Mathhammer for Blasts and Large Blasts

[Olesh]

Greetings!

 

Warp Angel introduced me to these forums some time ago; I'm that Tyranid player he is always playing against. :lol:

 

Earlier today, I was complaining how it's difficult to math out the average wounds when firing blast/large blast weapons. He told me that if I ever figured it out, to come here and brag. ;)

 

Actually, he told me to share it here on BoC. And here I am! BUT, and there's always a BUT, there does not exist a formula that is manageable. Sorry. No easy plugging in numbers to calculate the average hits. However, that's not the end. I developed a tool that calculates it for me; allows easy plugging in numbers and all that jazz.

 

However, I'm not allowed to distribute it because of the license of the program I used. :cuss: Personal use only and all that jazz.

 

What I CAN do is use the tool I developed to calculate on the behalf of others. So, if anyone has a question about the average wounds a given blast/large blast weapon will inflict on a given target, here's what I need from you to plug it in:

 

1.) The weapon used, i.e. "a plasma cannon at BS 4".

2.) The target unit, i.e., "imperial guardsmen"

3.) The model count of the target unit

4.) The distribution of the target unit.

 

#4 requires a bit more explanation. For example, are the models in a line? Are they in ranks? Are they forming a 3x3 square of models? How far apart are each of the models? Are we assuming a cover save? Are there two squads relatively near each other? and so forth.

 

I will try and work from a verbal description, picture, or diagram as accurately as possible, and post the results here (with images, if necessary) so that they can be referenced. Hopefully someone will find it useful to have accurate information on the averages of blast template weapons against various distributions of units.

[River Black]

If the distribution of the models is know then we can do it ourselves. The point is that the distribution is not known and with the new scatter rules the number hit is even more random. I believe the point of mathhammer is not to simply say “you will kill this many guys on average” but to be able to estimate the effectiveness of a unit and draw comparative conclusions from the data by using CONSTANT known statistics. The distribution is not constant so unless you release a spread sheet indicating casualties for 1 marine hit, then 2 marines hit, then 3 marines and so on the single statistic that you can provide with your blast formula doesn’t help me to much.

[Warp Angel]

You're missing the point.

 

The program can figure out effectiveness vs. a line of troops, a square, tight bunch, loose bunch, etc. Provide weapon S, AP, and the target T and Sv., and you supply the scenario and the program will figure out the average scatter, likely number of hits.

 

For example, it will tell you what the effectiveness of a template is against guard in COD (Close Order Drill) in one rank or COD in 2 ranks. And against a squad of genestealers in a V shape.

 

If you can plot things on a grid of some kind, the program can figure out the average number of wounds for that weapon type vs. that enemy troop deployment shape.

 

In theory, it'll also tell you what the most efficient anti-template formations are that you can deploy your troops in.

 

It's a brute force solution, but I'm planning on running enough simulations through it to come up with some "optimal" deployments.

[minigun762]

We'll I'd start off with a fairly standard request, the BattleCannon or equivalent.

 

1.) BattleCannon, S8 AP3

2.) IG/Defiler = BS3

3.) 10 man Tactical squad and 30 man Ork squad

4.) For Marines, 2 ranks of 5 Marines behind each other, spaced 1" apart, for Orks 3 ranks of 10 Orks behind each other, spaced 1" apart

[Olesh]

We'll I'd start off with a fairly standard request, the BattleCannon or equivalent.

 

1.) BattleCannon, S8 AP3

2.) IG/Defiler = BS3

3.) 10 man Tactical squad and 30 man Ork squad

4.) For Marines, 2 ranks of 5 Marines behind each other, spaced 1" apart, for Orks 3 ranks of 10 Orks behind each other, spaced 1" apart

 

The shots below are fired at BS3, with large blast templates, against models on standard 40mm bases.

 

A 2x5 set of marines, spaced 1" apart, in a rectangular grid like so...

 

O_O_X_O_O

_ _ _ _ _ _

O_O_O_O_O

 

where the X is the model the large blast template is centered over...

 

On a HIT result of the die, the center six models are struck.

On a scatter result of the die, 2.06623 models are struck on average.

The total average models hit per large template fired is 3.37749.

 

From here, the average wounds per model is (3.37749 * 5/6) = 2.81457 wounds, no save.

 

A 3x10 set of Orks, spaced apart in a rectangular grid with 1" between models like so...

O_O_O_O_O_O_O_O_O_O

_ _ _ _ _ _ _ _ _ _ _ _ _

O_O_O_O_X_O_O_O_O_O

_ _ _ _ _ _ _ _ _ _ _ _ _

O_O_O_O_O_O_O_O_O_O

 

where the X is the model the large blast template is centered over...

