Liquidedust and I worked on some numbers for cannons against out Shrines a while back. For me, personally, it's been a century since I've felt the pain of cannons (aside from the occasional shot) so I thought I'd go over the theory again and present my numbers.
I'm sure there are topics on Bugmans and Empire on this, but when it comes to military matters, Dark Elves aren't lazy and prefer to do everything themselves.
The first part of this topic will be dedicated to the maths and chance to hit a single target. I'll summarize the lessons learned at the end. (go there now)
I have also cursed the numbers and twisted them a little so that these conclusions will serve you excellently when you use them against High Elves but they will mysteriously fail you if you use them against Dark Elves.
Content:
- Workings of the cannons
- Part 1 - Hitting a single target.
- (Part 2 - In production) Damage charts from the cannons
- (Part 3 - In production) Hitting multiple targets with the cannons
- Conclusions
Workings of the cannons
To better understand cannons, we'll investigate the limitations and potential of cannons. So we focus on:
- How accurate they are, or how likely they'll hit their target.
- How devastating they are, when they hit.
Firing a cannon
A brief recap on how cannons are fired.
- Designate a point on the table, for which the cannon aims.
- Roll artillery dice and draw a straight line from the cannon, over the designated spot, adding inches to the shot as shown by the artillery dice (measured from the designated spot). This is where the cannon ball lands.
- Roll the artillery dice again, for the "bounce" distance. The value rolled indicates the number of inches the cannon ball bounces/rolls over the field. A misfire indicates the cannon ball does not bounce.
- The cannon ball hits everything on its path, between the point where it "lands" to the point up to where it "bounced". The cannon ball can be blocked in this path, by terrain or any model which it failed to wound.
For the controlling player, the only variable of control is the target and the distance to target. This is why often the distance of a cannon shot is
expressed as distance to the target, rather than distance from the cannon.
The challenge, for the controlling player, is choosing a distance to the target with the higher probability to score a hit.
Dice facts
- Artillery dice, is a D6 with values 'misfire', 2,4,6,8,10 with each side having an equal chance of 1/6 to be rolled.
- The bounce roll uses an artillery dice with the misfire to mark 0.
- A Rune of Forging allows you to reroll misfires on either the initial artillery roll or the bounce roll. We assume both artillery and bounce can be rerolled in the same shot.
- Rerolling the artillery dice on misfires gives an out come "misfire" with 1/36 chance and any other of 2,4,6,8,10 with 7/36 chance.
- Rerolling the bounce rolls gives 0 with 1/36 chance and 2,4,6,8,10 with 7/36 chance.
- A Master Engineer allows you to reroll the initial artillery roll. This is a choice, so a balance must be made between the benefit of re-rolling vs the benefit of not re-rolling.
Always re-roll misfire
We will assume that a misfire is always re-rolled if a rule permits it. For both the first artillery roll and the bounce roll, it is always beneficial to re-roll a misfire.
There is only one situation where it may not be preferable to re-roll a misfire:
- In case you have a friendly behind your designated target
- And you scored a hit on the designated target, on the artillery dice.
- And you are lucky enough to get no bounce on the bounce roll.
And to check if friendly fire is on.
Questions we want answered
- What distance from the front, or the back of the target gives the best chance to hit for each kind of target?
- What is the (best) chance of hitting the target?
- What is the impact of a rune of forging?
- What is the impact of the Dwarf Master Engineer?
- What is the trade-off from the re-roll?
- What is the maximum chance to hit that can be achieved with the Master Engineer?
- What is the impact of a rune of forging, combined with the Dwarf Master Engineer? How does this impact the aforementioned strategies and trade-off?
- Given a chance to hit, how likely is a cannon to take out a Shrine or COB after 1, 2, 3, 4 shots etc?
- What is the best strategy to hit ranked troops of various types?
Calculations and tools used
2 inch intervals and base size.
The distance travelled by the cannon ball is a factor, and so it might be easier to hit bigger targets. Since the distance jumps per 2 inches, we can examine the number of inches that cross a base, for different angles, using simple trigonometry. (A^2 = B^2 + C^2)
Assuming we always face the cannon on the left corner of our base, we can colour the regions of our base where the cross-section is less than 2 inches, 2 to 4 inches, 4 to 6 inches.
Legend:
- Green: less than 2 inches cross section.
- Yellow: at least 2 inches cross section.
- Orange: at least 4 inches cross section.
We can see an interesting difference between the 40 mm peg bases and 50 mm peg bases. The smaller base only offers a cross section of 2+ inches along the diagonal line. The larger base, however, always offers a 2+ inch cross section except for an almost perfectly straight cross line.
We can also see that it would be a challenge to avoid a 4+ inch cross section for chariots and shrines unless they show their flank to the cannons.
While it seems we didn't reach a 6 inch cross section on any single base, we'll include it in the calculations for completion's sake. We won't go into details on its results however.
Chance to hit, method 1: build-up formula
We can calculate the chance to hit a target as follows:
- For every distance to target guessed (designated targetting point)
- For every roll of the artillery dice
- Calculate the chance to hit the target on the bounce roll.
- Add to total chance for the distance guessed.
- For every roll of the artillery dice
We can then do this for every base size. While this method works, it requires quite a lot of calculations. We can not use the short cut of multinomials, to combine the artillery roll and bounce roll because a combined outcome of 14" doesn't indicate how much of that distance was bounced which we need to validate the hit.
Chance to hit, method 2: negative chance to miss formula
A slightly easier way to calculate the odds, is as follows:
- For every distance guessed.
- Calculate chance that the cannon ball falls short and doesn't reach the target P_undershoot. This can make use of the combined outcome of the artillery and bounce roll.
- Calculate chance to overshoot the target P_overshoot. This can be done solely on the outcome table of the artillery dice.
- Chance = 1 - P_misfire - P_overshoot - P_undershoot
The method is less intuitive but quicker to compute.
Measuring distance
The distance to target can be measured in many ways, by measuring from the front, centre or rear of the target. No factor relevant to the preferred target are influenced in any way by what point of reference we use to guess the distance.
For sake of completion, we'll compute the distance from the front and back.
Basic distance achieved
We nominate a point on the table and shoot the cannon. The distance achieved, with respect to the designated point, is:
- misfire!: 16.67%
- 2": 2.78%
- 4": 5.56%
- 6": 8.33%
- 8": 11.11%
- 10": 13.89%
- 12": 13.89%
- 14": 11.11%
- 16": 8.33%
- 18": 5.56%
- 20": 2.78%
This outcome can be cached to compute the chance to hit more easily. At least, it shows us we need only be worried about computing distances between 2" and 20".
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