Wednesday, November 18, 2009

Everyone Deserves a Second Chance - Calculating Re-rolls with Method 216

I don't get many things right the first time, in fact, I am told that a lot. - Ben Folds

So looks like I took another hiatus from 40K, luckily (a word I never use in conjunction with mathhammer) I'm back and brought an easy way to factor in re-rolls to Method 216.

Method 216 is a super easy and simple method for calculating expected kills in Warhammer 40K using some simple math in your head, on some paper, or even using the calculator on your cell phone.

If you don't know what Method 216 is, or just don't remember, then go learn all about it here Super Simple Method 216.

Now that you are a Method 216 expert, let's push it a step further and factor in the re-rolls.

Here's an example to start us off,
8 Guardian Jetbikes (with no Shuriken Cannon upgrades) are shooting at a Space Marine tactical squad.

The calculation for Method 216 would be,
(16 * 3 * 3 * 2) / 216 = 1.33~ Space Marines shot to pieces!

BUT WAIT A MINUTE, those Shuriken Cannons are twin linked, so how do I factor in the re-rolls?

For re-rolling failed hits and wounds all you have to do is add the Method 216 result from before times the re-roll multiplier in the chart below.

6+ 0.83
5+ 0.67
4+ 0.50
3+ 0.33
2+ 0.17

So since the Guardian Jetbikes needed a 4+ to hit before, you take the 1.33~ dead marines and add 1.33~ times 0.50.

1.33~ + (1.33 * 0.50) = 1.96~
An easier way to do with is simply 1.33~ * 1.50 = 1.96~ Dead Space Marines

If a Farseer had Doomed the Space Marine Tactical Squad, then the calculation would be,
1.33~ * 1.5 * 1.5 = 3~ Dead Space Marines

Now then for re-rolling saves you simply multiply the Method 216 result by the re-roll multiplier.

So if the 7 remaining marines were to rapid fire back (assuming they all just had boltguns) then you would have,
(14 * 4 * 3 * 2) / 216 = 1.56~ Eldar souls eaten by the warp

If the Farseer had cast Fortune on the Guardian Jetbikes then it would look like.
1.56~ * 0.33 = 0.51 Eldar souls eaten by the warp

Now go forth and mathhammer!!!

Friday, March 20, 2009

Mathhammer: Scatter dice Probability II

I've been asked since writing my previous article on scatter dice probability how to calculate the probability of a certain distance of scatter, not just a direct hit. This is actually pretty easy to calculate using the same forumla.

How do you calculate the prob of scattering 1 inch or less for a BS of 4?
- Use the same formula I showed you before, but use a BS of 5 instead of 4
- For 2 inches or less use a BS of 6 instead of a 4

That gives me the probability of 1 inch or less, how do I get the probability of just scattering 1 inch?
- Calculate the probability of scattering 1 inch or less, then subtract the probability of not scattering at all, to get the probability of just scattering 1 inch.

Lets go over some examples

To calculate the probability of a blast template scattering 2 inches or less with a BS of 4
- there are 15 ways to roll a 6 (BS of 4+2) or less on 2D6
P = ( 1/3 ) + ( ( 15 / 36 ) * 2/3 ) = .611 = 61% chance of scattering 2 inches or less with a BS of 4

To calculate the probability of a blast template scattering just 2 inches
- Subtract the probability of a blast template scattering 1 inch or less from the probability of scattering 2 inches or less
- P = 61% - 52% = 9%

You can check your results against these

BS of 2
scatter of 0 = 35%
scatter of 1 = 4%
scatter of 2 = 5%

BS of 3
scatter of 0 = 39%
scatter of 1 = 5%
scatter of 2 = 8%

BS of 4
scatter of 0 = 44%
scatter of 1 = 8%
scatter of 2 = 9%

- Hope this helps!

Thursday, March 19, 2009

Mathhammer: Scatter dice probability

“Our bombs are smarter than the average high school student. At least they can find Kuwait.” - A. Whitney Brown

A few days ago I started to think of an easy formula people could use for figuring out the probability that a blast weapon would scatter within a certain distance. Here is what I came up with.

How the scatter dice work
- You roll 1D6 in which two sides have a [hit] marker and four sides have an [arrow] for the direction the blast template scatters.
- You roll 2D6 and add the numbers together to see how far the blast template scatters if an [arrow] is rolled.
- Sometimes you can subtract the BS of the shooter from the 2D6 roll.

