Critical hit

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As you get to some of the higher experience levels, you will begin to score critical hits more and more often. There are five levels of critical hit, ranging from:

Type Magnitude
Critical hit types table
CRITICAL 2x damage
CRUSHING CRITICAL 4x damage
OBLITERATING CRITICAL 8x damage
ANNIHILATINGLY POWERFUL CRITICAL 16x damage
WORLD-SHATTERING CRITICAL 32x damage

Note: if you have a 10% chance to score a standard critical hit, then you have a 10% chance for a standard critical to become a crushing critical hit, and a 10% chance for a crushing critical to become an obliterating critical hit, etc.

In aethership vs. denizen combat, crew members can score critical hits while manning a battle turret. The chance of critical hits there is determined by one's skill in Aethercraft rather than level, however.

Note that certain denizens are immune to critical hits - notably city and commune guards and certain extraordinarily powerful denizens (i.e., the Demon Lords of Nil).

Maths

Expected Value

To calculate the expected value (aka the mean) of some attack a:
Let d = the base damage of the attack
Let c = the percentage critical hit chance

Type Damage Chance Expected Value
Expected Value Calculation
Normal hit d 1 - c d(1-c)
CRITICAL 2 * d c - c^2 2d(c-c^2)
CRUSHING CRITICAL 4 * d c^2 - c^4 4d(c^2-c^4)
OBLITERATING CRITICAL 8 * d c^4 - c^8 8d(c^4-c^8)
ANNIHILATINGLY POWERFUL CRITICAL 16 * d c^8 - c^16 16d(c^8-c^16)
WORLD-SHATTERING CRITICAL 32 * d c^16 32dc^16

So our expected value for this attack is the sum of rightmost column above
d(1-c) + 2d(c-c^2) + 4d(c^2-c^4) + 8d(c^4-c^8) + 16d(c^8-c^16) + 32dc^16
Which expands to
16dc^16 + 8dc^8 + 4dc^4 + 2dc^2 + dc + d
and we can factor this into
d((4c^8+1)^2 + 4c^4 + 2c^2 + c)