There are times, when you’re reading a Black Library book, when you think “I bet the math behind that is kinda interesting…”

I mean, probably not, but if you’re reading this blog, who are you to judge?

I’ve been catching up on the Horus Heresy on Audible, and had this thought triggered by the rather dramatic void battle at the conclusion of *Ruinstorm* by David Annandale. While most of the book is the sort of grand-scale horror that he’s known for, there’s actually some neat math that underpins how the final battle goes.

First, I’m talking about the end of a book. There are *obviously* spoilers here. But you’re warned.

## Lanchester’s Laws and the Math of Shooting People

Back in 1916, a man named F.W. Lanchester (who in his spare time was a major force in the automotive industry in Britain and basically founded the field of operation’s research…) started working on a mathematical representation of the aerial component of the Great War, looking for a way to analytically predict the outcome of a battle. Which would have obvious advantages to the side that could do it.

What he came up with were two fairly elegant equations to describe…well…combat.

Lanchester’s Linear Equation covers two oddly different situations – ancient combat, and indirect fire like artillery. Basically, situations where you can’t *really* aim, and while firepower kills people, ordinance might arrive in places where there isn’t anyone, etc. The winner is the one with the most people, and the expected casualties are the difference between the larger army size and the smaller one. Neat, but not what we’re going to talk about.

Lanchester’s Square Equation, on the other hand, presents an interesting situation – that of aimed fire. Which presents the interesting problem that not only can someone shoot at multiple things, but they *can be shot* by multiple things. A guy in a shield wall only really needs to worry about the guy across from him. A tank needs to worry about anyone who has them targeted. As it turns out, two sides (A and B) shooting at one another is expressed by a really elegant set of two differential equations:

$$\frac{dA}{dt}=-\beta*B$$

and

$$\frac{dB}{dt}=-\alpha*A$$

That is, the rate of change in the number of people in Army A is determined how many people in Army B there are and how lethal their guns are ($\beta$), and the rate of of change in the number of people in Army B is determined by how many people there are in Army A and how lethal *their* guns are ($\alpha$). From this relatively simple set of equations, we can see some cool things.

For example, lets consider 100 Guardsmen vs. 100 Marines, both in Rapid Fire range to get two shots. Because we know how this fight ends, but we can show it now with *math*!

Here, we set the lethality of the Marines to 0.593 – roughly the number of Guardsmen a single Marine should kill in one turn (2 shots from the Bolter * 0.6667 to hit * 0.6667 to wound * 0.6667 to get through the Guard’s armor), and the Guard to a dire 0.111.

Easy to see what’s happening here, right? The Guard get massacred, and as huge numbers of them die, their effectiveness goes down. But this isn’t exactly surprising, is it? Now let’s think about equal *points* of Guard. A standard Tactical Marine is 13 points, so 100 of them is 1300 points. That’s…325 heroic Cadians. Let’s see what the picture looks like now…

Quite the opposite result. The Guard, while suffering significant casualties, triumph, and the marines, despite having an almost 7-fold advantage in terms of firepower lose badly due to being outnumbered by 3.25 : 1. Which is the primary insight from Lanchester’s Square Equation – it’s far better to outnumber your foe than outgun them, and the disadvantage of being outnumbered isn’t linear.

This, at its core, is why spam works – not only are units being spammed often *better*, but taking three of something to do a job is more than three times as effective as taking one thing to do it. This is also the fundamental problem with elite armies – you have to get the Marine’s lethality up to 1.19, a close to 11-fold advantage, before the battle ends up being “close, but the Marines win”.

To put it in context, that’s giving every Marine a 3-shot Bolter, and every fourth Marine a 4-shot Bolter, and making all of those Bolters AP -1, for no additional points increase. Now of course, this isn’t how the game works – this assumes a blank, featureless board, no terrain, etc. And no randomness – but the core principle is there.

