Mathhammer : A Review of Resilience

Mathhammer : A Review of Resilience


DeeJay here with a mathhammer article talking about damage mitigation, or unit resilience, in the game of 40k.    If you like this article, there are many other great, similar, articles in the Tactics Corner!

How good is a scattershield on a wraithknight?   How much better are terminators to tactical marines?   Just how effective is FNP?    All these questions are answered by learning how a unit’s resilience to damage works.

Damage vs Resilience

Damage and Resilience are opposites, like hot and cold.     In order to properly understand resilience, we need to review damage.

Damage dealt is the process of placing wounds onto a unit with the intention of causing a unit to change it’s current state through the process of removal of wounds or hull points.     The upper boundary of ‘damage dealt’ is the amount of damage an army can deliver over the course of the entire game.   

This dealt damage is then mitigated by a unit’s resilience.    Resilience is defined in the game of 40k as “The capacity to reduce or ignore incoming damage”.     The amount of resilience a unit has determines the amount of damage that can be mitigated.

In contract to damage, resilience does have an upper limit — 100%.    When a unit has 100% damage reduction, the unit will not take any damage.   This can happen when shooting bolters at a wraithknight or land raider.

The unmitigated damage is then passed onto the unit, causing degraded performance of units, such as ‘weapon destroyed’, or could cause one or more models in the unit to be removed from play all together.  


Increasing Effects of Mitigation

As anyone who has ever played a tank in World of Warcraft can tell you, the difference between 60% and 80% damage mitigation is twice the damage, not 20%.   For example if 10 wounds of damage was dealt to a unit and they had a 60% damage mitigation, then four wounds of unmitigated damage would be passed onto the unit.    If the unit had 80% damage mitigation, then only two wounds of unmitigated damage would be passed on, resulting in half the overall damage done to the unit!

If we look at another example of one unit with an 80% damage mitigation, and another unit with 90% damage mitigation, and both units have 10 wounds of damage dealt — the first unit will suffer 2 wounds of unmitigated damage and the second unit will suffer one.   Once again, the unmitigated damage was cut in half, but this time we only needed to increase the mitigation by 10%

As we continue to get closer to a 100% damage reduction, we see that same effect.   90% to 95% is cuts damage by half.   95% to 97.5% cuts it again in half.    


How does this affect my 40k game?

The same concept applies to your 40k game, not just MMO’s.

Let’s say you are increasing the armor value of a unit from 5+ to 4+, such as giving carapace armor to IG vets.    If 10 wounds are placed on the vets, normally they would be taking ~6.66 wounds.   With the carapace armor they are taking only 5 wounds.   The unit with the flak armor took 33% more damage than the unit with the carapace armor.

What happens we continue this trend?   Let’s say you have a squad of Flesh Hounds, aka Khorne Dogs, and use the Grimoire of True Names on them?   Their save drops from a 5++ down to a 3++.    Given the same 10 wounds done to the unit, the Khrone Dogs without the Grimoire will take ~6.66 wounds, and the Khorne Dogs with the Grimoire will take ~3.33 wounds — or double the total damage done.   

In the case of the IG vets, not having the carapace armor resulted in a 32% increase in damage, where the dogs had a 100% increase in damage!    This is because as you get closer to 100% damage resistance, the effect becomes greater.    This concept is what I call the “Rule of Resilience” — as a unit get’s a better damage mitigation, the greater the benefits they receive from increasing that mitigation by the same percentage.

This chart illustrates the difference in damage increase dealt to a target by reducing its save.

Damage Increase Saving Throws
25% 5+ to 6+
33% 4+ to 5+
50% 3+ to 4+
100% 2+ to 3+


Time to Live

Another way to to visualize the effect described above is to talk about “Time to Live”   This is defined as the amount damage can be directed at a unit until it dies, given a state of average dice rolls.   

Yes, it is ‘possible’ that on scatter laser will kill 4 marines.  The chances for one scatter laser shot to kill a marine is 5/27, or 18.5% (2/3*5/6*1/3).

