The theory of critical hits

[Tengu]

Correct if I'm mistaken.

Against enemies without DR or with DR that you can pass, and without any effects that trigger on a critical hit, on the long run the critical hit range and multiplier of a weapon basically add a flat damage bonus:

x2/20: +5%
x2/19-20: +10%
x3/20: +10%
x2/18-20: +15%
x4/20: +15%

Improved Critical or a Keen weapon double the bonus.

[lesserarchangel]

That sounds right to me.

But what happens if you combine improved critical and a keen weapon?

[Anxe]

The bonus triples, duh. Two doubles make a triple.

[NEO|Phyte]

That sounds right to me.

But what happens if you combine improved critical and a keen weapon?

They don't stack, nothing happens.

[Iku Rex]

Tengu, you're right as long as a critical threat is always a hit. If it's not the threat range shrinks. For example, a falchion has an effective crit range of 19-20 if you need a 19 to hit your target.

[CharPixie]

A few things to consider:

Overkill is more likely with critical hits (especially the 4x style ones). So, wasted damage is more likely.

Critical hits have no effect outside of a small range of to hit numbers. So, statistically, they will produce far more outliers than the normal, predictable damage dice (or even the simple threshold that's the "to-hit").

Until a certain level, criticals can often mean instant down/death when at full hitpoints.

[martyboy74]

Also, don't forget that you have to confirm that critical. The numbers that you have only apply if you always confirm the critical (even on a natural 1). That's pretty unlikely.

To show the bonus, you need to know the attack bonus (which, in most cases, varies per attack), the armor class (varies when flatfooted, when touch attacks are in play, or when they have buffs or debuffs on them), and the threat range and multiplier. Also, fortification would wreak havoc with these.

[Lord Lorac Silvanos]

Tengu, you're right as long as a critical threat is always a hit. If it's not the threat range shrinks. For example, a falchion has an effective crit range of 19-20 if you need a 19 to hit your target.

... and the bonuses you have listed are actually a little lower, since you also need to confirm the critical threat.


EDIT: Ninja'ed

[Jasdoif]

And don't forget that bonus damage dice (like those from flaming weapons, holy weapons, sneak attack, etc.) aren't multiplied on a critical hit. Also, some weapon properties (like soulbreaker) are dependent on getting a critical hit, but not affected by the damage multiplier.

[Tengu]

Proportionally they are the same though, assuming that you confirm your critical 50% of the time both x2/19-20 and x3/20 add +5% damage, and x2/20 adds +2,5%.

The part Iku Rex mentioned is interesting though and I didn't put it under consideration earlier - however, if you can only hit the enemy on a natural 19, you only have 1% chance to actually crit, which is so small it's almost deniable.

Also, the damage bonus does not put some damage sources under consideration (sneak attack, energy weapons, et cetera), but everyone knows that I think.

EDIT: Ninja'ed!

[Iku Rex]

Also, don't forget that you have to confirm that critical. The numbers that you have only apply if you always confirm the critical (even on a natural 1). That's pretty unlikely.

No, as average damage on a hit they work as they are. The confirmation roll is included in the math.

On a hit you deal a minimum of [base] average damage. You get a chance to do an additional [base] damage [chance to crit]/[chance to hit] of the time. You'll confirm this [chance to hit] of the time. The neat thing is that [chance to hit] can be taken out of the equation.

Example: You have +19 to attack, your opponent has AC 30. You hit on 11-20 (50%). You threaten a crit on 15-20 (30%, keen falchion). As long as you've already hit, there's a 30/50 chance that the hit was a crit threat. And when you roll to confirm you've got a 50% chance to hit. Chance to deal double damage is (30/50)*50= 30%.

You can try it with other numbers. If chance to hit is 70% you get (30/70)*70 = 30% again. It's really quite tidy.

[Talya]

It's really quite tidy.

It does break down when your chance to hit is actually lower than the threat range of your weapon. (EG. You crit on a 15-20, but you need a natural 17 to hit.) At that point to maintain accuracy, you must treat the crit range as 17-20.

[Journey]

It's not a flat damage bonus, even on average, except for the simplest cases (no bonuses or penalties, same AC and same BAB, etc.). I posted this in one of the QA by the RAW threads when somebody asked a question like "which core weapon does the most damage on average?"

The question is essentially one of statistics, and the theory is actually quite messy. The formula, however, appears simple:

Basically, the average of the damage output of a given weapon after a number of trials will be:

<D> = h * <d>_base + c * h * <d>_crit
= h * (<d>_base + c*<d>_crit)

What the quantities are:
<d>: Average damage. For dice rolls, the average is (1+s)/2, where s is the number of sides (e.g. 1d8 has an average of (1+8)/2 = 4.5). The "base" value is the weapon's normal damage, and the "crit" value is the damage that is compiled during the critical part of the damage roll (in most cases, "crit" damage is just the crit modifier times the "base" damage).

h: The chance to hit the target, which depends on all bonuses (including BAB), the AC of the target and any penalties. For cases where only BAB and AC apply, the value of h is simple: (20 + BAB - AC)/20 (e.g. for a 1st level fighter with no bonuses against an unarmored opponent with no bonuses, this is (20 + 1 - 10)/20 = 11/20 = 0.55).

c: The critical threat chance, which is (21 - r)/20. The value "r" is the lowest possible value for which a threat can actually become a critical. This value is either the lowest number in the listed range (e.g. for a weapon with 19-20/x2, r = 19) or the lowest number for which the character scores a hit, if this is higher than r would be (e.g., for a 17-20 weapon, if a character can only possibly hit on an 18 or better, r = 18). This is, of course, if the DM doesn't rule that threats are negated when attacking enemies which can only be hit by values landing within the threat range anyway.

You can see that all but the simplest scenarios make this formula much more complicated. In principle one could construct a complete chart for any given set of expected AC, BAB, bonus, and penalty values for a given weapon. In practice this is a ridiculous undertaking, because even for basic weapons you have a list numbering the product of the number of AC, BAB, bonus, and penalties (in one point increments/decrements). For my fighter example, for example, calculating the chance over a range of 10 AC values (e.g. 11-20) and a range of 3 BAB, that's a list of 30 different expected damage numbers.

Essentially, any analysis of a weapon has to be done on a per-case basis, factoring in "reasonable" assumptions for expected modifiers to the various rolls, and the result will be a "best guess."

[Citizen Joe]

You don't need AC and BAB... all you need is net bonus. AC 20/+10 attack bonus is the same as AC 10/0 attack bonus. It is about a 40 point spread, from -19 to +19. Since a natural 1 always misses and a natural 20 always hits, net bonuses beyond those numbers won't change the statistics.

Also you don't need a value for damage, just use the 'average damage' as a variable. This generates a table about 40 lines long (net attack bonus) with about 10 columns (various critical ranges/multipliers). I think they are: 20/x2, 20/x3, 20/x4, 19-20/x2, 19-20/x3, 19-20/x4, 18-20/x2, 18-20/x3, 15-20/x2, 15-20/x3. So you'll have a bunch of fractional multipliers in a table that will dictate what portion of your average damage you would do each round. Plug in the numbers for each weapon you have to figure out how to optimize your combat vs. certain ACs.

I could see a warforged running these calculations, maybe... but for the typical adventurer, just use the weapon at hand.

[Ossian]

I think I have a question that migh have been solved by the math I read above. Unfortunately I am very scared by math, and I am not sure I understood the numbers correctly.

This is the situation: 1 player uses the elven thinblade, so dmg is 1d8 and threat range for criticals is 18-20.

What I know: if you roll a natural 20, you always hit AND threat a critical, no matter what the target AC is, even if it was 50.

What I do not know: if the d20 roll falls within the threat range, is this always a hit? AND is it also a critical?
This could be even worse if the player took "Improved Critical". Threat range becomes 15-20!!!

If anyone could put this in a plain and simple way, perhaps with an example, I'd appreciate!

Thanks a lot,

Ossian

[Jasdoif]

Only a natural 20 is an automatic hit. If your attack roll isn't a hit, you can't score a threat, regardless of what your threat range is.

