Attila's math desk, now open for business!

[Animastryfe]

I've got an attack bonus of two, and I rolled a two on the die, could you tell me what my actual attack roll is?

In all seriousness though, this could be a neat resource. I don't have any math questions right now, but no one in my group is actually all that good at math, so we get stumped pretty easily. Now we can just check this thread.

If you are able to form your question well, then try also using wolframalpha.com.

[AttilaTheGeek]

If you are able to form your question well, then try also using wolframalpha.com.

Wolfram is fantastic. It even does calculus for you.

[TuggyNE]

Instead of searching using your name click private messages and one of the menu choices allows you to pull up a list of subscriptions.

True, but that only works if you have it set to auto-subscribe to all threads you post in or start, which isn't the default. (And it's not the way I have it anymore either; I post in way too many threads for my inbox to survive that. ) It's also quite useless for searching on someone else's name.

[Story]

Sadly I didn't have it subscribed.

[Chilingsworth]

Ok, this might be an annoying request, but:

I just did some (very) basic analysis of the characters offered for a PbP I've applied to (just maximum, minimum, and average values for a variety of scores.) Others among the group would like a more through analysis of the data. Might you be able to provide a proper statistical analysis?

The table of links to the relevant character sheets

The first table I posted (concerning ability modifiers)

The second table I posted (concerning AC's, Save, and Attack bonuses)

[TypoNinja]

You had it right- you just multiply. For example, the odds of flipping heads, then tails, then rolling a natural 20 are (1/2)*(1/2)*(1/20), or 1/80.



I'm not sure of the question. Are you asking if there's a better way to spend gold than on Chaotic Outsider Bane?

Not necessarily the gold, but yes. That's my long term plan on a weapon, I'm quite happy with everything except the chaotic outsider bane, but it seems to be the best option left.

My DM has a MIC but I don't so my ability to just skim the book looking for goodies is limited, though I can happily use any material from it if I'm merely pointed at it.

So two 25%'s back to back would be .25x.25 for .0625 or 6.2%? I guess I expected another step.

Given that that is multiplication, it should work for any base % and any number of iterations yes?

[zimmerwald1915]

I apologize for this off-topic non sequitur, but I had to share that I first read the title as "Attila's Death Mask".

[TuggyNE]

I apologize for this off-topic non sequitur, but I had to share that I first read the title as "Attila's Death Mask".

Someone should homebrew one of those.

[Story]

I tried adding a signature, but it doesn't seem to apply retroactively.

[strider24seven]

Wolfram is fantastic. It even does calculus for you.

Amen to Mathematica products.

Now I pose some simple, straightforward questions:

1) I wish to accumulate a large amount of damage on a creature- let us say G-64, using darts. I may non-magical, standard PHB darts of any size, up to Colossal.

Assume the target's AC is 0 and has Regeneration 40 and DR 1/-.
Assume I have no bonus to damage (except strength bonus, obviously), a to-hit of +20, and I have 2 attacks per round.

Special: I can take 2 more attacks than the previous round if I take a cumulative -2 penalty on all of my attacks in the current round (so 2 attacks = -0 penalty, 4 attacks = -2 penalty, 6 attacks = -4 penalty, etc). I may never decrease my number of attacks in this manner, but may choose to instead reset the tempo to 2 attacks per round.

Note: Keep in mind that I must be able to lift the darts in order to throw them.

What is the shortest amount of time in which the given creature accumulates an average of G-64 damage? What is the smallest number of darts that I would have to use in that time? What is the smallest strength score I would require?

2) How much wood could a woodchuck chuck in a given round, given that a woodchuck could chuck wood?

[Story]

No matter what you do, it's going to take far more than the age of the universe. Grahm's Number is infinite for all practical purposes. Even an exponential growth rate won't help you.

Anyway, the best bet is to increase your attacks as fast as possible and eat the 95% miss rate. This will at least get you quadratic growth. Dart size, strength bonus, regeneration, and damage reduction are all irrelevant, since they only affect the time up to a constant multiple.

