This article is a follow-up to Episode 122. The MoM crew asked me to do two things:

**Prove John’s feelings with math.****Illustrate the breaking points in significance for DEF and ARM.**

That first bullet is going to be challenging. John is a complex guy. He’s like an onion: comprised of layers. I’ll do my best to construct a postulate based on those layers, but like an onion, when you get into one the wrong way… things might go wrong. ** So… caveat lector.**

**Axiom 1. John Demaris’ Match-Up’s ≡ Bad Match-Up’s**

No surprises here. Let’s keep digging.

**Axiom 2. John Demaris’ Menoth > John Demaris’ Skorne > Your Skorne**

Kind of a recently discovered nuance, although like an iceberg, I feel like we’re only seeing what’s above the water. The more ominous reality of this axiom lies below the surface. I’m not certain about the magnitudes here, but I think the inequality is pretty on point.

**Axiom 3. Perfect Memory + Preternatural Control of Dice + No Feelings = Keith = Robot (or Cylon)**

I think we all share this one. My vote comes down on the Cylon side of the fence. So say we all.

**Axiom 4. Looking into JVM’s Eyes : Swoon :: First time listening to the Beatles : Pensive**

Again, no surprises here, although I would have personally layered in “missing him like the deserts miss the rain”. No biggie. So…those are the four material tenets that I’ve heard play-out so far.

In all seriousness, I’m a huge fan of the MoM guys. John was an exceptionally welcoming force for me when I started the game. It meant the world to me at the time and I continually try to repay that kindness by trolling him into the ground. I suppose, if mimicry is the highest form of flattery, certainly trolling is just a shade under it.

The gang brings up a good point: **There are breaking points in DEF and ARM that are more significant than others in the game state. **So what are they and who do we solve for them?

There are two ways to illustrate this with math. One looks deals on the “threat by threat” basis, the other deals with the metrics in aggregate. I’m going to solve for this using the former method, but I’ll be sure to describe the later so you get to see both forms of logic at work.

Dice outcomes fall into a “bell curve”. This is also called a **Gaussian distribution**. The main point is that stuff near the average (mean) happens more frequently than stuff on the extreme ends (tails). Here is the distribution for outcomes on 3d6. I’ve elected to show this analysis with 3d6 because the pictures are prettier, but I’ll include the results for 2d6 as well.

Now, on the podcast, the MoM gents remarked that moving around the mean (10.5 on 3d6) really doesn’t do a whole hell of a lot. What you need to do is force your opponent’s “to hit” roll toward less likely tails of the bell curve. A way to objectively go for this point is to find the point of maximum “drop off” and force your opponent to need to make rolls in that highly unlikely range.

**This tiptoes into one of my favorite genres of Math; Calculus. **

What we’re trying to define is the first derivative of the 3d6 outcomes curve. That may have been an uncomfortable sentence, so I’ll boil that one down.

**A derivative is a function that is defined by something else.** One could say, its values are ‘derived’ by another function. Yea. Super creative math nerds at work on that one. One of the best ways to illustrate derivatives in action is by using position, velocity (speed) and acceleration. Let’s graph these for a car moving faster and faster across a five hour trip.

Position is… position. Ok. Easy start. Speed (velocity) is the change in position over the change in time. Ok. We’re still doing good. You can then go one step further and use those changes in velocity to define his acceleration over the five hour period shown. What I’ve graphed above then is a car at a constant acceleration of 2miles per hour every hour across a five hour trip.

These steps can also be shown by also by the following progression of formulae.

**Position = x ^{2} |x=time**

**Velocity (1 ^{st} derivative of position)= 2x**

**Acceleration(1 ^{st} derivative of velocity, 2^{nd} derivative of position) = 2**

Derivatives are used everywhere. Just ask your friendly local engineer about them. Great stuff. There are even financial markets (called derivatives, go figure) that operate solely on values derived by other underlying assets. Hell, there was even a derivative you could invest in that based its value on the sales of **David Bowie albums**. Go ahead. Google it. Wild stuff.

Back to the matter at hand; The MoM guys wanted to know where the maximum drop off would occur. If we take a look at the first derivative of the bell curve, we’ll be able to objectively say where that “most steep” point lies on 3d6.

Here’s the picture of the bell curve again in blue and its first derivative in red:

Here’s the answer: On the high side, 13.5. On the low side, 7.5.

What does that generally mean? It means that if your opponent is rolling 3d6 and they need less than an eight, it’s VERY likely to happen. Conversely, if you opponent is rolling 3d6 and they need a fourteen or higher, now your pushing into the strata of truly unlikely. For a 2d6 it’s five and nine.

What does THAT mean? This is a way to loosely define bending the game’s balance. Are you worried about POW8 blast damage with 2d6 rolls? Being ARM18 is markedly better than ARM17. Are you worried about hitting DEF18 troops? MAT10 is significantly better than MAT9. So buff up or just go bowling with trolls; that’s fun and effective too.

Ok. That’s a big meaty mountain of math and I really don’t want to push anyone too much further. As promised, I will briefly discuss the second method for solving for these breaking points. That said, this next part is complex and it’s going to move FAST. If enough people complain, I can go through the exercise more thoroughly and write a follow-up article on it. If not, the following will have to suffice.

Here you go…

What you would do is create a “true” supply and demand curve. Supply being the number of entities that you have to pierce high DEF (or ARM) and Demand being the number of targets you have would that require it.

To solve for this you need to define a Meta or pool of models. Yes. The values depend on what you interact with as a player. By example, my meta doesn’t really have Iron-Fleshed Kayazi. It never has. As a result, when people were bemoaning them and needed to deal with high DEF, I never felt that. Yea. Skorne LOVED those days.

Once you define the Meta you can build your Supply and Demand curves using the DEF, ARM, MAT/RAT and POW of each of the models. I would boil this down to either an attack or hp basis to mark ‘quantity’. My money is on hp being the better metric (to account for round-over-round persistence). I would also apply buffs at the “point of inflection” in battle. Buffs are generally going to be where they NEED to be assuming intelligent use. Again, another assumption.

Once you’ve constructed the curves you’ll see “breaking points” or points of inflection. These points would reflect exactly what the MoM gang wanted to know.

I hope this gave article gave you all a bit of assurance in defining meta-bending pieces mathematically and how to deal with them. **Thank you Greg** for forcing me to write this article quickly. I appreciate the mericeless voice only a friend can provide to do more, faster.

**Addendum:**

Per Pinegulf’s comments below I wanted to add a note for accuracy’s sake. The bell curves above are shown as continuous (containing all values between each integer). This is strictly speaking, not accurate. You can’t roll a 3.76. You only roll whole numbers. A discrete (based on specific values) graph would have been more accurate. Thank you Pinegulf. Here you go:

With that, don’t forget that two plus MAT is one quarter of the ‘battle’.