Pulling off four such shots lowers the odds to 625/531441, or 0.001%. (5/27*5/27*5/27*5/27).

I’m not saying it won’t happen in your lifetime. I am saying that odds like that are the reason the casinos are in business.

You should ignore extremes when estimating effects and plan your tactics on averages instead.

For example, we can estimate 203 MEQ bolter shots to kill 10 Plague Marines, while it takes 31 MEQ bolter shots to kill 10 orks not in cover. The “Time to Live” value for Plague Marines is significantly more than orks when shot at by bolter fire.

There are many elements that affect “Time to Live”. Let’s focus on the one for now– the armor save. Notice how the “Time to Live” value increases as the armor save increases.

Unit Time to Live
10 Marine Scouts 61 MEQ Bolters
10 Tactical Marines 91 MEQ Bolters
10 Tactical Terminators 181 MEQ Bolters

This chart should look very similar to the one earlier in the article describing damage taken vs armor saves.   That concept combined with ‘Time to Live’ illustrates how increasing armor saves can increase the unit’s time to live.

Increasing the armor save might be more than just bringing better units to the table.   It might be something as simple as going to ground in ruins.    Another concept is that night fighting helps units in ruins more than units on area terrain.

Strength/Toughness Alteration

Altering the STR/Toughness of a unit can have dramatic changes on the “Time to Live” value. Even one pip change can have significant changes.

Unit Time to Live
10 Tactical Marines 90 MEQ Bolters
10 Marine Bikes 135 MEQ Bolters
10 Mark of Nurgle CSM Bikers 270 MEQ Bolters
5 Mark of Nurgle Spawn 135 MEQ Bolters

There are a few takeaways here. One of which is that a single pip of toughness increase on the model gives a 50% increase in “Time to Live” when taking attacks from a weapon whose strength matched the previous pip!    

Multi-wound models are usually better in these cases, as they can take damage before they start losing their damage output.    4 wounds done to the tactical squads will result in a 40% damage output decrease, where clever placement of Marine Centurions could result in a 0% damage output decrease.

Notice how increasing the toughness to 6 with the Mark of Nurgle Bikes changed the “Time to Live” to 200% when compared the Centurions. The first pip change increased “Time to Live by 50% while the next increased by 100%. The nurgle bikes are nearly 300% as resilient as the TAC marines.

Look at the spawn. 15 wounds of T6 chaos spawn are as difficult to kill with MEQ bolters 5 Centurion Devastators – and half as difficult as the MoN bikers. This is due to the lack of armor save.    While toughness/strength comparisons make a big difference, the armor saves also play a role.

This is also why units such as Wraithknights are so powerful — it’s because their toughness is so high that many weapons cannot actually hurt them.     Increasing toughness or AV is also the only way in the game to reach 100% damage mitigation.     


Layering Resilience

Resilience can be added in layers to generate a greater effect.    Take your standard wraithlord with it’s toughness of an 8.   When it’s shot at by a White Scars Marine’s melta-gun, the chances of taking a wound look like this.

(⅔ to hit) * (½ to wound) * (1 shot) = ⅓ or 33.3% chance to wound.

Adding FNP

What happens if that same weapon is shot at a wraithknight with dual D cannons?    The chances of the weapons hitting and wounding are identical, but in this case the wraithknight gets a 5+ FNP save.

(⅔ to hit) * (½ to wound) * (⅔ failed FNP) * (1 shot) = 2/9 or 22.2% chance to wound.    

This means the wraithlord is taking 50% more damage than the wraithknight just because of the FNP save.    Where 10 melta gun shots would do 3.3 wounds to the wraithlord, the wraithknight will only take 2.2.    This is a 50% increase in the “Time to Live” on the same number of wounds on the wraithknight to the wraithlord.    In other words, even though the wraithlord three wounds, and the wraithknight has six — a total of three wraithlords would die under the same firepower as one wraithknight.

Adding Invulnerable Save

What happens if the wraithknight has a 5++ save?  

(⅔ to hit) * (½ to wound) * (⅔ failed save) * (⅔ failed FNP) * (1 shot) = 4/27 or 14.8% chance to wound.    