Which brings us to *Ruinstorm…*

## The Second Battle of Davin

One of the primary villains of *Ruinstorm* is “The Pilgrim”, a menacing, otherworldly entity that heralds the doom of worlds – and transforms them in very chaos-y, 40K ways, like fortresses that span entire star systems. As it turns out, The Pilgrim is actually the *Veritas Ferrum*, lost to the Imperium and having been transformed into a monsterous demon ship.

In the closing of the book, the *Veritas Ferrum* attacks the combined Blood Angels, Ultramarines and Dark Angel’s fleets with its own retainers of ex-Imperial ghost ships lost to the Warp. But that fleet is badly outnumbered by the Imperial fleet, and the commanders of the fleet, while intimidated by the size of the *Veritas Ferrum*, seem reasonably confident that they can handle it, albeit with losses.

And we, having now played with Lanchester’s Square Equation, agree. Being outnumbered is *bad*, and there’s no suggestion that, beyond being really, really huge, that the revenant ships are any *better*. And they’ve have to be a lot better, as we’ve seen.

(Technical note: There’s another set of equations, the aptly named ‘Salvo Equations’, that deal with naval combat, but that tends to be things like volleys of cruise missiles and counter-measure systems – Lanchester’s Square Equation feels more appropriate for how naval warfare is portrayed in Warhammer, especially when it breaks into the desperate point-blank range and boarding actions phase. The prow of the *Veritas Ferrum* is literally a giant mouth that can bite ships – how do you countermeasure *that?*)

Where David Annandale’s story collides with fun math is that when a revenant ship kills an Imperial ship, that now dead vessel joins the enemy fleet. Beyond the horror of seeing the brothers you failed to protect turn their guns on you, it poses and interesting question…what does the battle look like when one side can *turn* the other?

Mathematically, this looks something like

$$\frac{dImperium}{dt}=-\beta*Chaos$$

and

$$\frac{dChaos}{dt}=-\alpha*Imperium + \beta*Chaos$$

How does this change the battle? Quite dramatically. Let’s assume the ships are equally effective, for the sake of convenience, and give the Imperial fleets an even 1,000 ships (you could think of this as “Ship Points” where a frigate is one, and a strike cruiser might be 10, etc.). For a badly, but not comically outnumbered fleet, lets make the Pilgrim Fleet 650 ships. This, then, is the battle the fleet masters of the combined Legion fleets would have been expecting:

A handy victory. Yes, the loss of close to 200 ships would be sorely felt later when they try to relieve the heroic forces of the Imperial Fists currently holding Terra, but an engagement never really in doubt. Even a much worse 1000 vs. 800 ship matchup leaves a victorious Imperial fleet still *largely* intact.

But what about with this new set of equations, and the Imperial fleet losses capable of being corrupted?

The initial losses, toward the beginning of the battle smooth out the Pilgrim Fleet’s descent, and eventually, they actually outnumber the Imperial fleet despite being on much worse footing to start out with, and the engagement leads to disaster, and a Pilgrim Fleet that’s actually slightly larger than the one it started with. Now, the Pilgrim Fleet can’t be much smaller than this – 619 ships is the first scenario where they actually survive the battle, but even with smaller numbers, it turns the battle into something far more costly for the Imperium.

Of course, these equations don’t take into account that there are three Primarchs and some witchy-poo warp nonsense on the field as well…

So there you have it. *Ruinstorm*, but in differential equations.

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This is exactly why I come here. Love it.

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I came for the math

Stayed for the witchy-poo

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Do you want Dark Mechanicum? Because this is how you get Dark Mechanicum.

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Excellent post.

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Enlightening article, many thanks!

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Very cool. I had looked at the Lanchester laws in relation to 40K some years back. The only real conclusion I can recall was that concentration of force is good, which isn’t exactly a revelation.