[Matthew]

The below calculations should show you exactly how Critical Hits work from a basic math point of view:

+0 Damage Bonus

Spoiler

Show

[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 5.5 = 1,980) + (1 x 5.5 = 5.5) + (19 x 22 = 418)] = 2,403.5
1D12 20/x3 [(360 x 6.5 = 2,340) + (1 x 6.5 = 6.5) + (19 x 19.5 = 370.5)] = 2,717
1D12 19-20/x2 [(340 x 6.5 = 2,210) + (2 x 6.5 = 13) + (38 x 13 = 494)] = 2,717
1D10 18-20/x2 [(320 x 5.5 = 1,760) + (3 x 5.5 = 16.5) + (57 x 11 = 627)] = 2,403.5

2.
1D10 20/x4 [(360 x 5.5 = 1,980) + (1 x 5.5 = 5.5) + (19 x 22 = 418)] = 2,403.5
1D12 20/x3 [(360 x 6.5 = 2,340) + (1 x 6.5 = 6.5) + (19 x 19.5 = 370.5)] = 2,717
1D12 19-20/x2 [(340 x 6.5 = 2,210) + (2 x 6.5 = 13) + (38 x 13 = 494)] = 2,717
1D10 18-20/x2 [(320 x 5.5 = 1,760) + (3 x 5.5 = 16.5) + (57 x 11 = 627)] = 2,403.5

3.
1D10 20/x4 [(340 x 5.5 = 1,870) + (2 x 5.5 = 11) + (18 x 22 = 396)] = 2,277
1D12 20/x3 [(340 x 6.5 = 2,210) + (2 x 6.5 = 13) + (18 x 19.5 = 351)] = 2,574
1D12 19-20/x2 [(320 x 6.5 = 2,080) + (4 x 6.5 = 26) + (36 x 13 = 468)] = 2,574
1D10 18-20/x2 [(300 x 5.5 = 1,650) + (6 x 5.5 = 33) + (54 x 11 = 594)] = 2,277

4.
1D10 20/x4 [(320 x 5.5 = 1,760) + (3 x 5.5 = 16.5) + (17 x 22 = 374)] = 2,150.5
1D12 20/x3 [(320 x 6.5 = 2,080) + (3 x 6.5 = 19.5) + (17 x 19.5 = 331.5)] = 2,431
1D12 19-20/x2 [(300 x 6.5 = 1,950) + (6 x 6.5 = 39) + (34 x 13 = 442)] = 2,431
1D10 18-20/x2 [(280 x 5.5 = 1,540) + (9 x 5.5 = 49.5) + (51 x 11 = 561)] = 2,150.5

5.
1D10 20/x4 [(300 x 5.5 = 1,650) + (4 x 5.5 = 22) + (16 x 22 = 352)] = 2,024
1D12 20/x3 [(300 x 6.5 = 1,950) + (4 x 6.5 = 26) + (16 x 19.5 = 312)] = 2,288
1D12 19-20/x2 [(280 x 6.5 = 1,820) + (8 x 6.5 = 52) + (32 x 13 = 416)] = 2,288
1D10 18-20/x2 [(260 x 5.5 = 1,430) + (12 x 5.5 = 66) + (48 x 11 = 528)] = 2,024

6.
1D10 20/x4 [(280 x 5.5 = 1,540) + (5 x 5.5 = 27.5) + (15 x 22 = 330)] = 1,897.5
1D12 20/x3 [(280 x 6.5 = 1,820) + (5 x 6.5 = 32.5) + (15 x 19.5 = 292.5)] = 2,145
1D12 19-20/x2 [(260 x 6.5 = 1,690) + (10 x 6.5 = 65) + (30 x 13 = 390)] = 2,145
1D10 18-20/x2 [(240 x 5.5 = 1,320) + (15 x 5.5 = 82.5) + (45 x 11 = 495)] = 1,897.5

7.
1D10 20/x4 [(260 x 5.5 = 1,430) + (6 x 5.5 = 33) + (14 x 22 = 308)] = 1,771
1D12 20/x3 [(260 x 6.5 = 1,690) + (6 x 6.5 = 39) + (14 x 19.5 = 273)] = 2,002
1D12 19-20/x2 [(240 x 6.5 = 1,560) + (12 x 6.5 = 78) + (28 x 13 = 364)] = 2,002
1D10 18-20/x2 [(220 x 5.5 = 1,210) + (18 x 5.5 = 99) + (42 x 11 = 462)] = 1,771

8.
1D10 20/x4 [(240 x 5.5 = 1,320) + (7 x 5.5 = 38.5) + (13 x 22 = 286)] = 1644.5
1D12 20/x3 [(240 x 6.5 = 1,560) + (7 x 6.5 = 45.5) + (13 x 19.5 = 253.5)] = 1,859
1D12 19-20/x2 [(220 x 6.5 = 1,430) + (14 x 6.5 = 91) + (26 x 13 = 338)] = 1,859
1D10 18-20/x2 [(200 x 5.5 = 1,100) + (21 x 5.5 = 115.5) + (39 x 11 = 429)] = 1,644.5

9.
1D10 20/x4 [(220 x 5.5 = 1,210) + (8 x 5.5 = 44) + (12 x 22 = 264)] = 1,518
1D12 20/x3 [(220 x 6.5 = 1,430) + (8 x 6.5 = 52) + (12 x 19.5 = 234)] = 1,716
1D12 19-20/x2 [(200 x 6.5 = 1,300) + (16 x 6.5 = 104) + (24 x 13 = 312)] = 1,716
1D10 18-20/x2 [(180 x 5.5 = 990) + (24 x 5.5 = 132) + (36 x 11 = 396)] = 1,518

10.
1D10 20/x4 [(200 x 5.5 = 1,100) + (9 x 5.5 = 49.5) + (11 x 22 = 242)] = 1,391.5
1D12 20/x3 [(200 x 6.5 = 1,300) + (9 x 6.5 = 58.5) + (11 x 19.5 = 214.5)] = 1,573
1D12 19-20/x2 [(180 x 6.5 = 1,170) + (18 x 6.5 = 117) + (22 x 13 = 286)] = 1,573
1D10 18-20/x2 [(160 x 5.5 = 880) + (27 x 5.5 = 148.5) + (33 x 11 = 363)] = 1,391.5

11.
1D10 20/x4 [(180 x 5.5 = 990) + (10 x 5.5 = 55) + (10 x 22 = 220)] = 1,265
1D12 20/x3 [(180 x 6.5 = 1170) + (10 x 6.5 = 65) + (10 x 19.5 = 195)] = 1,430
1D12 19-20/x2 [(160 x 6.5 = 1040) + (20 x 6.5 = 130) + (20 x 13 = 260)] = 1,430
1D10 18-20/x2 [(140 x 5.5 = 770) + (30 x 5.5 = 165) + (30 x 11 = 330)] = 1,265

12.
1D10 20/x4 [(160 x 5.5 = 880) + (11 x 5.5 = 60.5) + (9 x 22 = 198)] = 1,138.5
1D12 20/x3 [(160 x 6.5 = 1,040) + (11 x 6.5 = 71.5) + (9 x 19.5 = 175.5)] = 1,287
1D12 19-20/x2 [(140 x 6.5 = 910) + (22 x 6.5 = 143) + (18 x 13 = 234)] = 1,287
1D10 18-20/x2 [(120 x 5.5 = 660) + (33 x 5.5 = 181.5) + (27 x 11 = 297)] = 1138.5

13.
1D10 20/x4 [(140 x 5.5 = 770) + (12 x 5.5 = 66) + (8 x 22 = 176)] = 1,012
1D12 20/x3 [(140 x 6.5 = 910) + (12 x 6.5 = 78) + (8 x 19.5 = 156)] = 1,144
1D12 19-20/x2 [(120 x 6.5 = 780) + (24 x 6.5 = 156) + (16 x 13 = 208)] = 1,144
1D10 18-20/x2 [(100 x 5.5 = 550) + (36 x 5.5 = 198) + (24 x 11 = 264)] = 1,012

14.
1D10 20/x4 [(120 x 5.5 = 660) + (13 x 5.5 = 71.5) + (7 x 22 = 154)] = 885.5
1D12 20/x3 [(120 x 6.5 = 780) + (13 x 6.5 = 84.5) + (7 x 19.5 = 136.5)] = 1,001
1D12 19-20/x2 [(100 x 6.5 = 650) + (26 x 6.5 = 169) + (14 x 13 = 182)] = 1,001
1D10 18-20/x2 [(80 x 5.5 = 440) + (39 x 5.5 = 214.5) + (21 x 11 = 231)] = 885.5

15.
1D10 20/x4 [(100 x 5.5 = 550) + (14 x 5.5 = 77) + (6 x 22 = 132)] = 759
1D12 20/x3 [(100 x 6.5 = 650) + (14 x 6.5 = 91) + (6 x 19.5 = 117)] = 858
1D12 19-20/x2 [(80 x 6.5 = 520) + (28 x 6.5 = 182) + (12 x 13 = 156)] = 858
1D10 18-20/x2 [(60 x 5.5 = 330) + (42 x 5.5 = 231) + (18 x 11 = 198)] = 759

16.
1D10 20/x4 [(80 x 5.5 = 440) + (15 x 5.5 = 82.5) + (5 x 22 = 110)] = 632.5
1D12 20/x3 [(80 x 6.5 = 520) + (15 x 6.5 = 97.5) + (5 x 19.5 = 97.5)] = 715
1D12 19-20/x2 [(60 x 6.5 = 390) + (30 x 6.5 = 195) + (10 x 13 = 130)] = 715
1D10 18-20/x2 [(40 x 5.5 = 220) + (45 x 5.5 = 247.5) + (15 x 11 = 165)] = 632.5

17.
1D10 20/x4 [(60 x 5.5 = 330) + (16 x 5.5 = 88) + (4 x 22 = 88)] = 506
1D12 20/x3 [(60 x 6.5 = 390) + (16 x 6.5 = 104) + (4 x 19.5 = 78)] = 572
1D12 19-20/x2 [(40 x 6.5 = 260) + (32 x 6.5 = 208) + (8 x 13 = 104)] = 572
1D10 18-20/x2 [(20 x 5.5 = 110) + (48 x 5.5 = 264) + (12 x 11 = 132)] = 506