Again, we find that there is an increase of “Time to Live” by 50%.      It also means that the wraithlord will take 225% more damage than the wraithknight with a scattershield!

Increasing FNP Save

Another great example of this is the greater rewards of a 4+ FNP on daemons.    The same melta-gun shooting at a daemon prince will have the following profile.

(⅔ to hit) * (5/6 to wound) * (⅔ failed save) * (1 shot) = 10/27 or ~37% chance to wound.    

If you get the 4+ FNP, the result will like this.

(⅔ to hit) * (5/6 to wound) * (⅔ failed save) * (½ failed FNP) * (1 shot) = 5/27 or ~18.5% chance to wound.   The daemon without the FNP takes double the damage and has half the “Time to Live”

How Can I Layer Mitigation

There are four key ways to improve mitigation.  The first way is to reduce the chance of your model being hit.   This can be done through causing incoming weapons to take ‘snap shots’ to hit your unit, reduce their ballistic skill to 1 through tools like ‘Invisibility’, or to increase your weapon skill to reduce their chances of hitting.

Another way to layer mitigation is to improve the toughness of your unit(s).   This can be when building your army, by picking tougher units, or can be done in game through psychic powers such as ‘Iron Arm’.

Improving saving throws also have major effects on mitigation.    If your unit is hit by a weapon with an AP value that exceeds their armor save, then no armor save is allowed.    Severely reduces the mitigation of the unit, and if standing in ruins gives a 4+ cover save, then the unit receiving damage will have twice the time to live than the unit not in cover.    If the unit also has ‘Shrouding’ affecting them while they are in ruins, they will they will have 600% the “Time to Live” as the unit out in the open!

The final ‘layering’ effect is through FNP or Reanimation Protocols.   When this save is allowed, it is effectively another armor save, and even a small value of 5+ will have a big impact on the units overall “Time to Live”

Anything that allows you to ‘reroll’ the saving throw or FNP roll gives a nice advantage.    Daemon gifts, psychic powers (Precog, Fortune) or army special rules (Ravenwing, Tzeentch re-roll 1’s)  can all give this ability.    The rerolls have greater effects the better the normal save is.   In other words, rerolling a 5++ results in a successful save chance of 55%.   Rerolling a 3+ for Ravenwing gives a 88.89% chance of saving.    


Built in Limits

The STR/Toughness effects can only be applied when modifying STR/Toughness within 2 pips of each other. A scatter laser does not care if you are a marine or a Fire Warrior.  In both cases the weapon will be wounding on a 2+.

This means that when facing armies with large amounts of high strength weapons, armies that bank a lot on higher toughness units will find themselves at a disadvantage when compared to lower toughness units.

This is why today, where STR 6+ weapons are commonplace, you are seeing less armies that spam plague marines.   Those armies put to much of a premium on the cost of the toughness, which is not worth it when high STR weapons are commonplace.   When bolters were the main weapons, those units fared much better.

This is also why units such as ‘warp spiders’ are effectively just as durable as tactical marines when your opponent has brought nothing but Eldar Windriders with scatter lasers.   

This is also why ‘psybolt’ ammo was such a bad idea back in the old GK codex.   Making a lascannon STR 10 has much less of an impact than giving storm bolters and autocannons +1 STR.   This is because the lascannon is usually out of the 2 point pip range, where the autocannons and storm bolters were always in that sweet spot to maximize effect.

Abusing the Extremes

We know that as we get closer to 100% in resilience we see a greater effect, and we know that layering levels of resistance also increases the “Time to Live”, so we push these envelopes to maximize the levels of resilience.

Anytime you can reroll a 2+ save, you are going from a ⅚ save to 35/36 save, or a 97.2% damage mitigation.    The ITC wisely changed that to a 11/12 damage mitigation, or 91% damage mitigation.   That ruling increased the damage done to units with a 2+ rerollable by over 300% — and fixed one of the biggest ‘breaks’ in the game.

