Make ready the guns!

So I thought it’s about time I do another post on some game math. This time, I’m going to talk about cannons and how to make them the most effective.

For those uninitiated, I’ll give you a quick rundown on how cannons work. In Warhammer, there is a special die called an Artillery Die. This is a standard, 6 sided die but instead of the numbers 1 through 6, it has a Misfire symbol and the numbers 2,4,6,8 and 10. When it comes to shooting, you pivot your cannon and nominate any point directly in front of the cannon within its range (usually 60″ or further). You then roll the Artillery Die and that is how much the cannon overshoots that point by. You then roll the Artillery Die again and that is how far the ball bounces. If a Misfire is rolled on the first roll, the cannon has malfunctioned in some way and may even blow up. If a Misfire is rolled on the second roll, the ball just digs into the ground. To determine the models hit, you look at every model who base is crossed by the line between the point of impact and the point at the end of the bounce. Multiple wound models have a chance to stop the cannon ball if they are not killed and units can suffer, at most, one casualty per rank.

Right from the get-go it is easy to see a few things. Firstly; there is a ⅙ chance to roll a Misfire. If you don’t roll a Misfire, then the average roll of each die is 6″. If you were to consider the Misfire result as a 0 for the bounce, the average roll is 5″.*  This is where things start to get interesting as it is not actually possible to roll 5″.

The next point where things get interesting is that Warhammer is rather fickle with measuring systems. In the game, all measurements are done in inches, usually whole inches. The size of the bases on the models is metric. Standard infantry are on 20mmx20mm bases, large infantry, like orcs, are on 25mmx25mm bases and monstrous infantry are usually on 40mmx40mm bases. This tops out with the biggest monsters being on, I believe, 100mmx150mm bases.

100mm is approximately 4″. Approximately is important though. If you were to aim exactly at the point on the front of a 100mm base from directly infront of it and rolled a 4, you would over shoot by 0.073″. An ideal opponent wouldn’t care and it’s very rare that you are shooting from directly in front anyway. A quick application of the pythagorean theorem tells us that on a 100mmx150mm base, the longest line from corner to corner on this base is ~180mm 7.096″.

Here’s a quick table of some bases sizes and their diagonals in millimetres and inches.

These numbers don’t mean a whole lot out of context and it’s not like there’s anything here you couldn’t have worked out for yourself. Let’s talk about where is the best place to aim to hit things. I’ll start with hitting single models.

Let’s consider aiming for the front of the model. Literally at the earliest point on the base. Here are the probabilities of hitting.

…

Ok, it’s missing. Truth be told, I started this post months ago and since then have thought about it a bit more and there is a much, much simpler approach owing to the discrete nature of the dice. Suppose you are shooting at a single model. Drawing a line across the model from the point of the cannon, you are guaranteed one point on that line. If the base is long enough, you can fit two points 2″ apart, if it’s longer still, you can fit three points 2″ apart each (a 4″ line). If it’s monstrously large, you might even fit four points 2″ apart (a 6″ line). Now let’s begin with the simple case where the base is small enough such that you can only have one point hitting it. How far in front of the model should you aim? If you aim more than 10″ in front of the front of the model, you cannot hit it on the full, if you aim less than 10″ from the back of the model, there is a chance you overshoot. If you aim at it such that you will hit on a roll of a 10, let’s consider the chance you hit.

P(hit single spot) = P(hit on the full) + P(bounce through)
P(hit single spot) = ⅙ + P(bounce through)

Ok, let’s look at the probability of bouncing through. Essentially all this means is that the sum of the two dice is greater than or equal to 10 because of where we’ve aimed. We could roll a 2 followed by an 8 or a 10, a 4 followed by a 6, 8 or 10, a 6 followed by a 4, 6, 8 or a 10 or an 8 followed by a 2, 4, 6, 8 or 10. This gives us 14/36 ways that we hit.

P(hit single spot) = ⅙ + 14/36
P(hit single spot) = 20/36