 

On a HIT result of the die, the center nine models are struck.

On a scatter result of the die, 4.02591 models are struck on average.

The total average models hit per large template fired is 5.683.

 

From here, the average wounds per model is (5.683 * 5/6) = 4.735833 wounds, no save.

[Reinholt]

This is actually not all that hard to compute given a "standardized" formation, if you will.

 

However, speaking as someone with a formal background in mathematics and statistics, the problem is that you are most likely going to encounter units in one of a few specific formations:

 

1 - Line

2 - Loose "blob"

3 - Tightly packed "blob"

 

I've actually got a couple hundred photographs of games played (I take snapshots of the board after each player turn and sometimes during turns for particularly interesting moves so that I can go back and look at them later with a digital camera), and those are far, far and away the most common deployments you will see.

 

The line is very easy to compute; you use a bit of math about the probability of scatter in any given direction (for instance, assuming equal probability along each 1 degree interval of a 360 degree possibility), if the thing scatters at all, and then compare it to the line of models (taking into consideration the spacing of the models, which is usually pretty regular for lines).

 

The blobs, on the other hand, are a <deleted by the Inquisition> nightmare, because small changes in the precise deployment of the blob means that you are actually going to experience large changes in the number of models hit. There are some critical points with regard to spacing and shape that are not intuitive or easy to compute (though you can get at them with some ideas like neighborhood analysis from topology, but I'm going to suggest that's a bit much for casual use); given the staggering variety of things that you run into, compared with the necessity of considering permutations based on terrain, movement restrictions, deliberate attempts to muck with template placement, and the like, blobs are going to be pretty intractable to deal with.

 

That's not taking into account multi-level terrain or any of that nonsense, which is when things go from bad to worse.

 

This is one of those times when you can quantify the basic formations, but the problem is that basic formations are rarely what you encounter on the actual field of battle... as a mathematician, I formally advise you to beware of mathematics in this case.

 

Addendum: I use ~2 and ~4 as my "rule of thumb" hit rates for small and large blast templates, respectively, but it will vary wildly depending on unit size, distribution, and the like (and, of course, you can always scatter into other units...)

[Khorneeq]

Quite interesting...

 

1) Plasma Cannon, BS 4

2) Rhino 3"x5"

 

And some clarifications: Shot is coming from the side as I'm assuming this is most probable. Bearing in mind rear armour AV 10, I would like to know if there is a better spot to place a blast marker than the centre point?

[Warp Angel]

This is actually not all that hard to compute given a "standardized" formation, if you will.

 

However, speaking as someone with a formal background in mathematics and statistics, the problem is that you are most likely going to encounter units in one of a few specific formations:

 

1 - Line

2 - Loose "blob"

3 - Tightly packed "blob"

 

I've actually got a couple hundred photographs of games played (I take snapshots of the board after each player turn and sometimes during turns for particularly interesting moves so that I can go back and look at them later with a digital camera), and those are far, far and away the most common deployments you will see.

 

The line is very easy to compute; you use a bit of math about the probability of scatter in any given direction (for instance, assuming equal probability along each 1 degree interval of a 360 degree possibility), if the thing scatters at all, and then compare it to the line of models (taking into consideration the spacing of the models, which is usually pretty regular for lines).

 

The blobs, on the other hand, are a <deleted by the Inquisition> nightmare, because small changes in the precise deployment of the blob means that you are actually going to experience large changes in the number of models hit. There are some critical points with regard to spacing and shape that are not intuitive or easy to compute (though you can get at them with some ideas like neighborhood analysis from topology, but I'm going to suggest that's a bit much for casual use); given the staggering variety of things that you run into, compared with the necessity of considering permutations based on terrain, movement restrictions, deliberate attempts to muck with template placement, and the like, blobs are going to be pretty intractable to deal with.

 

That's not taking into account multi-level terrain or any of that nonsense, which is when things go from bad to worse.

 

This is one of those times when you can quantify the basic formations, but the problem is that basic formations are rarely what you encounter on the actual field of battle... as a mathematician, I formally advise you to beware of mathematics in this case.

 

Addendum: I use ~2 and ~4 as my "rule of thumb" hit rates for small and large blast templates, respectively, but it will vary wildly depending on unit size, distribution, and the like (and, of course, you can always scatter into other units...)

 

So give him a blob layout on a grid and see what you get. He can do more than just straight lines. Hollow square? Circle? V? Packed in one spot (up against terrain) but spaced well over the rest of the unit....