Here is what we know
- 1 out of 3 times the template doesn't scatter at all due to the [hit] marker.
- 2 out of 3 times the template doesn't scatter if the 2D6 roll is less than or equal to your BS.

So far the formula looks like this
- P=Probability that blast templates won't scatter
- P = ( 1/3 ) + ( chance that 2D6 - BS = 0 )
- the ( 1 / 3 ) is because one out of three times we roll a [hit] on the scatter dice

How do you figure out the odds of a 2D6 roll?
- There are 36 (6*6) different possibilities when rolling 2D6
- Here is a chart that shows how many possibilities there are for each result on a 2D6 Roll

2D6 ROLL --- # Possibilities
2 --- 1
3 --- 2
4 --- 3
5 --- 4
6 --- 5
7 --- 6
8 --- 5
9 --- 4
10 --- 3
11 --- 2
12 --- 1

Now what does the formula look like?
- P=Probability that blast templates won't scatter
- POSS=Number of possibilities 2D6 will be equal to or less than your BS
- P = ( 1/3 ) + ( ( POSS / 36 ) * 2/3 )

Let's go through some examples

What is the probability that a blast template won't scatter, with a BS 4 shooter?
- There are 6 ways to roll a 4 or less on 2D6, 1 way to roll 2, 2 ways to roll 3, and 3 ways to roll 4.
- So plug in the values and you get
P = ( 1/3 ) + ( ( 6/36 ) * 2/3 ) = 0.44 = 44% chance the template won't scatter

What is the probability that a blast template will scatter 2 inches or less with a BS 3 shooter?
- A BS 3 shooter needs to roll a 5 or less on 2D6 to scatter 2 inches or less.
- Thee are 10 ways to roll 5 or less on 2D6, 1 way to roll 2, 2 ways to roll 3, 3 ways to roll 4, and 4 ways to roll 5.
- So plug in the values and you get
P = ( 1/3 ) + ( (10/36) * 2/3 ) = 0.518 = 52% chance the template won't scatter more than 2 inches

What is the probability that my deep striking Necron lord will scatter 8 inches or less?
- You need to roll 8 or less on 2D6 to scatter 8 inches or less.
- There are 26 ways to roll 8 or less on 2D6.
- So plug in the values and you get
P = ( 1/3 ) + ( (26/36) * 2/3 ) = 0.814 = 81% chance of not deep striking more than 8 inches

- I hope this helps everyone, and of course probability for blast template scatter will be in FarseerMobile and FarseerOnline when they are released. Let me know what you think!

Monday, March 16, 2009

FarseerMobile (Mathhammer Calculator) for Blackberry First Beta Complete

The first beta of FarseerMobile for Blackberry is complete and was sent out to the beta testers tonight. What makes the Blackberry version so useful is you can actually use it during a game as getting the results is super fast, all you have to do is plug in three to four numbers and there you have it. I used it while putting together a Necron army list to play against my brothers IG army, and I was very surprised especially by the results of comparing destroyers with heavy destroyers by their expected damage results against vehicles. I ended up taking two units of regular destroyers instead of the heavy destroyers.

I especially like the feature of being able to select each result on the damage chart you want combined into a single probability. The math for combining these is a lot more complicated than just adding the probability of each damage result, as it's a binomial probabilty, so it's really nice to get that number by just checking a few boxes.

I'm working on a screen for blast scatter, and leadership probability, should be done in a week or so. I could still use a few more beta testers, so again if you are interested, just email me and I'll get you setup.

- As soon as the Blackberry release version is complete I'll be starting on a web based version, which shouldn't be too hard as most of the code from the BB version will be reused.

Thursday, March 12, 2009

WANTED: FarseerMobile beta testers

Do you play Warhammer 40K? Do you own a Blackberry? If so you might be interested in beta testing FarseerMobile, a new free Blackberry application I've developed for making quick mathhammer calculations. It's simple to use, very fast, and you don't even need to have any mathhammer skills to use it. I plan on having calculations for shooting at infantry, shooting at vehicles, rolled distance moves such as moving through cover etc, and passing or failing moral checks.