[TypoNinja]

2) How much wood could a woodchuck chuck in a given round, given that a woodchuck could chuck wood?

A woodchuck would chuck as much wood as a woodchuck could chuck, if a woodchuck could chuck wood.

[warmachine]

A character who has Second Chance and a second attack due to +6 BAB should re-roll a missed first attack, rather than try a second attack, as the second attack is at relative -5. However, having a third attack due to +11 or better BAB may allow more hits instead. Assuming no other feats, under what conditions should a character try a second and third attack, rather than re-roll the first?

[Greenish]

Assuming you don't have to confirm your critical hits, how much damage do you have to do with each attack for Keen to be better than Flaming (etc.)?

[yougi]

Imagine an ability that allows you to reroll your attack roll, but force you to keep the second result: how much of a bonus is that equivalent to? (wow, I suck at this syntax thing...) What about if you could roll twice and keep the best roll?

Also, is there a formula to calculate how to optimize your Power Attack in 3.5? I mean, I know there's a way to calculate it, but I mean a way that would be simple enough to do the math at the table?

[AttilaTheGeek]

A character who has Second Chance and a second attack due to +6 BAB should re-roll a missed first attack, rather than try a second attack, as the second attack is at relative -5. However, having a third attack due to +11 or better BAB may allow more hits instead. Assuming no other feats, under what conditions should a character try a second and third attack, rather than re-roll the first?

Second Chance with three iterative attacks is useful if rolling a 6 (on the d20) on your first attack would hit. It makes sense because two attacks at a lower bonus are better only if you're likely to hit with both of them. I did the math out on paper, and I'm kind of rushed at the moment, so I'll post it later.

Assuming you don't have to confirm your critical hits, how much damage do you have to do with each attack for Keen to be better than Flaming (etc.)?

If you don't have to confirm crits, Keen becomes much better and competitive. I'll see if I can get you an answer tonight.

Imagine an ability that allows you to reroll your attack roll, but force you to keep the second result: how much of a bonus is that equivalent to? (wow, I suck at this syntax thing...) What about if you could roll twice and keep the best roll?

Also, is there a formula to calculate how to optimize your Power Attack in 3.5? I mean, I know there's a way to calculate it, but I mean a way that would be simple enough to do the math at the table?

For the second question, I remember writing a formula about a year ago; I'll see if I can dig it up. I'll work on the first question tonight or tomorrow morning.

[AttilaTheGeek]

Assuming you don't have to confirm your critical hits, how much damage do you have to do with each attack for Keen to be better than Flaming (etc.)?

I think I'll just put all the maths in a spoiler from now on so that if you're looking for an answer, you can just find it quickly.

Spoiler

Show

It (of course) depends on the crit range of the weapon. But we start with an inequality:

1/20 * (Crit multipliers added) * (Weapon damage) > 3.5

By "crit multipliers", I mean the number of possible extra weapon damage terms added. For example, an 18-20/x2 crit range adds three extra weapon damage components (on an 18, 19, or 20), as does a 20-20/x4 range (on 20, times three). In other words, the crit multipliers of the range is equal to (21 minus the minimum needed to crit) times (the crit multiplier (x2, x3, or x4) -1), because a crit of x1 isn't a crit at all.

From there, we just divide 3.5 by the number of crit multipliers and multiply by 20 to solve for weapon damage.

{table=head]Crit range|Crit multipliers|Damage
20-20/x2|1|70
19-20/x2|2|35
18-20/x2|3|23.3
20-20/x3|2|35
19-20/x3|4|17.5
20-20/x4|3|23.3[/table]

Keen is better than Flaming (or another +1d6 energy enhancement) if your weapon damage is greater than the number in the table and you don't have to confirm crits.

[AttilaTheGeek]

Also, is there a formula to calculate how to optimize your Power Attack in 3.5? I mean, I know there's a way to calculate it, but I mean a way that would be simple enough to do the math at the table?