[Olesh]

That's true. You can sit down and do it by hand, and probably get fairly close, but that's extraordinarily time consuming. I wanted an accurate tool that could handle any squad configuration, and from start to finish was quick to input and compute.

 

The wonderful thing about the tool is that it doesn't care, really, what the actual shape of the formation is. Given a shapeless blob, an uneven line stretching 15 inches, a flying V, or a squad of gaunts in the shape of the letter "H", it'll calculate it quickly and accurately.

 

That's why I mentioned pictures or drawings. You can calculate "non-standard" formations easily if you have a picture, or a plot of the models' locations on a sheet of graph paper.

 

The only thing the tool does, by the way, is determine hits. The wounds/average failed saves is a known quantity and easy to compute.

 

Khorneeq, you handed me an interesting problem.

 

Plasma Cannon at BS 4 vs Rhino Side Armor.

 

Naturally, a HIT result from the die is a direct hit.

However, because a plasma cannon is half strength unless the hole in the template is directly over the vehicle, partial results are ineffective even against rear armor. So aiming for the center of the model looks like the simplest solution.

 

When aiming at the center, scatter at BS 4 will still average a hit ~69% of the time (0.690756). Adding in the chance of rolling a HIT result and your average chance of hitting is ~79% (0.7938).

 

But what if I put the template an inch from the corner of the rhino and let it scatter from there? It still hits 54% of the time (0.541458) on average. That's BS4 for you; 1/6 of all your scatter rolls become hits regardless of direction, and almost 8/9 of the time if it scatters in the right direction you're still hitting. Even if it's slightly outside the 90" arc, a 5 or 6 on the scatter die can still have the hole in the template over the model.

 

Aiming at the center for best chance, however, you're hitting 0.7938 times on average, and have a 50% chance of getting a glance or a pen if you do hit. So your total chance of getting a glance or a pen on a rhino's side armor with a plasma cannon is 39.69% (0.3969) per shot. Of that chance, you've got a 13.23% chance of glancing and a 23.46% chance of getting a penetrating hit. Your total chance of destroying the rhino in a single shot with a plasma cannon through wrecking or exploding it is a measly 7.82% (0.2346 * 1/3).

[Reinholt]

So give him a blob layout on a grid and see what you get.

 

The problem is not computing any one blob layout (easy); the problem is computing infinity possible blob layouts when considering spacing, lines of sight, adjacent units / terrain, placement on the board (did you scatter the center of the template off the board edge?), and so on.

 

Any individual discrete situation is easy (assuming you accurately capture it). The problem is that I don't know in advance what every shape my opponent will use throughout the course of a game and their context. In other words, I can't predict the future, so I don't know which discrete situations to pick (though I use a rule of thumb, as stated above, to capture typical results for the most common ones, but there is still a HUGE amount of variety).

 

Conversely, the continuous set of all situations is very hard, and as stated above, small and apparently superficial changes in blob shape actually can produce dramatic changes in the result of the shot.

 

What I'm saying is that it's easy to compute any given individual situation, but when you find a way to compute all infinity situations for me so I can take an accurate average, let me know.

 

:P

 

Edit - to be clear here, I have a bunch of statistical packages that can do exactly what the OP is talking about on my computer at work; the math is not actually all that hard. You could even do it by hand, if you were enough of a glutton for punishment, or more likely, probably whip up something in Excel or Access to do it for you, if you wanted it to be accessible to anyone with office. The problem is that even being able to do this with 100% accuracy is of very limited utility. You can't monte-carlo infinity possibilities, and once you get beyond the usual suspects (rough blob, rough line, number of models all with good visibility to unit), the variation becomes so immense that knowing you'll hit between zero and a lot is not a useful result in advance. I knew that already.

[minigun762]

Thanks, that makes sense to me. Less than I would have liked but thats good to know.

My main goal is to use it to compare the effectiveness of blast/ordnance weapons to normal weapons for killing power. In the case of the Marines and Battle Cannon, 2.8 dead Marines is not much greater than 4 Krak Missiles fired @ BS4 = 2.2 dead Marines.

[Olesh]

It's actually 72 lines of code. That's it.

 

And the tool provides a sufficiently accurate result within seconds. How much accuracy do you need? A degree? A hundredth of a degree? Accurate to a degree is sufficient to me (the above examples were all accurate to 1/10th a degree).

 

Yes, you could develop the tool in an excel spreadsheet if you really wanted to.

 

My eventual goal for this project is to make a GUI and streamline the tool use to the point where you could run it on the fly, in the middle of a game (if so inclined), and get accurate averages.