18.
1D10 20/x4 [(40 x 5.5 = 220) + (17 x 5.5 = 93.5) + (3 x 22 = 66)] = 379.5
1D12 20/x3 [(40 x 6.5 = 260) + (17 x 6.5 = 110.5) + (3 x 19.5 = 58.5)] = 429
1D12 19-20/x2 [(20 x 6.5 = 130) + (34 x 6.5 = 221) + (6 x 13 = 78)] = 429
1D10 18-20/x2 [(0 x 5.5 = 0) + (51 x 5.5 = 280.5) + (9 x 11 = 99)] = 379.5

19.
1D10 20/x4 [(20 x 5.5 = 110) + (18 x 5.5 = 99) + (2 x 22 = 44)] = 253
1D12 20/x3 [(20 x 6.5 = 130) + (18 x 6.5 = 117) + (2 x 19.5 = 39)] = 286
1D12 19-20/x2 [(0 x 6.5 = 0) + (36 x 6.5 = 234) + (4 x 13 = 52)] = 286
1D10 18-20/x2 [(0 x 5.5 = 0) + (36 x 5.5 = 198) + (4 x 11 = 44)] = 242

20.
1D10 20/x4 [(0 x 5.5 = 0) + (19 x 5.5 = 104.5) + (1 x 22 = 22)] = 126.5
1D12 20/x3 [(0 x 6.5 = 0) + (19 x 6.5 = 123.5) + (1 x 19.5 = 19.5)] = 143
1D12 19-20/x2 [(0 x 6.5 = 0) + (19 x 6.5 = 123.5) + (1 x 13 = 13)] = 136.5
1D10 18-20/x2 [(0 x 5.5 = 0) + (19 x 5.5 = 104.5) + (1 x 11 = 11)] = 115.5

Average Damage = (126.5 x 209 =26,438.5) / (8,000) = 3.3048125
Average Damage = (143 x 209 = 29,887) / (8,000) = 3.735875
Average Damage = 136.5 + (143 x 208 =) / (8,000) = 3.7350625
Average Damage = 115.5 + 242 + (126.5 x 206 = ) / 8,000 = 3.3020625

+10 Damage Bonus

Spoiler

Show

[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 15.5 = 5580) + (0 x 15.5 = 0) + (20 x 62 = 1240)] = 6,820
1D12 20/x3 [(360 x 16.5 = 5940) + (0 x 16.5 = 0) + (20 x 49.5 = 990)] = 6,930
1D12 19-20/x2 [(340 x 16.5 = 5610) + (0 x 16.5 = 0) + (40 x 33 = 1320)] = 6,930
1D10 18-20/x2 [(320 x 15.5 = 4960) + (0 x 15.5 = 0) + (60 x 31 = 1860)] = 6,820

2.
1D10 20/x4 [(360 x 15.5 = 5580) + (1 x 15.5 = 15.5) + (19 x 62 = 1178)] = 6,773.5
1D12 20/x3 [(360 x 16.5 = 5940) + (1 x 16.5 = 16.5) + (19 x 49.5 = 940.5)] = 6,897
1D12 19-20/x2 [(340 x 16.5 = 5610) + (2 x 16.5 = 33) + (38 x 33 = 1254)] = 6,897
1D10 18-20/x2 [(320 x 15.5 = 4960) + (3 x 15.5 = 46.5) + (57 x 31 = 1767)] = 6,773.5

3.
1D10 20/x4 [(340 x 5.5 = ) + (2 x 5.5 = ) + (18 x 22 = )] =
1D12 20/x3 [(340 x 6.5 = ) + (2 x 6.5 = ) + (18 x 19.5 = )] =
1D12 19-20/x2 [(320 x 6.5 = ) + (4 x 6.5 = ) + (36 x 13 = )] =
1D10 18-20/x2 [(300 x 5.5 = ) + (6 x 5.5 = ) + (54 x 11 = )] =

4.
1D10 20/x4 [(320 x 5.5 = ) + (3 x 5.5 = ) + (17 x 22 = )] =
1D12 20/x3 [(320 x 6.5 = ) + (3 x 6.5 = ) + (17 x 19.5 = )] =
1D12 19-20/x2 [(300 x 6.5 = ) + (6 x 6.5 = ) + (34 x 13 = )] =
1D10 18-20/x2 [(280 x 5.5 = ) + (9 x 5.5 = ) + (51 x 11 = )] =

5.
1D10 20/x4 [(300 x 5.5 = ) + (4 x 5.5 = ) + (16 x 22 = )] =
1D12 20/x3 [(300 x 6.5 = ) + (4 x 6.5 = ) + (16 x 19.5 = )] =
1D12 19-20/x2 [(280 x 6.5 = ) + (8 x 6.5 = ) + (32 x 13 = )] =
1D10 18-20/x2 [(260 x 5.5 = ) + (12 x 5.5 = ) + (48 x 11 = )] =

6.
1D10 20/x4 [(280 x 15.5 = 4340) + (5 x 15.5 = 77.5) + (15 x 62 = 930)] = 5,347.5
1D12 20/x3 [(280 x 16.5 = 4620) + (5 x 16.5 = 82.5) + (15 x 49.5 = 742.5)] = 5,445
1D12 19-20/x2 [(260 x 16.5 = 4290) + (10 x 16.5 = 165) + (30 x 33 = 990)] = 5,445
1D10 18-20/x2 [(240 x 15.5 = 3720) + (15 x 15.5 = 232.5) + (45 x 31 = 1395)] = 5,347.5

7.
1D10 20/x4 [(260 x 5.5 = ) + (6 x 5.5 = ) + (14 x 22 = )] =
1D12 20/x3 [(260 x 6.5 = ) + (6 x 6.5 = ) + (14 x 19.5 = )] =
1D12 19-20/x2 [(240 x 6.5 = ) + (12 x 6.5 = ) + (28 x 13 = )] =
1D10 18-20/x2 [(220 x 5.5 = ) + (18 x 5.5 = ) + (42 x 11 = )] =

8.
1D10 20/x4 [(240 x 5.5 = ) + (7 x 5.5 = ) + (13 x 22 = )] =
1D12 20/x3 [(240 x 6.5 = ) + (7 x 6.5 = ) + (13 x 19.5 = )] =
1D12 19-20/x2 [(220 x 6.5 = ) + (14 x 6.5 = ) + (26 x 13 = )] =
1D10 18-20/x2 [(200 x 5.5 = ) + (21 x 5.5 = ) + (39 x 11 = )] =

9.
1D10 20/x4 [(220 x 5.5 = ) + (8 x 5.5 = ) + (12 x 22 = )] =
1D12 20/x3 [(220 x 6.5 = ) + (8 x 6.5 = ) + (12 x 19.5 = )] =
1D12 19-20/x2 [(200 x 6.5 = ) + (16 x 6.5 = ) + (24 x 13 = )] =
1D10 18-20/x2 [(180 x 5.5 = ) + (24 x 5.5 = ) + (36 x 11 = )] =

10.
1D10 20/x4 [(200 x 5.5 = ) + (9 x 5.5 = ) + (11 x 22 = )] =
1D12 20/x3 [(200 x 6.5 = ) + (9 x 6.5 = ) + (11 x 19.5 = )] =
1D12 19-20/x2 [(180 x 6.5 = ) + (18 x 6.5 = ) + (22 x 13 = )] =
1D10 18-20/x2 [(160 x 5.5 = ) + (27 x 5.5 = ) + (33 x 11 = )] =

11.

1D10 20/x4 [(180 x 15.5 = 2790) + (10 x 15.5 = 155) + (10 x 62 = 620)] = 3,565
1D12 20/x3 [(180 x 16.5 = 2970) + (10 x 16.5 = 165) + (10 x 49.5 = 495)] = 3,630
1D12 19-20/x2 [(160 x 16.5 = 2640) + (20 x 16.5 = 330) + (20 x 33 = 660)] = 3,630
1D10 18-20/x2 [(140 x 15.5 = 2170) + (30 x 15.5 = 465) + (30 x 31 = 930)] = 3,565

12.
1D10 20/x4 [(160 x 5.5 = ) + (11 x 5.5 = ) + (9 x 22 = )] =
1D12 20/x3 [(160 x 6.5 = ) + (11 x 6.5 = ) + (9 x 19.5 = )] =
1D12 19-20/x2 [(140 x 6.5 = ) + (22 x 6.5 = ) + (18 x 13 = )] =
1D10 18-20/x2 [(120 x 5.5 = ) + (33 x 5.5 = ) + (27 x 11 = )] =

13.
1D10 20/x4 [(140 x 5.5 = ) + (12 x 5.5 = ) + (8 x 22 = )] =
1D12 20/x3 [(140 x 6.5 = ) + (12 x 6.5 = ) + (8 x 19.5 = )] =
1D12 19-20/x2 [(120 x 6.5 = ) + (24 x 6.5 = ) + (16 x 13 = )] =
1D10 18-20/x2 [(100 x 5.5 = ) + (36 x 5.5 = ) + (24 x 11 = )] =

14.
1D10 20/x4 [(120 x 5.5 = ) + (13 x 5.5 = ) + (7 x 22 = )] =
1D12 20/x3 [(120 x 6.5 = ) + (13 x 6.5 = ) + (7 x 19.5 = )] =
1D12 19-20/x2 [(100 x 6.5 = ) + (26 x 6.5 = ) + (14 x 13 = )] =
1D10 18-20/x2 [(80 x 5.5 = ) + (39 x 5.5 = ) + (21 x 11 = )] =