I can't find any flaws in the logic or the math, and the formula seems right, but I think I remember coming up with something different before. Math spoiler is here. Warning: calculus involved.

Spoiler

Show

I began with three variables: how much you power attack for, P, how much weapon damage you do on a hit, W, and the natural d20 roll you'd need to hit the target, A. (If A is small, then you're likely to hit, because you need to roll above that small number).

For a two-handed weapon, I defined a function f of W, A, and P and rearranged for P.

f (W, A, P) = (1/20)*(21-A-P)*(W+2P)
= (1/20)* [21W-AW-WP+42P-2AP-2P^2]
= (1/20)* [(-2)P^2 + (42-W-2A)P + (21W-AW)]

Now, since we're trying to solve the value of P that maximizes f, we take the partial derivative of f with respect to P and set it to 0. It's a maximum because the function doesn't have a minimum, and we don't have to worry about saddle points because we're holding W and A constant.

∂f/∂p = (1/20)* [ -4P + 42 - W - 2A] = 0

And from here, we solve for P in terms of W and A. The 1/20 multiplies out because the other side is 0.

42-W-2A = 4P
P = (42-W-2A)/4

For a 2H weapon, power attack = (42 - weapon damage - two times d20 roll needed to hit) divided by 4. For example, if you need to roll at least an 8 on your d20 to hit, and you hit for 20 damage, then (42-20-8)/4 = 3.5, which rounds down to 3. It makes intuitive sense to an optimizer, because if you hit for a million, you don't want to risk a 1/20 chance of missing for only 2 extra damage, and it would actually be better to reduce damage for an increased chance to hit, so the formula outputs a negative number. But if you know you're going to hit anyway, but not for much, then power attack adds a lot.

For a 1H weapon, the math is similar.

Spoiler

Show

f (W, A, P) = (1/20)*(21-A-P)*(W+P)
= (1/20)* [21W-AW-PW+21P-AP-P^2]
= (1/20)* [(-1)P^2 + (21-W-A)P + (21W-AW)]
∂f/∂p = (1/20)*[-2P+21-W-A]=0
2P=21-W-A
P=(21-W-A)2

For a 1H weapon, power attack = (21 - weapon damage - d20 roll needed to hit) divided by 2.

Edit: I found my error in the 2H calculations. the formula is now " - two times d20 roll needed to hit", as opposed to " - d20 roll needed to hit".

[yougi]

I can't find any flaws in the logic or the math, and the formula seems right, but I think I remember coming up with something different before. Math spoiler is here. Warning: calculus involved.

Spoiler

Show

I began with three variables: how much you power attack for, P, how much weapon damage you do on a hit, W, and the natural d20 roll you'd need to hit the target, A. (If A is small, then you're likely to hit, because you need to roll above that small number).

For a two-handed weapon, I defined a function f of W, A, and P and rearranged for P.

f (W, A, P) = (1/20)*(21-A-P)*(W+2P)
= (1/20)* [21W-AW-WP+42P-AP-2P^2]
= (1/20)* [(-2)P^2 + (42-W-A)P + (21W-AW)]

Now, since we're trying to solve the value of P that maximizes f, we take the partial derivative of f with respect to P and set it to 0. It's a maximum because the function doesn't have a minimum, and we don't have to worry about saddle points because we're holding W and A constant.

∂f/∂p = (1/20)* [ -4P + 42 - W - A] = 0

And from here, we solve for P in terms of W and A. The 1/20 multiplies out because the other side is 0.

42-W-A = 4P
P = (42-W-A)/4

For a 2H weapon, power attack = (42 - weapon damage - d20 roll needed to hit) divided by 4. For example, if you need to roll at least an 8 on your d20 to hit, and you hit for 20 damage, then (42-20-8)/4 = 3.5, which rounds down to 3. It makes intuitive sense to an optimizer, because if you hit for a million, you don't want to risk a 1/20 chance of missing for only 2 extra damage, and it would actually be better to reduce damage for an increased chance to hit, so the formula outputs a negative number. But if you know you're going to hit anyway, but not for much, then power attack adds a lot.