However, a 91% damage mitigation is still complete rockstar.   Anytime you can pick up the dice and reroll your save, you will find a significant increase in damage resilience.    That’s part of what makes ‘Fortune’ such a great power.  Even when just using the 3+ save of the armor, the effective save is 8/9 or 88.8%.    

Given the ITC changes, this means that there is not that great of a difference between a 3+ rerollable and a 2+ rerollable.   The 3+ rerollable will only take 23% more damage — a fairly small percentage when we are working this close to the extreme boundaries.    Therefore, if you can find a way to get a 3+ rerollable, it will greatly increase the unit’s damage resilience.

Being able to reroll 1st, such as being a daemon of Tzeentch is another good way to increase damage resilience.   As with the other damage resilience increases, the closer the base save is to the boundary, the greater the effect.   Rerolling ones on a 2+ is the most extreme example, but even rerolling 1s on a 3+ is very good, providing a ~75% damage resilience.

Also, remember how layering protection greatly increased the “Time to Live” values.   Spend the points to put that 4+ invulnerable save on your stormsurge — the math shows that it’s worth it!   Boosting your riptide to a 3+ invulnerable save doubles it’s “Time to Live” when being shot at by AP2 weapons.

Reducing your opponent’s ability to deal damage, such as forcing your opponent to make snap shots against you will have a huge impact. (Nurgle warlord trait ‘PlaugeFly Hive’)   Conversely, this is also why powers such as ‘Doom’ or twin linked weapons are so effective — because they lower the resilience of the target.   I know it’s odd to think that a twin-linked weapon in that capacity.  

Other ways to remove layers of mitigation are to have the AP weapons that ignore the armor saves, or to use “Ignores Cover” weapons.    When these two traits are combined, they have devastating effect as they just strip away your opponents mitigation.


It’s All About Ratios

A lot of understanding mathhammer is understanding the ratios of effectiveness vs point cost.    Sure, you might be able to make an extremely durable unit by taking 10 Grey Knight Paladins with an Apothecary and two Brother Captains in Terminator armor,but you are looking at a 800+ point unit.

How much more durable is that paladin squad, expressed as a percentage, than an equal number of GK Terminators?   How does it compare to two wraithknights?   120 orks?

The idea here is that you just can’t take a unit and load it up with bling and expect it to kick ass, as it might be greatly outclassed due to ‘point bloat’ in a unit.

However, going cheap on a squad also can have significantly reduce that squad’s durability.   The 20 point upgrade in the aforementioned paladins increases their “Time to Live” by 50%, which in a 550 points unit is a no-brainer of an upgrade!

The concept here is ‘Resilience per Point’ of your unit.   How good is that unit when compared to others on a per-point ratio?


Using the Right Tools

Another concepts of understanding unit resilience is knowing to use the right tool for the job.   If you know a meltagun has a 14.8% to wound the wraithknight, but a grav gun on a bike has an 67% of inflicting a wound, then you know that the grav guns are tools to use!


These are the key takeaways.

  • Continued increases in durability have greater effects.
  • Layering levels of mitigation has increasingly greater effects.
  • Levels of layering are (to hit) (to wound) (armor save) (FNP save)
  • STR/Toughness values only matter within 2 pips
  • Rerolling mitigation dice can result in extremely broken units
  • Rerolling hits and wounds or removing cover can greatly lower a unit’s resilience
  • Think in ratios and percentages when making units.  

Mathhammer is all about percentages, and since Frontline Gaming sells Games Workshop product at up to 25% off,  you can apply some mathhammer to your bank account.

If you have any input or comments, please add them below!


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6 years ago

> Those armies put to much of a premium on the cost of the toughness, which is not worth it when high STR weapons are commonplace. When bolters were the main weapons, those units fared much better.

Which is to say, never in the history of the game. 😛

Pretty good article overall in terms of being an introduction of the math of how things work out. The only major concepts I might add to it are penetration and diminishing returns.

Penetration would be the idea that no type of damage mitigation is perfect; all of them can be negated by some sort of effect or another. Low AP weapons, weapons with the Ignores Cover rule, weapons that don’t inflict wounds, etc, can all bypass various sorts of defenses, so when analyzing resilience it pays to know what your defenses will work against and what they won’t, especially in tournaments these days where things like Hunter’s Eye are common.