15.
1D10 20/x4 [(100 x 5.5 = ) + (14 x 5.5 = ) + (6 x 22 = )] =
1D12 20/x3 [(100 x 6.5 = ) + (14 x 6.5 = ) + (6 x 19.5 = )] =
1D12 19-20/x2 [(80 x 6.5 = ) + (28 x 6.5 = ) + (12 x 13 = )] =
1D10 18-20/x2 [(60 x 5.5 = ) + (42 x 5.5 = ) + (18 x 11 = )] =

16.
1D10 20/x4 [(80 x 15.5 = 1240) + (15 x 15.5 = 232.5) + (5 x 62 = 310)] = 1,782.5
1D12 20/x3 [(80 x 16.5 = 1320) + (15 x 16.5 = 247.5) + (5 x 49.5 = 247.5)] = 1,815
1D12 19-20/x2 [(60 x 16.5 = 990) + (30 x 16.5 = 495) + (10 x 33 = 330)] = 1,815
1D10 18-20/x2 [(40 x 15.5 = 620) + (45 x 15.5 = 697.5) + (15 x 31 = 465)] = 1,782.5

17.
1D10 20/x4 [(60 x 5.5 = ) + (16 x 5.5 = ) + (4 x 22 = )] =
1D12 20/x3 [(60 x 6.5 = ) + (16 x 6.5 = ) + (4 x 19.5 = )] =
1D12 19-20/x2 [(40 x 6.5 = ) + (32 x 6.5 = ) + (8 x 13 = )] =
1D10 18-20/x2 [(20 x 5.5 = ) + (48 x 5.5 = ) + (12 x 11 = )] =

18.
1D10 20/x4 [(40 x 5.5 = ) + (17 x 5.5 = ) + (6 x 22 = )] =
1D12 20/x3 [(40 x 6.5 = ) + (17 x 6.5 = ) + (6 x 19.5 = )] =
1D12 19-20/x2 [(20 x 6.5 = ) + (34 x 6.5 = ) + (12 x 13 = )] =
1D10 18-20/x2 [(0 x 5.5 = ) + (51 x 5.5 = ) + (9 x 11 = )] =

19.
1D10 20/x4 [(20 x 5.5 = ) + (18 x 5.5 = ) + (2 x 22 = )] =
1D12 20/x3 [(20 x 6.5 = ) + (18 x 6.5 = ) + (2 x 19.5 = )] =
1D12 19-20/x2 [(0 x 6.5 = ) + (36 x 6.5 = ) + (4 x 13 = )] =
1D10 18-20/x2 [(0 x 5.5 = ) + (36 x 5.5 = ) + (4 x 11 = )] =

20.
1D10 20/x4 [(0 x 15.5 = 0) + (19 x 15.5 = 294.5) + (1 x 62 = 62)] = 356.5
1D12 20/x3 [(0 x 16.5 = 0) + (19 x 16.5 = 313.5) + (1 x 49.5 = 49.5)] = 363
1D12 19-20/x2 [(0 x 16.5 = 0) + (19 x 16.5 = 313.5) + (1 x 33 = 33)] = 346.5
1D10 18-20/x2 [(0 x 15.5 = 0) + (19 x 15.5 = 294.5) + (1 x 31 = 31)] = 325.5

(356.5 x 209 = ) / (8,000) =
(363 x 209 = ) / (8,000) =
346.5 + (363 x 208 = ) / (8,000) =
325.5 + X + (356.5 x 206 = ) / 8,000 =

 
+20 Damage Bonus

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[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 25.5 = 9180) + (0 x 25.5 = 0) + (20 x 102 = 2040)] = 11,220
1D12 20/x3 [(360 x 26.5 = 9540) + (0 x 26.5 = 0) + (20 x 79.5 = 1590)] = 11,130
1D12 19-20/x2 [(340 x 26.5 = 9010) + (0 x 26.5 = 0) + (40 x 53 = 2120)] = 11,130
1D10 18-20/x2 [(320 x 25.5 = 8160) + (0 x 25.5 = 0) + (60 x 51 = 3060)] = 11,220

2.
1D10 20/x4 [(360 x 25.5 = 9180) + (1 x 25.5 = 25.5) + (19 x 102 = 1938)] = 11,143.5
1D12 20/x3 [(360 x 26.5 = 9540) + (1 x 26.5 = 26.5) + (19 x 79.5 = 1510.5)] = 11,077
1D12 19-20/x2 [(340 x 26.5 = 9010) + (2 x 26.5 = 53) + (38 x 53 = 2014)] = 11,077
1D10 18-20/x2 [(320 x 25.5 = 8160) + (3 x 25.5 = 76.5) + (57 x 51 = 2907)] = 11,143.5

6.
1D10 20/x4 [(280 x 25.5 = 7140) + (5 x 25.5 = 127.5) + (15 x 102 = 1530)] = 8,797.5
1D12 20/x3 [(280 x 26.5 = 7420) + (5 x 26.5 = 132.5) + (15 x 79.5 = 1192.5)] = 8,745
1D12 19-20/x2 [(260 x 26.5 = 6890) + (10 x 26.5 = 265) + (30 x 53 = 1590)] = 8,745
1D10 18-20/x2 [(240 x 25.5 = 6120) + (15 x 25.5 = 382.5) + (45 x 51 = 2295)] = 8,797.5

11.
2D4 20/x4 [(180 x 25 = 4500) + (10 x 25 = 250) + (10 x 100 = 1000)] = 5,750
1D10 20/x4 [(180 x 25.5 = 4590) + (10 x 25.5 = 255) + (10 x 102 = 1020)] = 5,865
1D12 20/x3 [(180 x 26.5 = 4770) + (10 x 26.5 = 265) + (10 x 79.5 = 795)] = 5,830
1D12 19-20/x2 [(160 x 26.5 = 4240) + (20 x 26.5 = 530) + (20 x 53 = 1060)] = 5,830
1D10 18-20/x2 [(140 x 25.5 = 3570) + (30 x 25.5 = 765) + (30 x 51 = 1530)] = 5,865
2D4 18-20/x2 [(140 x 25 = 3500) + (30 x 25 = 750) + (30 x 50 = 1500)] = 5,750

16.
1D10 20/x4 [(80 x 25.5 = 2040) + (15 x 25.5 = 382.5) + (5 x 102 = 510)] = 2,932.5
1D12 20/x3 [(80 x 26.5 = 2120) + (15 x 26.5 = 397.5) + (5 x 79.5 = 397.5)] = 2,915
1D12 19-20/x2 [(60 x 26.5 = 1590) + (30 x 26.5 = 795) + (10 x 53 = 530)] = 2,915
1D10 18-20/x2 [(40 x 25.5 = 1020) + (45 x 25.5 = 1147.5) + (15 x 51 = 765)] = 2,932.5

20.
1D10 20/x4 [(0 x 25.5 = 0) + (19 x 25.5 = 484.5) + (1 x 102 = 102)] = 586.5
1D12 20/x3 [(0 x 26.5 = 0) + (19 x 26.5 = 503.5) + (1 x 79.5 = 79.5)] = 583
1D12 19-20/x2 [(0 x 26.5 = 0) + (19 x 26.5 = 503.5) + (1 x 53 = 53)] = 556.5
1D10 18-20/x2 [(0 x 25.5 = 0) + (19 x 25.5 = 484.5) + (1 x 51 = 51)] = 535.5

+30 Damage Bonus

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[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 35.5 = 12,780) + (0 x 35.5 = 0) + (20 x 142 = 2,840)] = 15,620
1D12 20/x3 [(360 x 36.5 = 13,140) + (0 x 36.5 = 0) + (20 x 109.5 = 2,190)] = 15,330
1D12 19-20/x2 [(340 x 36.5 = 12,410) + (0 x 36.5 = 0) + (40 x 73 = 2,920)] = 15,330
1D10 18-20/x2 [(320 x 35.5 = 11,360) + (0 x 35.5 = 0) + (60 x 71 = 4,260)] = 15,620

2.
1D10 20/x4 [(360 x 35.5 = 12,780) + (1 x 35.5 = 35.5) + (19 x 142 = 2,698)] = 15,513.5
1D12 20/x3 [(360 x 36.5 = 13,140) + (1 x 36.5 = 36.5) + (19 x 109.5 = 2,080.5)] = 15,257
1D12 19-20/x2 [(340 x 36.5 = 12,410) + (2 x 36.5 = 73) + (38 x 73 = 2,774)] = 15,257
1D10 18-20/x2 [(320 x 35.5 = 11,360) + (3 x 35.5 = 106.5) + (57 x 71 = 4,047)] = 15,513.5

6.
1D10 20/x4 [(280 x 35.5 = 9,940) + (5 x 35.5 = 177.5) + (15 x 142 = 2,130)] = 12,247.5
1D12 20/x3 [(280 x 36.5 = 10,220) + (5 x 36.5 = 182.5) + (15 x 109.5 = 1,642.5)] = 12,045
1D12 19-20/x2 [(260 x 36.5 = 9,490) + (10 x 36.5 = 365) + (30 x 73 = 2,190)] = 12,045
1D10 18-20/x2 [(240 x 35.5 = 8,520) + (15 x 35.5 = 532.5) + (45 x 71 = 3,195)] = 12,247.5

