MathHammer
Overview
MathHammer at its most basic refers to the practice of calculating odds of unit A killing unit B. In the abstract, it refers to the application of statistics to judge army composition decisions.
As the name implies it's frequently used in 40k and Warhammer fantasy, but also applies to other RPGs. It can be very useful in determining the value of a unit, especially when factoring in points costs.
D6 mathhammer, the 216 method (7th Edition)
Being built around 3 successive rolls of 1d6 (To hit, To Wound, Save), 40k lends itself to a very quick and effective method for getting a % chance of killing a model, without having to resort to spreadsheets. The 216 method (6^3) simply totals up the chances of getting a result you want, then divides them by the total number of results that can occur, which for rolling 3d6 is 216.
This example will use a Chaos Marine attacking a Guardsman in close combat
- Total the number of roll results that result in a hit - 3+ so 4
- Total the number of roll results that result in a wound - 3+ so 4
- Total the number of roll results that result in a FAILED save - 5+ so 4
- Multiply them together: 4x4 = 16, 16x4 = 64, this is the number of possible dice rolls on 3d6 that result in what you want, i.e. a casualty.
- divide the number you got from step 4 by 216, in this case 0.29~, which is the % of a kill.
This gives a result of one attack having a slightly worse than a 1 in 3 chance of killing a guardsman in close combat.
For instances where one of the rolls is ignored, such as with a bolt pistol against a humble flak armour save, it becomes the 36 method (6^2), since there are 36 possible rolls for 2d6:
- Total the number of roll results that result in a hit - 3+ so 4
- Total the number of roll results that result in a wound - 3+ so 4
- Multiply them together: 4x4 = 16,
- Divide the number from step 4 by 36, in this case .44~
So a marine with a bolt pistol has a slightly worse than 1 in 2 chance of killing a guardsman.
With these odds readily at hand, the decision to charge a guardsman unit with marines suddenly doesn't have any merit whatsoever, and conversely the guardsmen are probably in a better position if they charge the marines.
D6 mathhammer, the 216 method (8th Edition)
Standard Bolters no longer have AP, so the Sternguard Special Issue Bolter is possibly the only way to replicate the .44~ from a bolt weapon used in the example above at 18 points per model + gun. Standard Bolters now have the same 33~% chance per shot as close combat attacks, but have the advantage that you can rapid fire and then charge, leading to 1 marine at 13 points having a combined rapid fire + close combat likelihood of guardsman murder of approximately 87%.
12 points spent on 3 basic guardsmen under the same conditions have a 50~% chance of killing said marine with 9 s3 attacks (( 3 * 2 * 2 )/ 216), A guardsman with an overcharged plasma gun (11 points) gets an agreeable 75~% ((( 3 * 5 * 5 )/216) * 2) + 1 close combat attack, and a guardsman heavy weapon team with a heavy bolter ((( 3 * 4 * 3 )/216) * 3) at 14 points a slightly disappointing 72~% factoring even with 1 rapid fire lasgun and 2 close combat attacks.
It should be noted however this completely discounts the non-uncommon occurrence of hot dice, providing merely average performance grades. With more low value shots and bodies in play, the impact of a swing in your favor increases.
The drawback illustrated
Since 7th edition games are won or lost in the army composition stage, not in the actual rolling of the dice, working from pure statistics is schmuck bait. Mathhammer can lead you down trails of endless disappointment if you leave out the other factors which, per game, can change wildly. It's vitally important to account for numbers such as shot volume, which can behave counter-intuitively - 1 shot at BS 4 is 1/4 the amount of shots hitting, on average, as 4 shots at BS 4, but nearly 1/2 (43.75%) the odds of at least one shot hitting.
For example, these odds are at per-attack basis, so a marine with a bolt pistol has a 44% chance of killing a guardsman, a marine with a rapid-firing bolter has a ~44%+~44% chance, a little less than an 69% chance (odds of both missing would be 56%x56%=31%), almost a guaranteed kill on a single guardsman, and a reasonable chance to kill two of them (19% chance).
A Chaos Marine that's assault-optimized can get as many as 4 attacks on the charge, giving the guardsman roughly a .70~ chance of survival per attack, meaning 70% to survive the first melee attack, 49.5% to survive the second, 34.8% against the third, and 24.5% against the fourth. The single Chaos Marine from this example could reasonably be expected to kill 1 to 2 guardsman in a round of shooting his bolter, or firing a bolt pistol followed by a charge, 2 to 3 guardsman in a turn, though could kill as many as 5 even though it's statistically unlikely.
Bearing in mind, if, say, a squad of 5 marines is going against a squad of 20 guardsman, that's a little less than a 2% chance to be hurt by a shot, per shot, on overwatch, but with 40 shots means a 74% chance that one of those marines goes down.
Why you would want to do this
So why exactly would you want to recreate statistics 101 in your hobby time? Because GW didn't. Running Mathhammer against your army list will show up the truly junk options that were crammed in your codex, and in turn the absolutely insane power breaks that have no bearing on units points cost. It can however be very useful in avoiding trap units, and putting together viable counters if you are struggling with army composition.
Beware!
Mathhammer, like employing the power of the warp, can be a force for good when used with discipline and restraint. But reckless mathhammer is a step on the path to damnation.