 

But I'm absolute rubbish at programming, so no user interface. Just a clunky tool that's fun to use.

 

Also, in response to minigun762...

 

A line of models is one of the least vulnerable formations you can get. The closer the formation gets to resembling a filled circle, the more likely it is that a blast or large blast template will hit while scattering. For a line, there is a relatively small arc where a scatter more than 1 or 2 inches will still hit models. If the marines are closer to the aforementioned "shapeless blob", and if they're packed closer together, a large blast will hit more of them. If they are spread further apart, you are less likely to scatter off them entirely but a HIT will get fewer models. And so forth.

[minigun762]

A line of models is one of the least vulnerable formations you can get. The closer the formation gets to resembling a filled circle, the more likely it is that a blast or large blast template will hit while scattering. For a line, there is a relatively small arc where a scatter more than 1 or 2 inches will still hit models. If the marines are closer to the aforementioned "shapeless blob", and if they're packed closer together, a large blast will hit more of them. If they are spread further apart, you are less likely to scatter off them entirely but a HIT will get fewer models. And so forth.

 

So is the case of having 2 ranks more similar to having a line or a blob?

 

I would assume that having your army arranged in a circle formation would be the worst, as it most closely matches the blast marker and allows for some scatter in any direction. An "X" formation would seem to accomplish similar results.

 

The other cases would be the "V" that was mentioned as well as the standard line.

 

Could you do the same Battle Cannon example at 1" differences with an "X" shape, "V" shape, single/double line as well as the blob to give us an example of what is the best deployment to counter pie plates?

[Marlow]

I am interested to see how many hits a Plasma Cannon gets when fired with BS 4 against other Marines

Can you give results for the "X" shape, "V" shape, single/double line assuming figures are spaced out 1" apart

as well as clumped into a blob to maximise cover save.

[Olesh]

A line of models is one of the least vulnerable formations you can get. The closer the formation gets to resembling a filled circle, the more likely it is that a blast or large blast template will hit while scattering. For a line, there is a relatively small arc where a scatter more than 1 or 2 inches will still hit models. If the marines are closer to the aforementioned "shapeless blob", and if they're packed closer together, a large blast will hit more of them. If they are spread further apart, you are less likely to scatter off them entirely but a HIT will get fewer models. And so forth.

 

So is the case of having 2 ranks more similar to having a line or a blob?

 

I would assume that having your army arranged in a circle formation would be the worst, as it most closely matches the blast marker and allows for some scatter in any direction. An "X" formation would seem to accomplish similar results.

 

The other cases would be the "V" that was mentioned as well as the standard line.

 

Could you do the same Battle Cannon example at 1" differences with an "X" shape, "V" shape, single/double line as well as the blob to give us an example of what is the best deployment to counter pie plates?

 

I'll edit this post with examples in a little bit; I'm about to grab dinner, but yes, having two ranks (as opposed to one) increases your odds of getting hit by scatter. Your base odds of being hit by a HIT result are the same in any case (1/3), with ballistic skill modifying that up slightly (a BS4 is the equivalent of getting a HIT result 50% of the time overall.)

 

The absolute worst formation you tend to see on a regular basis in play is the standard deep strike formation of seven terminators - you have a center model, with six models arranged around it in a hexagon which describes a filled rough circle shape. An actual literal, unfilled circle (with no models in the center) is similar to a double line, depending somewhat on the size of the circle, because you have to place the blast marker on a model. For a hollow circle, the minimum and worst size formation has a diameter equal or less than a diameter of 9 - Ballistic Skill in inches. Anything larger than that is less likely to recieve scatter.

 

I will already tell you that a single line, spread out, is the best deployment to counter pie plates, but you have to consider more than just one individual squad. If you have a row of marines, with another squad of marines behind them, it's similar to having got a 2x10 formation. Also realize that 2/3 of all scatter die rolls fall between 5-9 inches on the scatter dice. At BS3 you can expect most pie plates to fall between 2 and 6 inches away.

 

Edit: I'm doing these at 1 degree angle increments, so the accuracy is slightly less.

10 man squad in an X-shape = 2.8196 hits on average.

10 man squad in a wide angle V shape = 1.71842 hits on average. (!)

10 man squad in a straight line = 1.93544 hits on average.

10 man squad in a double line = 3.38132 hits on average. (!)

10 man squad in a rough blob = 2.96263 hits on average.