I also seem to recall that the name for the Lanchester Square equation comes because there are at least some circustances (I believe mostly when α and β are close or equal) where it can be approximated fairly well by just comparing the squares of the forces involved, rather than actually working out the Differential Equation. That would put the Guardsmen’s advantage at something like 12.5:1, which fits reasonably closely with the SM needing an 11:1 advantage in lethality to make up for it.

Also, that said, things actually work out even worse for the SM in your earlier example. Guardsmen have a 5+ Sv, and Bolters don’t have any AP, so there’s only a 2/3 chance to get through their Armour, not 5/6. I get their actual lethality as 0.592 repeating.

It would also be reasonable to put another modifier on that last equation, to account for Imperial ships that are destroyed to thoroughly to be turned. I mean, if the Chaos forces can get a working ship out of the remnants left by a full on drive explosion, they can make working ships out of interstellar dust, and that’s just game over no matter how you slice it.

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Could have sworn boaters were AP -1, but you’re right, the actual math is even worse. I’ll edit that in. And yeah, the name comes from just what you suggested.

It’s likely reasonable to add that, you’re correct, although the only ship in the story that manages to do so essentially goes “On my position, fire for effect” with it’s own point-blank range cyclonic torpedoes.

One might, for the sake of easy math, assume the revenant ships are firing to badly maim, rather than achieve reactor kills.

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Bolters were AP-1 the last time they used Save Modifiers, back in 2nd Ed. And the Bolt Rifles that Primaris Intercessors use are AP-1. But not regular 8th Ed Bolters.

For the modifer on the math, I was thinking just a .9 multiplier or something on the positive term in the Chaos fleet’s equation. nothing too complicated.

Also, I just realized that this modification of Lanchester’s equations would also work for Zombie Apocalypse situations.

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Agreed. An additional parameter on the “new Chaos ships” equation is a pretty straightforward extension that would let you model “This isn’t quite the right fit”.

Funny you should mention zombies – I’ve done some zombie modeling work in my professional life. It’s sort of a fun mix of epidemic models, predator-prey models, and Lanchester’s equations.

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Man, I really wish I hadn’t burnt out on academia before I made it through to a Masters in math. So many cool jobs available with one more piece of paper 😉

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I mean, I read this blog (mostly) for the math. I feel like plenty of other folks do to. I don’t think you need to apologize for getting mathy!

So how does it look if only half of the “destroyed” Imperium ships resurrect to Chaos? I’d imagine some suffer catastrophic hits, and others see what’s happening to their brethren and try to scuttle themselves before the conversion can happen.

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I’ll post a followup, since two people have asked about this.

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http://variancehammer.com/2019/04/07/only-in-death-following-up-on-lanchesters-laws/

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It sure looks bad for the Marines there, but things do get a lot better if you use the Beta Bolter rule, and especially if you substitute the much more efficient Intercessors for Tacticals — as they have 2 wounds, and AP-1 and are less than 1.5 times the cost. In addition their weapon is 30″ range. If you can maximize your utilization of range, I actually have the Intercessors winning; and if you use the IF Warlord trait that makes your IF infantry have cover, they dominate.

The other thing I noticed running this equation is that, if IG forces start off outside of 30″ and move into 30″, they absolutely should forgo the shooting phase and Move Move Move until they reach Rapid Fire range, where the math is more in their favor.

It gives weapons like the Thunder Fire Cannon + Tremor Shells (1/2 movement strat) a great deal of power. If the Space Marine player can trap the vast swarm of Guardsmen somewhere between 30″ and 24″ for a turn it can be a beautiful massacre.

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There are definitely ways for the marine player to mitigate some of these problems – see also “Be Raven Guard” in this case.

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So, plugging in some random numbers I got marines vs. guard to balance (roughly) at 45 guard vs 20 marines. Therefore, marines should be 9 points each 😉

(caveat: I did this on wolfram alpha so who knows whether it’s working properly..)

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That is of course assuming that guard are appropriately pointed to begin with…