11.
2D4 20/x4 [(180 x 35 = 6,300) + (10 x 35 = 350) + (10 x 140 = 1,400)] = 8,050
1D10 20/x4 [(180 x 35.5 = 6,390) + (10 x 35.5 = 355) + (10 x 142 = 1,420)] = 8,165
1D12 20/x3 [(180 x 36.5 = 6,570) + (10 x 36.5 = 365) + (10 x 109.5 = 1,095)] = 8030
1D12 19-20/x2 [(160 x 36.5 = 5,840) + (20 x 36.5 = 730) + (20 x 73 = 1,460)] = 8030
1D10 18-20/x2 [(140 x 35.5 = 4,970) + (30 x 35.5 = 1065) + (30 x 71 = 2,130)] = 8,165
2D4 18-20/x2 [(140 x 35 = 4,900) + (30 x 35 = 1050) + (30 x 70 = 2,100)] = 8,050

16.
1D10 20/x4 [(80 x 35.5 = 2,840) + (15 x 35.5 = 532.5) + (5 x 142 = 710)] = 4,082.5
1D12 20/x3 [(80 x 36.5 = 2,920) + (15 x 36.5 = 547.5) + (5 x 109.5 = 547.5)] = 4,015
1D12 19-20/x2 [(60 x 36.5 = 2,190) + (30 x 36.5 = 1095) + (10 x 73 = 730)] = 4,015
1D10 18-20/x2 [(40 x 35.5 = 1,420) + (45 x 35.5 = 1,597.5) + (15 x 71 = 1,065)] = 4,082.5

20.
1D10 20/x4 [(0 x 35.5 = 0) + (19 x 35.5 = 674.5) + (1 x 142 = 142)] = 816.5
1D12 20/x3 [(0 x 36.5 = 0) + (19 x 36.5 = 693.5) + (1 x 109.5 = 109.5)] = 803
1D12 19-20/x2 [(0 x 36.5 = 0) + (19 x 36.5 = 693.5) + (1 x 73 = 73)] = 766.5
1D10 18-20/x2 [(0 x 35.5 = 0) + (19 x 35.5 = 674.5) + (1 x 71 = 71)] = 745.5


(816.5 x 209 = ) / (8,000) = 21.3310625
(803 x 209 = ) / (8,000) = 20.978375
766.5 + (803 x 208 = ) / (8,000) = 20.9738125
745.5 + X + (816.5 x 206 = ) / 8,000 =

+40 Damage Bonus

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[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 45.5 = 16,380) + (0 x 45.5 = 0) + (20 x 182 = 3,640)] = 20,020
1D12 20/x3 [(360 x 46.5 = 16,740) + (0 x 46.5 = 0) + (20 x 139.5 = 2,790)] = 19,530
1D12 19-20/x2 [(340 x 46.5 = 15,810) + (0 x 46.5 = 0) + (40 x 93 = 3,720)] = 19,530
1D10 18-20/x2 [(320 x 45.5 = 14,560) + (0 x 45.5 = 0) + (60 x 91 = 5,460)] = 20,020

2.
1D10 20/x4 [(360 x 45.5 = 16,380) + (1 x 45.5 = 45.5) + (19 x 182 = 3,458)] = 19,883.5
1D12 20/x3 [(360 x 46.5 = 16,740) + (1 x 46.5 = 46.5) + (19 x 139.5 = 2,650.5)] = 19,437
1D12 19-20/x2 [(340 x 46.5 = 15,810) + (2 x 46.5 = 93) + (38 x 93 = 3,534)] = 19,437
1D10 18-20/x2 [(320 x 45.5 = 14,560) + (3 x 45.5 = 136.5) + (57 x 91 = 5,187)] = 19,883.5

6.
1D10 20/x4 [(280 x 45.5 = 12,740) + (5 x 45.5 = 227.5) + (15 x 182 = 2,730)] = 15,697.5
1D12 20/x3 [(280 x 46.5 = 13,020) + (5 x 46.5 = 232.5) + (15 x 139.5 = 2.092.5)] = 15,345
1D12 19-20/x2 [(260 x 46.5 = 12,090) + (10 x 46.5 = 465) + (30 x 93 = 2,790)] = 15,345
1D10 18-20/x2 [(240 x 45.5 = 10,920) + (15 x 45.5 = 682.5) + (45 x 91 = 4,095)] = 15,697.5

11.
1D10 20/x4 [(180 x 45.5 = 8,190) + (10 x 45.5 = 455) + (10 x 182 = 1,820)] = 10,465
1D12 20/x3 [(180 x 46.5 = 8,370) + (10 x 46.5 = 465) + (10 x 139.5 = 1,395)] = 10,230
1D12 19-20/x2 [(160 x 46.5 = 7,440) + (20 x 46.5 = 930) + (20 x 93 = 1,860)] = 10,230
1D10 18-20/x2 [(140 x 45.5 = 6,370) + (30 x 45.5 = 1,365) + (30 x 91 = 2,730)] = 10,465

16.
1D10 20/x4 [(80 x 45.5 = 3,640) + (15 x 45.5 = 682.5) + (5 x 182 = 910)] = 5,232.5
1D12 20/x3 [(80 x 46.5 = 3,720) + (15 x 46.5 = 697.5) + (5 x 139.5 = 697.5)] = 5,115
1D12 19-20/x2 [(60 x 46.5 = 2,790) + (30 x 46.5 = 1,395) + (10 x 93 = 930)] = 5,115
1D10 18-20/x2 [(40 x 45.5 = 1,820) + (45 x 45.5 = 2047.5) + (15 x 91 = 1,365)] = 5,232.5

20.
1D10 20/x4 [(0 x 45.5 = 0) + (19 x 45.5 = 864.5) + (1 x 182 = 182)] = 1,046.5
1D12 20/x3 [(0 x 46.5 = 0) + (19 x 46.5 = 883.5) + (1 x 139.5 = 139.5)] = 1,023
1D12 19-20/x2 [(0 x 46.5 = 0) + (19 x 46.5 = 883.5) + (1 x 93 = 93)] = 976.5
1D10 18-20/x2 [(0 x 45.5 = 0) + (19 x 45.5 = 864.5) + (1 x 91 = 91)] = 955.5

+50 Damage Bonus

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[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 55.5 = 19,980) + (0 x 55.5 = 0) + (20 x 222 = 4,440)] = 24,420
1D12 20/x3 [(360 x 56.5 = 20,340) + (0 x 56.5 = 0) + (20 x 169.5 = 3,390)] = 23,730
1D12 19-20/x2 [(340 x 56.5 = 19,210) + (0 x 56.5 = 0) + (40 x 113 = 4,520)] = 23,730
1D10 18-20/x2 [(320 x 55.5 = 17,760) + (0 x 55.5 = 0) + (60 x 111 = 6,660)] = 24,420

2.
1D10 20/x4 [(360 x 55.5 = 19,980) + (1 x 55.5 = 55.5) + (19 x 222 = 4,218)] = 24,253.5
1D12 20/x3 [(360 x 56.5 = 20,340) + (1 x 56.5 = 56.5) + (19 x 169.5 = 3,220.5)] = 23,617
1D12 19-20/x2 [(340 x 56.5 = 19,210) + (2 x 56.5 = 113) + (38 x 113 = 4,294)] = 23,617
1D10 18-20/x2 [(320 x 55.5 = 17,760) + (3 x 55.5 = 166.5) + (57 x 111 = 6,327)] = 24,253.5

6.
1D10 20/x4 [(280 x 55.5 = 15,540) + (5 x 55.5 = 277.5) + (15 x 222 = 3,330)] = 19,147.5
1D12 20/x3 [(280 x 56.5 = 15,820) + (5 x 56.5 = 282.5) + (15 x 169.5 = 2,542.5)] = 18,645
1D12 19-20/x2 [(260 x 56.5 = 14,690) + (10 x 56.5 = 565.0) + (30 x 113 = 3,390)] = 18,645
1D10 18-20/x2 [(240 x 55.5 = 13,320) + (15 x 55.5 = 832.5) + (45 x 111 = 4,995)] = 19,147.5

11.
1D10 20/x4 [(180 x 55.5 = 9,990) + (10 x 55.5 = 555) + (10 x 222 = 2,220)] = 12,765
1D12 20/x3 [(180 x 56.5 = 10,170) + (10 x 56.5 = 565) + (10 x 169.5 = 1,695)] = 12,430
1D12 19-20/x2 [(160 x 56.5 = 9,040) + (20 x 56.5 = 1,130) + (20 x 113 = 2,260)] = 12,430
1D10 18-20/x2 [(140 x 55.5 = 7,770) + (30 x 55.5 = 1,665) + (30 x 111 = 3,330)] = 12.765

16.
1D10 20/x4 [(80 x 55.5 = 4,440) + (15 x 55.5 = 832.5) + (5 x 222 = 1,110)] = 6,382.5
1D12 20/x3 [(80 x 56.5 = 4,520) + (15 x 56.5 = 847.5) + (5 x 169.5 = 847.5)] = 6,215
1D12 19-20/x2 [(60 x 56.5 = 3,390) + (30 x 56.5 = 1,695) + (10 x 113 = 1,130)] = 6,215
1D10 18-20/x2 [(40 x 55.5 = 2,220) + (45 x 55.5 = 2,497.5) + (15 x 111 = 1,665)] = 6,382.5

