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So I was playing around in the D&D character builder and made a bugbear barbarian that used an execution axe. The Execution axe has a normal damage die of 1d12 but the bugbear racial ability bumps that to 2d6. The brutal weapon property says to "reroll any die that displays a value equal to or lower than the brutal value given for the weapon". The Brutal Value for the execution axe is 2, does this mean that on that the lowest a bugbear wielding an execution axe can roll for damage is 6?

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Yes.

Brutal applies equally to all dice in a weapon's damage die and if it goes from a d12 to 2d6 then it's extra painful.

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Yes, that's right.
2d6 brutal 2 = 6-12 (average 9) damage per [W].

If he used an oversized Mordenkrad or Gouge that is normally "2d6 brutal 1" it becomes "2d8 brutal 1" for an even more ridiculous 4-16 (average 10) damage per [W] but without the "high crit" property of the Execution Axe.

Bugbears are a perennial favorite for optimizing damage to the hilt due to the over-sized thing, but none of my DMs have ever allowed them as a playable race. =P

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