Diminishing returns is essentially the flip side of the effect of resilience scaling geometrically at the top end of the scale- being 99.8% resistant to Bolters is not actually all that different from 97% resistant from an in-game perspective; the vast majority of them will patter harmlessly off you either way and most opponents won’t even both trying, even though technically the 99.8% resistance is a hundred and fifty times harder to kill than the 97%. But functionally, at a certain point a particular tactic becomes so inefficient that it is no longer a viable option and opponents will simply not use it, hence the diminishing returns. If your unit already has Invisibility, Shrouding, and Endurance up, adding in Sanctuary is probably not going to change the way your opponent’s try to hurt you very much.

6 years ago
Reply to  DeeJay

Love the article. I think the “3+ re-rollable taking 23% more damage than a 2+/4+” is kind of misleading though. The difference between 9 wounds out of 100 and 11 wounds out of 100….not a lot 🙂 It technically is 23% but that makes it sound so much more significant haha

6 years ago

“This means that when facing armies with large amounts of high strength weapons, armies that bank a lot on higher toughness units will find themselves at a disadvantage when compared to lower toughness units.”

This is one of the big reasons horde orks don’t work anymore. In 5th, there wasn’t this abundance of S6 shooting on high BS platforms.

Nathan Fluger
Nathan Fluger
6 years ago
Reply to  DeeJay

It used to be that just T4 and snagging a cover save was enough for Orks to be incredibly durable, well above their points value. People would marvel at putting out an army’s worth (in 5th ed) of firepower and downing only like 10 boyz. “But I shot 50 bolters at them!” “Yeah, and that’s about what you should’ve killed: 60 pts”

6 years ago
Reply to  Nathan Fluger

Well, that was because in 5E everyone was bringing more anti-tank weapons and fewer anti-infantry weapons, simply because the meta was very different. Weapons like Scatter Lasers _existed_, people simply weren’t making all that much use of them.

Of course, the flip side was that you had all the joy of No Retreat to deal with. Kill ten Boyz and get a bonus ten dead ones absolutely FREE!

6 years ago

Great article, DeeJay!

6 years ago

Great overview, but some of the examples have some issues. Centurions have 2+Armour as well as the T5 and the extra Wound, for instance, which doesn’t seem to have been factored in. Also, 135 MEq Bolters only averages 5 Nurgle Spawn dead, not 10.

Also, I think that Resilience per Point needs more emphasis. Per Model, Terminators are unquestionably more resilient than Tactical Marines. But when you factor in that the Termies cost 2.5 times as much, it becomes clear that they’re actually less resilient, Point for Point, against everything except AP3 firepower.

Metagame considerations are very important, too. You touch on it a bit with the mention of how the Strength of common firepower has gone up, but things like the proliferation of Ignores Cover and Grav change the values of things like Shrouded or good Armour Saves immensely.

6 years ago
Reply to  DeeJay

I’m definitely looking forward to the ‘per point’ articles. It’s a factor that confounds me on say, adding stim injectors or drones to a unit of crisis suits.

6 years ago
Reply to  DeeJay

Excellent. I look forward to the “Per Point” articles as well.

6 years ago

DeeJay thank you for another good article to read. RL been a little busy lately, so when ppl post insights to 40k they are a valuable read for me. (I’ve only played about 10 games since 5th edition). Last time I tried to get a game in, I had to go rescue my drunk friends who got stranded an hour away from my area lol. Sometimes, the phones gotta just get turned off.

6 years ago

Nice article! I think the math is easier to follow if you just stick to percentages or decimals though, instead of using all the fractions and percentages together, forcing people to convert them back and forth to really understand. 0.333 * 0.666 = 0.221, or 22% chance is easier to follow along with, in my opinion than (2/6) * (4/6) = 8/36 or 22%. Plus decimals has the added benefit of being able to quickly be calculated with a basic calculator! 😀

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