20.
1D10 20/x4 [(0 x 55.5 = 0) + (19 x 55.5 = 2004.5) + (1 x 222 = 222)] = 2,226.5
1D12 20/x3 [(0 x 56.5 = 0) + (19 x 56.5 = 2023.5) + (1 x 169.5 = 169.5)] = 2,193
1D12 19-20/x2 [(0 x 56.5 = 0) + (19 x 56.5 = 2023.5) + (1 x 113 = 113)] = 2,136.5
1D10 18-20/x2 [(0 x 55.5 = 0) + (19 x 55.5 = 2004.5) + (1 x 111 = 111)] = 2,115.5

+100 Damage Bonus

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[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
1D10 20/x4 [(360 x 105.5 = 37,980) + (0 x 25.5 = 0) + (20 x 422 = 8,440)] = 46,420
1D12 20/x3 [(360 x 106.5 = 38,340) + (0 x 26.5 = 0) + (20 x 319.5 = 6,390)] = 44,730
1D12 19-20/x2 [(340 x 106.5 = 36,210) + (0 x 26.5 = 0) + (40 x 213 = 8,520)] = 44,730
1D10 18-20/x2 [(320 x 105.5 = 33,760) + (0 x 25.5 = 0) + (60 x 211 = 12,660)] = 46,420

2.
1D10 20/x4 [(360 x 105.5 = 37,980) + (1 x 105.5 = 105.5) + (19 x 422 = 8,018)] = 46,103.5
1D12 20/x3 [(360 x 106.5 = 38,340) + (1 x 106.5 = 106.5) + (19 x 319.5 = 6,070.5)] = 44,517
1D12 19-20/x2 [(340 x 106.5 = 36,210) + (2 x 106.5 = 213) + (38 x 213 = 8,094)] = 44,517
1D10 18-20/x2 [(320 x 105.5 = 33,760) + (3 x 105.5 = 316.5) + (57 x 211 = 12,027)] = 46,103.5

6.
1D10 20/x4 [(280 x 105.5 = 29,540) + (5 x 105.5 = 527.5) + (15 x 422 = 6,330)] = 36,397.5
1D12 20/x3 [(280 x 106.5 = 29,820) + (5 x 106.5 = 532.5) + (15 x 319.5 = 4,792.5)] = 35,145
1D12 19-20/x2 [(260 x 106.5 = 27,690) + (10 x 106.5 = 1,065) + (30 x 213 = 6,390)] = 35,145
1D10 18-20/x2 [(240 x 105.5 = 25,320) + (15 x 105.5 = 1,582.5) + (45 x 211 = 9,495)] = 36,397.5

11.
1D10 20/x4 [(180 x 105.5 = 18,990) + (10 x 105.5 = 1,055) + (10 x 422 = 4,220)] = 24,265
1D12 20/x3 [(180 x 106.5 = 19,170) + (10 x 106.5 = 1,065) + (10 x 319.5 = 3,195)] = 23,430
1D12 19-20/x2 [(160 x 106.5 = 17,040) + (20 x 106.5 = 2,130) + (20 x 213 = 4,260)] = 23,430
1D10 18-20/x2 [(140 x 105.5 = 14,770) + (30 x 105.5 = 3,165) + (30 x 211 = 6,330)] = 24,265

16.
1D10 20/x4 [(80 x 105.5 = 8,440) + (15 x 105.5 = 1,582.5) + (5 x 422 = 2,110)] = 12,132.5
1D12 20/x3 [(80 x 106.5 = 8,520) + (15 x 106.5 = 1,597.5) + (5 x 319.5 = 1,597.5)] = 11,715
1D12 19-20/x2 [(60 x 106.5 = 6,390) + (30 x 106.5 = 3,195) + (10 x 213 = 2,130)] = 11,715
1D10 18-20/x2 [(40 x 105.5 = 4,220) + (45 x 105.5 = 4,747.5) + (15 x 211 = 3,165)] = 12,132.5

20.
1D10 20/x4 [(0 x 105.5 = 0) + (19 x 105.5 = 2004.5) + (1 x 422 = 422)] = 2,426.5
1D12 20/x3 [(0 x 106.5 = 0) + (19 x 106.5 = 2023.5) + (1 x 319.5 = 319.5)] = 2,343
1D12 19-20/x2 [(0 x 106.5 = 0) + (19 x 106.5 = 2023.5) + (1 x 213 = 213)] = 2,236.5
1D10 18-20/x2 [(0 x 105.5 = 0) + (19 x 105.5 = 2004.5) + (1 x 211 = 211)] = 2,215.5

63.3923125
61.210875
61.1975625

+0 Damage Bonus

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[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
2D4 20/x4 [(360 x 5 = 1,800) + (1 x 5 = 5) + (19 x 20 = 380)] = 2,185
2D6 20/x3 [(360 x 7 = 2,520) + (1 x 7 = 7) + (19 x 21 = 399)] = 2,926
2D6 19-20/x2 [(340 x 7 = 2,380) + (2 x 7 = 14) + (38 x 14 = 532)] = 2,926
2D4 18-20/x2 [(320 x 5 = 1,600) + (3 x 5 = 15) + (57 x 10 = 570)] = 2,185

2.
2D4 20/x4 [(360 x 5 = 1,800) + (1 x 5 = 5) + (19 x 20 = 380)] = 2,185
2D6 20/x3 [(360 x 7 = 2,520) + (1 x 7 = 7) + (19 x 21 = 399)] = 2,926
2D6 19-20/x2 [(340 x 7 = 2,380) + (2 x 7 = 14) + (38 x 14 = 532)] = 2,926
2D4 18-20/x2 [(320 x 5 = 1,600) + (3 x 5 = 15) + (57 x 10 = 570)] = 2,185

3.
2D4 20/x4 [(340 x 5 = 1,700) + (2 x 5 = 10) + (18 x 20 = 360)] = 2,070
2D6 20/x3 [(340 x 7 = 2,380) + (2 x 7 = 14) + (18 x 21 = 378)] = 2,772
2D6 19-20/x2 [(320 x 7 = 2,240) + (4 x 7 = 28) + (36 x 14 = 504)] = 2,772
2D4 18-20/x2 [(300 x 5 = 1,500) + (6 x 5 = 30) + (54 x 10 = 540)] = 2,070

4.
2D4 20/x4 [(320 x 5 = 1,600) + (3 x 5 = 15) + (17 x 20 = 340)] = 1,955
2D6 20/x3 [(320 x 7 = 2,240) + (3 x 7 = 21) + (17 x 21 = 357)] = 2,618
2D6 19-20/x2 [(300 x 7 = 2,100) + (6 x 7 = 42) + (34 x 14 = 476)] = 2,618
2D4 18-20/x2 [(280 x 5 = 1,400) + (9 x 5 = 45) + (51 x 10 = 510)] = 1,955

5.
2D4 20/x4 [(300 x 5 = 1,500) + (4 x 5 = 20) + (16 x 20 = 320)] = 1,840
2D6 20/x3 [(300 x 7 = 2,100) + (4 x 7 = 28) + (16 x 21 = 336)] = 2,464
2D6 19-20/x2 [(280 x 7 = 1,960) + (8 x 7 = 56) + (32 x 14 = 448)] = 2,464
2D4 18-20/x2 [(260 x 5 = 1,300) + (12 x 5 = 60) + (48 x 10 = 480)] = 1,840

6.
2D4 20/x4 [(280 x 5 = 1,400) + (5 x 5 = 25) + (15 x 20 = 300)] = 1,725
2D6 20/x3 [(280 x 7 = 1,960) + (5 x 7 = 35) + (15 x 21 = 315)] = 2,310
2D6 19-20/x2 [(260 x 7 = 1,820) + (10 x 7 = 70) + (30 x 14 = 420)] = 2,310
2D4 18-20/x2 [(240 x 5 = 1,200) + (15 x 5 = 75) + (45 x 10 = 450)] = 1,725

7.
2D4 20/x4 [(260 x 5 = 1,300) + (6 x 5 = 30) + (14 x 20 = 280)] = 1,610
2D6 20/x3 [(260 x 7 = 1,820) + (6 x 7 = 42) + (14 x 21 = 294)] = 2,156
2D6 19-20/x2 [(240 x 7 = 1,680) + (12 x 7 = 84) + (28 x 14 = 392)] = 2,156
2D4 18-20/x2 [(220 x 5 = 1,100) + (18 x 5 = 90) + (42 x 10 = 420)] = 1,610

8.
2D4 20/x4 [(240 x 5 = 1,200) + (7 x 5 = 35) + (13 x 20 = 260)] = 1,495
2D6 20/x3 [(240 x 7 = 1,680) + (7 x 7 = 49) + (13 x 21 = 273)] = 2,002
2D6 19-20/x2 [(220 x 7 = 1,540) + (14 x 7 = 98) + (26 x 14 = 364)] = 2,002
2D4 18-20/x2 [(200 x 5 = 1,000) + (21 x 5 = 105) + (39 x 10 = 390)] = 1,495

