WoW Mathematics: Destructive Shadow interference

During the Blood Prince Council encounter in Icecrown Citadel, players may find a number of Dark Nuclei floating around. These are vital for the tank tanking Prince Keleseth, for they each offer a stacking Shadow damage reduction. Against conventional wisdom, this damage reduction stacks multiplicatively, so full immunity cannot be achieved from a stack of three (as is the case with additive stacking).

Aim: To calculate the damage taken from Empowered Shadow Lance at a particular level of Shadow Resonance.

Summary:

• With Shadow Resonance of an order of a stack of n, the Keleseth tank will take 80,000(0.65)ⁿ damage from an Empowered Shadow Lance.
• Where the given proportion is r, multiplicative stacking refers to the repeated multiplication of its one's complement (1 - r).
• It is impossible for the Keleseth tank to gain full immunity from Shadow damage.

The Blood Princes are after your blood! Prince Keleseth is to the left.

The given:

• On average, an unmitigated Empowered Shadow Lance will inflict 80,000 Shadow damage.
• Any given Dark Nucleus will reduce Shadow damage taken by 35% (a proportion figure of 0.35).

The detail:
The proportion of Shadow damage that is not mitigated by a Dark Nucleus' Shadow Resonance and is therefore taken is:

1 - 0.35 = 0.65. This is the one's complement of the proportion 0.35.

Were the Keleseth tank only to have Shadow Resonance of order 1, from the Empowered Shadow Lance, they would take:

80,000 × 0.65 = 52,000 damage. Needless to say, it would kill a level 80 warlock tank.

Consider the case of Shadow Resonance of order 2. With the first damage reduction already applied, the second damage reduction would act on the remaining 52,000 HP of unmitigated damage. So, with Shadow Resonance of order 2, the Keleseth tank would take:

52,000 × 0.65 = (80,000 × 0.65) × 0.65 = 80,000 × 0.65² = 33,800 damage. This is the process of multiplicative stacking.

Where D is the damage from Empowered Shadow Lance, the general formula is:

D = 80,000(0.65)ⁿ

Consider a ranged tank with 30,000 HP that is tanking Prince Keleseth. To be able to survive one Empowered Shadow Lance, they must suffer less than 30,000 HP in damage from it:

30,000 = 80,000(0.65)ⁿ. Rearranging the formula, n ~ 2.2768527. For D < 30,000 , n must be rounded up to n ~ 3 (Shadow Resonance of order 3).