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Message: <blockquote data-quote="Esker" data-source="post: 7908216" data-attributes="member: 6966824">Ah, so in the first derivation, K had represented the ratio between a monster's hitpoints and the AoE damage per target, whereas in the second it became the ratio between a monster's hitpoints and the party's damage.Let's use the second definition, and redo the first formula, also changing M to be the actual number of monsters, and ignoring the floor and ceiling bits in order to be more general (if a bit more idealized).So now it takes K turns to kill each monster with no contribution from the caster, and so the first monster will get K turns, the second 2K, etc., for a total of K(1+2+...+M) monster turns in total, which is KM(M+1)/2.The AoE doing PD damage to each target reduces each monster's HP from KD to KD-PD = (K-P)D, in other words, changing K to K'=K-P. Therefore, the monsters will now get K'M(M+1)/2 = KM(M+1)/2 - PM(M+1)2 turns, and the AoE has saved PM(M+1)/2 turns in total.A single target burst doing PD damage to one target is shaving off P/K monsters worth of hitpoints, which is P rounds worth. This accelerates every monster's demise therefore by P rounds (I had this wrong before in my late-night delirium; assuming efficient spillover, the chance of pushing a monster's demise back a round doesn't depend on the monster's HP; it depends on the number of additional rounds worth of damage you're supplying), and so we've prevented a total of MP monster-turns. Now, reassuringly, this is a special case of AoE damage if there's only one monster.The ratio in impact, therefore, between doing PD AoE damage to each of M monsters and doing PD damage to a single target is (M+1)/2; whereas scaling up single target by a factor of M would increase impact proportionally, so AoE damage is (M+1)/2M as efficient as an equivalent amount of single target damage.This has [USER=72555]@NotAYakk[/USER]'s derivation as a (sort of) special case: instead of fixing the rest of the party's baseline damage to one monster's worth of HP, we've allowed monster HP and background party damage to vary separately, with their ratio controlled by the constant K, and expressed the AoE damage in terms of the party's damage per round, rather than in terms of a monster's HP.</blockquote>

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