 

It's important to note that at 1" apart, these marines are only half as spread out as they could be. The #1 defense for protecting yourself against pie plates is spread out! In the double line, a direct HIT result on the scatter die meant that 6 marines were hit. In the straight line and the V shape, only three marines could be hit at one time - maybe four, on a lucky scatter between models. The rough blob was a little bit more spread out, because it's not a rough blob if everyone's exactly 1" apart, and consequently a HIT result only got 5 marines.

 

Generally speaking, your gut feeling is going to be correct. To protect yourself against blast templates, you spread out (and stay spread out), and stay in straight lines. Particularly against large blasts, you're not going to avoid being hit - you minimize the effects of being hit instead.

 

Knowing what kind of averages you're going to get firing at different shapes, though, helps you judge where your templates are going to be most effective. An Ork player firing a BS2 small blast template at a 5-man squad of marines that just disembarked is frequently going to scatter far enough that it misses the squad completely; it becomes an all-or-nothing situation. Against a larger group of spread out marines, however, you can be reasonably assured of hitting SOMETHING.

[minigun762]

Edit: I'm doing these at 1 degree angle increments, so the accuracy is slightly less.

10 man squad in an X-shape = 2.8196 hits on average.

10 man squad in a wide angle V shape = 1.71842 hits on average. (!)

10 man squad in a straight line = 1.93544 hits on average.

10 man squad in a double line = 3.38132 hits on average. (!)

10 man squad in a rough blob = 2.96263 hits on average.

 

It's important to note that at 1" apart, these marines are only half as spread out as they could be.

 

Great work Olesh. Its good to know what is and isn't safe for models with incoming ordnance. I'm surprised by the V shape, but I must say I'll be using that more often.

 

I mentioned the 1" spread simply because its a compromise.

Against Ordnance you want to be fully spread out, but against assaults, or moving into cover, you usually want to bunch up, so I had to split the difference.

 

Very good point about who is firing at you. A Marine Vindicator is probably going to drop it right on your head more often than not so spread out to minimize damage. An Ork blast might miss you entirely, so bunch up if you're feeling lucky.

[Olesh]

Great work Olesh. Its good to know what is and isn't safe for models with incoming ordnance. I'm surprised by the V shape, but I must say I'll be using that more often.

 

I mentioned the 1" spread simply because its a compromise.

Against Ordnance you want to be fully spread out, but against assaults, or moving into cover, you usually want to bunch up, so I had to split the difference.

 

Very good point about who is firing at you. A Marine Vindicator is probably going to drop it right on your head more often than not so spread out to minimize damage. An Ork blast might miss you entirely, so bunch up if you're feeling lucky.

 

I'd like to point out that this is just the effectiveness of a position for the purposes of avoiding a blast template. A V-shaped formation is going to be just as vulnerable as a line to regular fire (because only one model has to be visible and within range of the shooting unit) and it's possible that some or most of the troops in the V will themselves not be in a firing position. Particularly during deployment, a line is effective at maximizing shooting of the unit while minimizing the unit's particular risk.

 

With my Tyranids, I start out with lines of units, with an inverted "open box" of gaunts to protect the rest of my models (some of whom are themselves providing cover for other models). This formation typically only lasts for the first turn or two of the game, as I lose gaunts or crack the whip (use models on the "tail" end to keep an objective while I maintain a long "whip" across the battlefield to stay in synapse and/or provide cover.)

 

Unless I'm in an assault, or jockying for space behind/in limited cover, I always keep models as spread out as possible. Tyranids are very vulnerable to blasts and templates so it behooves me to minimize my exposure while taking as much advantage from the situation as possible.

 

One thing I've noticed about many players, particularly Guard and Eldar players, is that they do not spread their units very far out. Coherency is 2", and the standard 25mm Marine base is another inch. From model to model, a 10-man squad of marines can cover almost two feet of table space, allowing you to keep an objective and still deploy firepower forward or across the table (albeit in a less concentrated fashion).

 

A quick and dirty estimation of a unit's vulnerability to blast templates works like this - find the "center" of the unit, which is the most likely target for a template to be placed. Measure out a rough circle around that unit with a radius equal to 7.5 inches - the opponent's ballistic skill. (For Guardsmen, this would be 4.5 inches). Anything that touches the invisible circle you draw has the highest likelihood of being directly under a blast template (and therefore hit). Moving closer (or, more preferably) away from the center of the circle reduces the likelihood of being hit by scatter.

 

Edit: A marine vindicator has approximately a 72% chance of hitting the model it targets. You want to spread out, as you said, but even against Ork gunnery you still take fwer wounds on average by spreading out (with even small squads). The only time, defensively, that bunching up is potentially worth the risk is against blasts that are fired indirectly, because the models' ballistic skill does not reduce the scatter.