So far the beta only includes calculations for shooting at infantry, here is how it works.

Here is what you input,
- Number of shots
- Dice roll needed to hit
- Dice roll needed to wound
- Dice roll needed for target to make an armor / cover / inv save
- Optional dice roll needed for a second save such as FNP or WBB

Here is what it calculates,
- Expected number of unsaved wounds for a single shot
- Expected number of unsaved wounds totaled for all shots
- Probability of the shooting resulting in 0 unsaved wounds
- Probability of the shooting resulting in 1+ unsaved wounds
- Probability of the shooting resulting in 2+ unsaved wounds etc.....

If you are interested in becoming a beta tester just send me an email to benjamin.cauldwell ( at ) with the following information,
- Your name
- How long you have been playing 40K and which army do you play the most
- Which model of Blackberry you have

I plan on having a web based version in the future, but I want to focus on the Blackberry version for now since I have one, and it's so portable.

Hope to hear from some eager beta tester soon!

Friday, January 23, 2009

40K and the Assault

"The bayonet has always been the weapon of the brave and the chief tool of victory" - Napoleon Bonaparte

Napoleon Bonaparte is my absolute favorite historical figure of all time. Of course the biggest reason behind this is his ability to apply his strategies to destroy his enemies over and over again on the field of battle. Most people when they hear Napoleon think of his defeat at Waterloo, but don't think about how he completely out generaled the finest leaders Europe had to offer battle after battle. At Waterloo I think Napoleon rolled way too many 1's, it happens to all of us sometimes.

Napoleon's winning strategy tended to look like this...

1) - Out maneuver your opponent so that the majority of your army is fighting a minority of your enemies army.

2) - Soften up your enemy with long range firepower.

3) - Engage your enemy with just about everything you have to further soften them up.

4) - Assault with your elites and watch your enemy run for the hills.

The point is, shooting with your big guns and your gun line does a lot, but nothing ends a battle quite like some good assaulting.

Here is why assaulting in 40K is the grand daddy of ways to defeat your opponent...

1) - Your enemy is always in range!

2) - When you charge you get an extra attack!

3) - Causing a single wound can destroy a unit with a failed leadership test and a sweeping advance!

4) - You get to attack during both yours and your opponent's turns!

5) - You get to move D6 inches after you destroy your opponent, even if it's not your turn!

When you assault you should be throwing 2-3 times the number of points as the enemy unit costs at them. The point is NOT to sit there turn after turn hacking away at each other. The point is to hit them and hit them hard to force a leadership test they will fail, so that just a few wounds results in that unit no longer existing, in just one round of combat! Having a multiple of your units in the combat ensures you win the combat, and that you also have a good chance sweeping advancing the losers.

That's all for now, don't forget to fix bayonets!

Wednesday, January 14, 2009

Super Simple 'Method 216'

Do not worry about your problems with mathematics, I assure you mine are far greater. - Albert Einstein

Today I'm going to explain to you a super simple method I've developed for quickly calculating how many of something you can expect to kill by shooting or close combat. I've named it 'Method 216' for reasons that will become very obvious to you when I'm done explaining how it works.

Before we get into 'Method 216', let's go over some very basic things about probability. Very often we are going to need to know how many sides of a D6 will give us a successful roll of the dice. This is quite easy to figure out. Let's say a Space Marine is shooting his bolter at a Necron warrior 18" away. The Space Marine needs a 3+ on a D6 in order to score a hit, so there are 4 sides of the D6 that will result in a successful roll, because a 3, 4, 5, or 6 will result in a hit. So if I were to ask you how many sides of a D6 will result in a successful roll in the previous example, you would say 4. The same would be true for rolling to wound, if a bolter with a S4 will wound a Necron warrior with a T4 on a 4+ then you would say 3 sides of a D6 will result in a successful roll.

Take note that in the case of your opponent making an armor / cover save it's a little different because a successful roll for your opponent is a failure for you. So because the Necron warrior needs a 3+ to make his armor save, then a 1 or a 2 will result in a successful dice roll for you. So if asked how many sides of a D6 will result in a successful dice roll for you when the Necron warrior rolls for his save, you would say 2.

Now then here is how the 'Method 216' works...

