in a standard D&D gaming session. However, a second observation occurs to us: in the central part of that curve, at distances from around 20 to 40 yards (that is, ignoring the parts that are close to 0% or 100%; i.e., the part with maximal rate-of-change), the curve is practically a straight line.Let's find an approximating line for that "critical" part of the curve. Our regression formula generates the points (20, 0.28) and (40, 0.73) -- so, this is the region where hit-or-miss chances vary from about 25% to about 75%. Solving for an equation of a line through those points (using Wolfram Alpha or good ol' college algebra) gives: y = 0.0225x − 0.17. Note the slope m = 0.0225, which means the chance to hit drops by 2.25% per yard on that region. Converting to feet we get 0.0225/3 = 0.0075, so: 0.75% per foot, or

**7.5% per 10 feet**. Note that this is freakishly close to the 7.6% per 10 feet figure we saw in the Milks spear-throwing experiment a few weeks ago.In conclusion: It seems like our data and multiple models are telling us that there's a consistent dropoff in hit rates of around 7.5% per 10 feet, in the part of the range where it matters (neither a near-automatic hit or miss). This is why in the last few months in my D&D game I've shaved this number off to 5% and simply said there's a

**−1 chance to hit per 10 feet**, on a d20 attack roll. But how to account for the extended upper and lower parts of the sigmoid S-curve distribution (where the chances are almost, but not quite, 0% or 100%)? Well, the classic rule to auto-miss on natural "1" and auto-hit on "20" (or something close to that: say they count as −10 or +30) does a fair job of recreating the rest of that model.(P.S. Keep in mind that the exact hit-or-miss numbers shown above assume a single unmoving, undefended, man-size target of radius 2 feet or so. In practice, we need all kinds of extra modifiers to account for aware, defensive men in the field; shooting at a clustered army of bodies; and so forth. But from what we can tell the specific range modifiers increments would be generally consistent regardless of other considerations.)

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