9.
2D4 20/x4 [(220 x 5 = 1,100) + (8 x 5 = 40) + (12 x 20 = 240)] = 1,380
2D6 20/x3 [(220 x 7 = 1,540) + (8 x 7 = 56) + (12 x 21 = 252)] = 1,848
2D6 19-20/x2 [(200 x 7 = 1,400) + (16 x 7 = 112) + (24 x 14 = 336)] = 1,848
2D4 18-20/x2 [(180 x 5 = 900) + (24 x 5 = 120) + (36 x 10 = 360)] = 1,380

10.
2D4 20/x4 [(200 x 5 = 1,000) + (9 x 5 = 45) + (11 x 20 = 220)] = 1,265
2D6 20/x3 [(200 x 7 = 1,400) + (9 x 7 = 63) + (11 x 21 = 231)] = 1,694
2D6 19-20/x2 [(180 x 7 = 1,260) + (18 x 7 = 126) + (22 x 14 = 308)] = 1,694
2D4 18-20/x2 [(160 x 5 = 800) + (27 x 5 = 135) + (33 x 10 = 330)] = 1,265

11.
2D4 20/x4 [(180 x 5 = 900) + (10 x 5 = 50) + (10 x 20 = 200)] = 1,150
2D6 20/x3 [(180 x 7 = 1,260) + (10 x 7 = 70) + (10 x 21 = 210)] = 1,540
2D6 19-20/x2 [(160 x 7 = 1,120) + (20 x 7 = 140) + (20 x 14 = 280)] = 1,540
2D4 18-20/x2 [(140 x 5 = 700) + (30 x 5 = 150) + (30 x 10 = 300)] = 1,150

12.
2D4 20/x4 [(160 x 5 = 800) + (11 x 5 = 55) + (9 x 20 = 180)] = 1,035
2D6 20/x3 [(160 x 7 = 1,120) + (11 x 7 = 77) + (9 x 21 = 189)] = 1,386
2D6 19-20/x2 [(140 x 7 = 980) + (22 x 7 = 154) + (18 x 14 = 252)] = 1,386
2D4 18-20/x2 [(120 x 5 = 600) + (33 x 5 = 165) + (27 x 10 = 270)] = 1,035

13.
2D4 20/x4 [(140 x 5 = 700) + (12 x 5 = 60) + (8 x 20 = 160)] = 920
2D6 20/x3 [(140 x 7 = 980) + (12 x 7 = 84) + (8 x 21 = 168)] = 1,232
2D6 19-20/x2 [(120 x 7 = 840) + (24 x 7 = 168) + (16 x 14 = 224)] = 1,232
2D4 18-20/x2 [(100 x 5 = 500) + (36 x 5 = 180) + (24 x 10 = 240)] = 920

14.
2D4 20/x4 [(120 x 5 = 600) + (13 x 5 = 65) + (7 x 20 = 140)] = 805
2D6 20/x3 [(120 x 7 = 840) + (13 x 7 = 91) + (7 x 21 = 147)] = 1,078
2D6 19-20/x2 [(100 x 7 = 700) + (26 x 7 = 182) + (14 x 14 = 196)] = 1,078
2D4 18-20/x2 [(80 x 5 = 400) + (39 x 5 = 195) + (21 x 10 = 210)] = 805

15.
2D4 20/x4 [(100 x 5 = 500) + (14 x 5 = 70) + (6 x 20 = 120)] = 690
2D6 20/x3 [(100 x 7 = 700) + (14 x 7 = 98) + (6 x 21 = 126)] = 924
2D6 19-20/x2 [(80 x 7 = 560) + (28 x 7 = 196) + (12 x 14 = 168)] = 924
2D4 18-20/x2 [(60 x 5 = 300) + (42 x 5 = 210) + (18 x 10 = 180)] = 690

16.
2D4 20/x4 [(80 x 5 = 400) + (15 x 5 = 75) + (5 x 20 = 100)] = 575
2D6 20/x3 [(80 x 7 = 560) + (15 x 7 = 105) + (5 x 21 = 105)] = 770
2D6 19-20/x2 [(60 x 7 = 420) + (30 x 7 = 210) + (10 x 14 = 140)] = 770
2D4 18-20/x2 [(40 x 5 = 200) + (45 x 5 = 225) + (15 x 10 = 150)] = 575

17.
2D4 20/x4 [(60 x 5 = 300) + (16 x 5 = 80) + (4 x 20 = 80)] = 460
2D6 20/x3 [(60 x 7 = 420) + (16 x 7 = 112) + (4 x 21 = 84)] = 616
2D6 19-20/x2 [(40 x 7 = 280) + (32 x 7 = 224) + (8 x 14 = 112)] = 616
2D4 18-20/x2 [(20 x 5 = 100) + (48 x 5 = 240) + (12 x 10 = 120)] = 460

18.
2D4 20/x4 [(40 x 5 = 200) + (17 x 5 = 85) + (3 x 20 = 60)] = 345
2D6 20/x3 [(40 x 7 = 280) + (17 x 7 = 119) + (3 x 21 = 63)] = 462
2D6 19-20/x2 [(20 x 7 = 140) + (34 x 7 = 238) + (6 x 14 = 84)] = 462
2D4 18-20/x2 [(0 x 5 = 0) + (51 x 5 = 255) + (9 x 10 = 90)] = 345

19.
2D4 20/x4 [(20 x 5 = 100) + (18 x 5 = 90) + (2 x 20 = 40)] = 230
2D6 20/x3 [(20 x 7 = 140) + (18 x 7 = 126) + (2 x 21 = 42)] = 308
2D6 19-20/x2 [(0 x 7 = 0) + (36 x 7 = 252) + (4 x 14 = 56)] = 308
2D4 18-20/x2 [(0 x 5 = 0) + (36 x 5 = 180) + (4 x 10 = 40)] = 220

20.
2D4 20/x4 [(0 x 5 = 0) + (19 x 5 = 95) + (1 x 20 = 20)] = 115
2D6 20/x3 [(0 x 7 = 0) + (19 x 7 = 133) + (1 x 21 = 21)] = 154
2D6 19-20/x2 [(0 x 7 = 0) + (19 x 7 = 133) + (1 x 14 = 14)] = 147
2D4 18-20/x2 [(0 x 5 = 0) + (19 x 5 = 95) + (1 x 10 = 10)] = 105

(209 x 115 = 24,035) / (20 x 400 = 8,000) = 3.004375
(209 x 154 = 32,186) / (20 x 400 = 8,000) = 4.02325

+10 Damage Bonus

Spoiler

Show

[Normal Number of Hits x Average Damage] + [Unconfirmed Critical Hits x Average Damage] + [Confirmed Critical Hits x (Average Damage x Multiplier)

1.
2D4 20/x4 [(360 x 5 = ) + (1 x 5 = ) + (19 x 20 = )] =
2D6 20/x3 [(360 x 7 = ) + (1 x 7 = ) + (19 x 21 = )] =
2D6 19-20/x2 [(340 x 7 = ) + (2 x 7 = ) + (38 x 14 = )] =
2D4 18-20/x2 [(320 x 5 = ) + (3 x 5 = ) + (57 x 10 = )] =

2.
2D4 20/x4 [(360 x 5 = ) + (1 x 5 = ) + (19 x 20 = )] =
2D6 20/x3 [(360 x 7 = ) + (1 x 7 = ) + (19 x 21 = )] =
2D6 19-20/x2 [(340 x 7 = ) + (2 x 7 = ) + (38 x 14 = )] =
2D4 18-20/x2 [(320 x 5 = ) + (3 x 5 = ) + (57 x 10 = )] =

3.
2D4 20/x4 [(340 x 5 = ) + (2 x 5 = ) + (18 x 20 = )] =
2D6 20/x3 [(340 x 7 = ) + (2 x 7 = ) + (18 x 21 = )] =
2D6 19-20/x2 [(320 x 7 = ) + (4 x 7 = ) + (36 x 14 = )] =
2D4 18-20/x2 [(300 x 5 = ) + (6 x 5 = ) + (54 x 10 = )] =

4.
2D4 20/x4 [(320 x 5 = ) + (3 x 5 = ) + (17 x 20 = )] =
2D6 20/x3 [(320 x 7 = ) + (3 x 7 = ) + (17 x 21 = )] =
2D6 19-20/x2 [(300 x 7 = ) + (6 x 7 = ) + (34 x 14 = )] =
2D4 18-20/x2 [(280 x 5 = ) + (9 x 5 = ) + (51 x 10 = )] =

5.
2D4 20/x4 [(300 x 5 = ) + (4 x 5 = ) + (16 x 20 = )] =
2D6 20/x3 [(300 x 7 = ) + (4 x 7 = ) + (16 x 21 = )] =
2D6 19-20/x2 [(280 x 7 = ) + (8 x 7 = ) + (32 x 14 = )] =
2D4 18-20/x2 [(260 x 5 = ) + (12 x 5 = ) + (48 x 10 = )] =

6.
2D4 20/x4 [(280 x 5 = ) + (5 x 5 = ) + (15 x 20 = )] =
2D6 20/x3 [(280 x 7 = ) + (5 x 7 = ) + (15 x 21 = )] =
2D6 19-20/x2 [(260 x 7 = ) + (10 x 7 = ) + (30 x 14 = )] =
2D4 18-20/x2 [(240 x 5 = ) + (15 x 5 = ) + (45 x 10 = )] =