(H * W * S) / 216 = average number of kills

H = # of sides on a D6 that will result in a hit

W = # of sides on a D6 that will result in a wound

S = # of sides on a D6 that will result in your opponent failing his armor / cover save

Let's go over a few examples...

1) Space marine firing at a Necron warrior 18" away.

SM needs a 3+ to hit, a 4+ to wound, and Necron warrior needs a 3+ to save.

3+ to hit = 4 sides of a D6 to hit

4+ to wound = 3 sides of a D6 to wound

3+ armor save = 2 sides of a D6 to kill

Plug in the numbers and you get (4 * 3 * 2) / 216 = 0.111111

So the SM on average should kill a Necron warrior about 11% of the time, if you have a unit of 5 marines firing, then just multiply the .111111 by 5 to get .555555. Or in other words the 5 marines can expect to kill .555 Necrons per round of shooting.

2) 3 Necron Heavy Destroyers firing at a squad of Space Marines.

Necrons need a 3+ to hit, a 2+ to wound, and Space Marine does not get a save (S9 weapon vs T4).

3+ to hit = 4 sides of a D6 to hit

2+ to wound = 5 sides of a D6 to wound

no save = 6 sides of a D6 to kill

Plug in the numbers and you get (4 * 5 * 6) / 216 = .556

So each Heavy Destroyer should kill a SM 55.6% of the time, or as a squadron of 3, they should kill .556 * 3 = 1.667 Space Marines per round of shooting.

3) 12 Necron warriors shooting at a unit of Terminators less than 12" away.

Necrons need 3+ to hit, 4+ to wound, and Terminators need a 2+ to save. Necrons get double number of shots since they are within rapid fire range.

3+ to hit = 4 sides of a D6 to hit

4+ to wound = 3 sides of a D6 to wound

2+ armor save = 1 side of a D6 to kill

Plug in the numbers and you get (4 * 3 * 1) / 216 = .0555

24 shots * .0555 = 1.33 Terminators per round of shooting

Once you have done this a dozen or so times it should become very easy, and you should be able to figure it out during a game quickly using the calculator that comes with your cell phone.

That's it for today, hope this helps you crush your foes in battle, let me know if you have any questions. In the future I'll post some more advanced Mathhammer concepts.

- By the way, the number 216 that you divide by comes from the fact that to kill someone three dice rolls have to take place, hit, wound, and save. So 6*6*6 = 216. Chaos players might prefer to call this 'Method 666'.

What is Mathhammer?

Probability is the very guide of life. - Cicero

Mathhammer is using mathematics and probability to give yourself an advantage over your opponent in Warhammer and Warhammer 40K. The Warhammer games are full of numbers and probability, which makes understanding the math of the game a very powerful tool.

By learning Mathhammer you can apply simple methods to assist you in making rational decisions before, during and after a game.

Here are just some examples of what Mathhammer can be used for...
- Deciding which units / weapons to take against certain opponents (should I take Necron destroyers or heavy destroyers against Space Marines?)
- Making decisions during the game to maximize shooting and close combat (should I shoot at that unit or charge it?)
- Working out if something that happened during the game was probable or not (should my landraider be afraid of a single scarab base because this game it was destroyed by one?)

This is what Mathhammer is NOT!!!
- A way to predict exactly what will happen
- A tool that will make all your decisions for you
- An actual hammer you can hit things with

In my next post I'll explain my super simple 'Method 216', a quick and easy method that will be the base of the majority of your Mathhammer calculations. It will take just a few minutes to learn even if you suck at math and can be used to calculate most things in just a few seconds using a cell phone's calculator.

Tuesday, January 13, 2009


Hello everyone and welcome to Mathhammer 4 Morons!

First I'd like to thank my brother Alex for creating the 'Mathhammer 4 Morons' logo, I think he did a fantastic job and as always I am very jealous of his artistic ability.

At the end of December I decided I wanted to get back into wargaming, specifically 40K. Being as I'm a computer programmer by day I couldn't help but think that an army of souless mechanical Necrons would suit me the best. So New Years Eve I purchased a Necron boxed army and have been busy ever since putting everything together, thus ending my 10 year hiatus from the hobby. I forgot how much work this was!

Although I hope to cover everything Warhammer with my blog, I'll probally be focusing on Mathhammer content.

Check back soon for a very simple tutorial on Mathhammer and some updates on my Necron army.