7.
2D4 20/x4 [(260 x 5 = ) + (6 x 5 = ) + (14 x 20 = )] =
2D6 20/x3 [(260 x 7 = ) + (6 x 7 = ) + (14 x 21 = )] =
2D6 19-20/x2 [(240 x 7 = ) + (12 x 7 = ) + (28 x 14 = )] =
2D4 18-20/x2 [(220 x 5 = ) + (18 x 5 = ) + (42 x 10 = )] =

8.
2D4 20/x4 [(240 x 5 = ) + (7 x 5 = ) + (13 x 20 = )] =
2D6 20/x3 [(240 x 7 = ) + (7 x 7 = ) + (13 x 21 = )] =
2D6 19-20/x2 [(220 x 7 = ) + (14 x 7 = ) + (26 x 14 = )] =
2D4 18-20/x2 [(200 x 5 = ) + (21 x 5 = ) + (39 x 10 = )] =

9.
2D4 20/x4 [(220 x 5 = ) + (8 x 5 = ) + (12 x 20 = )] =
2D6 20/x3 [(220 x 7 = ) + (8 x 7 = ) + (12 x 21 = )] =
2D6 19-20/x2 [(200 x 7 = ) + (16 x 7 = ) + (24 x 14 = )] =
2D4 18-20/x2 [(180 x 5 = ) + (24 x 5 = ) + (36 x 10 = )] =

10.
2D4 20/x4 [(200 x 5 = ) + (9 x 5 = ) + (11 x 20 = )] =
2D6 20/x3 [(200 x 7 = ) + (9 x 7 = ) + (11 x 21 = )] =
2D6 19-20/x2 [(180 x 7 = ) + (18 x 7 = ) + (22 x 14 = )] =
2D4 18-20/x2 [(160 x 5 = ) + (27 x 5 = ) + (33 x 10 = )] =

11.
2D4 20/x4 [(180 x 5 = ) + (10 x 5 = ) + (10 x 20 = )] =
2D6 20/x3 [(180 x 7 = ) + (10 x 7 = ) + (10 x 21 = )] =
2D6 19-20/x2 [(160 x 7 = ) + (20 x 7 = ) + (20 x 14 = )] =
2D4 18-20/x2 [(140 x 5 = ) + (30 x 5 = ) + (30 x 10 = )] =

12.
2D4 20/x4 [(160 x 5 = ) + (11 x 5 = ) + (9 x 20 = )] =
2D6 20/x3 [(160 x 7 = ) + (11 x 7 = ) + (9 x 21 = )] =
2D6 19-20/x2 [(140 x 7 = ) + (22 x 7 = ) + (18 x 14 = )] =
2D4 18-20/x2 [(120 x 5 = ) + (33 x 5 = ) + (27 x 10 = )] =

13.
2D4 20/x4 [(140 x 5 = ) + (12 x 5 = ) + (8 x 20 = )] =
2D6 20/x3 [(140 x 7 = ) + (12 x 7 = ) + (8 x 21 = )] =
2D6 19-20/x2 [(120 x 7 = ) + (24 x 7 = ) + (16 x 14 = )] =
2D4 18-20/x2 [(100 x 5 = ) + (36 x 5 = ) + (24 x 10 = )] =

14.
2D4 20/x4 [(120 x 5 = ) + (13 x 5 = ) + (7 x 20 = )] =
2D6 20/x3 [(120 x 7 = ) + (13 x 7 = ) + (7 x 21 = )] =
2D6 19-20/x2 [(100 x 7 = ) + (26 x 7 = ) + (14 x 14 = )] =
2D4 18-20/x2 [(80 x 5 = ) + (39 x 5 = ) + (21 x 10 = )] =

15.
2D4 20/x4 [(100 x 5 = ) + (14 x 5 = ) + (6 x 20 = )] =
2D6 20/x3 [(100 x 7 = ) + (14 x 7 = ) + (6 x 21 = )] =
2D6 19-20/x2 [(80 x 7 = ) + (28 x 7 = ) + (12 x 14 = )] =
2D4 18-20/x2 [(60 x 5 = ) + (42 x 5 = ) + (18 x 10 = )] =

16.
2D4 20/x4 [(80 x 5 = ) + (15 x 5 = ) + (5 x 20 = )] =
2D6 20/x3 [(80 x 7 = ) + (15 x 7 = ) + (5 x 21 = )] =
2D6 19-20/x2 [(60 x 7 = ) + (30 x 7 = ) + (10 x 14 = )] =
2D4 18-20/x2 [(40 x 5 = ) + (45 x 5 = ) + (15 x 10 = )] =

17.
2D4 20/x4 [(60 x 5 = ) + (16 x 5 = ) + (4 x 20 = )] =
2D6 20/x3 [(60 x 7 = ) + (16 x 7 = ) + (4 x 21 = )] =
2D6 19-20/x2 [(40 x 7 = ) + (32 x 7 = ) + (8 x 14 = )] =
2D4 18-20/x2 [(20 x 5 = ) + (48 x 5 = ) + (12 x 10 = )] =

18.
2D4 20/x4 [(40 x 5 = ) + (17 x 5 = ) + (3 x 20 = )] =
2D6 20/x3 [(40 x 7 = ) + (17 x 7 = ) + (3 x 21 = )] =
2D6 19-20/x2 [(20 x 7 = ) + (34 x 7 = ) + (6 x 14 = )] =
2D4 18-20/x2 [(0 x 5 = 0) + (51 x 5 = ) + (9 x 10 = )] =

19.
2D4 20/x4 [(20 x 5 = ) + (18 x 5 = ) + (2 x 20 = )] =
2D6 20/x3 [(20 x 7 = ) + (18 x 7 = ) + (2 x 21 = )] =
2D6 19-20/x2 [(0 x 7 = 0) + (36 x 7 = ) + (4 x 14 = )] =
2D4 18-20/x2 [(0 x 5 = 0) + (36 x 5 = ) + (4 x 10 = )] =

20.
2D4 20/x4 [(0 x 5 = 0) + (19 x 5 = ) + (1 x 20 = )] =
2D6 20/x3 [(0 x 7 = 0) + (19 x 7 = ) + (1 x 21 = )] =
2D6 19-20/x2 [(0 x 7 = 0) + (19 x 7 = ) + (1 x 14 = )] =
2D4 18-20/x2 [(0 x 5 = 0) + (19 x 5 = ) + (1 x 10 = )] =

It should just confirm what Journey was saying.

[Zaggab]

A lot of numbers

Please tell me that you cheated on that post and used some kind of computer program to type that out automagically. Otherwise, I'm scared...

[Matthew]

Be scared, I did as much as I could in my head with a word document before resorting to a calculator to double check... it was a good mental excercise

To be fair, I did it a long while ago and the formula for finding the next iteration is actually pretty simple and consistant. This is probably the third time I have had cause to post it here.

[Yakk]

Yep. A way to think about this is "crit pips".

19x2 has "2 crit pips" -- on a 19 or 20, you threaten an extra x1 damage.
18x3 has "3 crit pips".
20x3 has "2 crit pips".

etc.

The advantage of broad crits is that they boost variance less than narrow criticals, and in a game where players have an advantage over their opponents, lower variance is good. The advantage of narrow crits is that they work even when you can barely hit the target.

Each crit pip is worth +5% to the non-crit damage that gets multiplied by critical multipliers.

The impact of either keen or imp. crits is doubling the number of crit pips. Mechanically, an ability that boosted crit multiplier as follows:
x2 -> x3
x3 -> x5
x4 -> x7
would have the same impact on average damage as keen/imp crit. One would have to be careful to avoid letting it stack with keen/imp crit.

...

A boost from 2 to 3 crit pips usually comes at the cost of 1 to 2 less average swing damage. Assuming keen/imp. crit, that 1 pip is worth +10% damage.

For this to be a wash, you need to do somewhere between 10 and 20 average damage per swing (depending on the weapon).

Going from: non-keen low-crit weapon with +2 damage
to keen high-crit weapon:
(X-2)*1.3 = (X)*1.1
.2X = 2.6
X = 13
which is pretty low, but doesn't take into account the cost of getting imp. crit or keen. Feats and +1 enchantments aren't free!

If the keen/imp crit is "worth" +3 damage
(X-5)*1.3 = X*(1.1)
.2 X = 6.5
X = 32.5

Getting keen all by itself is worth it at:
(X-3)*1.2 = X*1.1
.1X = 3.6
X = 36

Now, a more generous analysis (longsword->scimitar) and (a feat/+1 enchantment is worth 2 damage):
(X-3)*1.3 = X*1.1
.2 X = 3.9
X = 19.5

Lastly, not using imp. crit/keen, when is it worth swapping to a high-crit weapon?
2 damage cost for 1 pip:
(X-2)*1.15 = X*1.1
.05 X = 2.3
X = 46

1 damage cost for 1 pip:
(X-1)*1.15 = X*1.1
0.05 X = 1.15
X = 23

Conclusion:
Your average damage has to be above 20 for it to be worth it, on average, to go for imp. crit/keen effects. Two hand weapons (where you typically give up 2 damage) require an average damage of 30 or higher for it to be worth it.

Generally it is best to get both a crit-monkey weapon (with 3 pips) at the same time as you get keen/imp. crit.

In effect, imp. crit/keen sucks at low levels, as do crit monkey weapons. At high levels, they are worth it.

In my opinion, this is sup-par and an indication of mechanics that need tuning.

Fun, eh?