While I did not understand the math behind it at all, I believe this did not account for critical hits. Could that be calculated?

[Greenish]

{table=head]Crit range|Crit multipliers|Damage
20-20/x2|1|70
19-20/x2|2|35
18-20/x2|3|23.3
20-20/x3|2|35
19-20/x3|4|17.5
20-20/x4|3|23.3[/table]

Keen is better than Flaming (or another +1d6 energy enhancement) if your weapon damage is greater than the number in the table and you don't have to confirm crits.

Cheers!

While I did not understand the math behind it at all, I believe this did not account for critical hits. Could that be calculated?

In addition, what would "reverse PA" (sacrifice damage for accuracy) formula look like? Say, with +1 to hit for each -3 to damage? (Why yes, that just happens to coincide with Legend's Precise Strike. )

[AttilaTheGeek]

While I did not understand the math behind it at all, I believe this did not account for critical hits. Could that be calculated?

Exception: Extra damage over and above a weapon’s normal damage, such as that dealt by a sneak attack or the special ability of a flaming sword, is not multiplied when you score a critical hit.

Would this not apply to Power Attack as well?

In addition, what would "reverse PA" (sacrifice damage for accuracy) formula look like? Say, with +1 to hit for each -3 to damage?

It's exactly the same. If the formula is negative, then using reverse PA is better; if it's positive, then it's more optimal to power attack. However, I do think you have a typo there- according to power attack's description, each -1 to hit only gives +2 to damage with a two-handed weapon.

[Greenish]

Would this not apply to Power Attack as well?

It's generally interpreted to refer to effect that add extra dice of damage, not to fixed damage.

It's exactly the same. If the formula is negative, then using reverse PA is better; if it's positive, then it's more optimal to power attack. However, I do think you have a typo there- according to power attack's description, each -1 to hit only gives +2 to damage with a two-handed weapon.

Not a typo. I was hoping for something relevant to Legend's Precise Strike (since it's not like such an ability even exists in 3.5 or PF).

[thethird]

For a 2H weapon, power attack = (42 - weapon damage - two times d20 roll needed to hit) divided by 4.

So 42 isn't just the meaning of life.

[AttilaTheGeek]

So 42 isn't just the meaning of life.

Actually, I found a small error in that formula. It's "minus two times d20 roll needed to hit", if you could add that into the quote? I don't want people to get confused.

[yougi]

It's generally interpreted to refer to effect that add extra dice of damage, not to fixed damage.

Pretty much that.

And aside from crits, I don't think your calculations included iterative attacks, right? (wow, is that getting complicated or what?)

And with a 1H, would it be the same formula, but divided by 8?

[AttilaTheGeek]

It's generally interpreted to refer to effect that add extra dice of damage, not to fixed damage.

Ahh. Well, then.

*sound of crunching numbers*

*crunch crunch crunch*

Edit to add:

Pretty much that.

And aside from crits, I don't think your calculations included iterative attacks, right? (wow, is that getting complicated or what?)

And with a 1H, would it be the same formula, but divided by 8?

That's what I initially expected for 1H weapons, but actually the numbers are slightly different. I edited that formula in.

I think I'll now need five or six different variables: P for how much you power attack for, W for weapon damage, A for d20 roll needed to hit (the A comes from AC), I for number of iterative attacks, and C for damage added by crits. Instead of editing into that old post, I'll just make a new one. And possibly K for how much power attack is added in- K=1 for 1H weapons, K=2 for 2H weapons, and K=3 for Greenish's Precise Strike.

[thethird]

And then you can make a revenant blade frenzied berserker and check that for the lulz.

[AttilaTheGeek]

And then you can make a revenant blade frenzied berserker and check that for the lulz.

*is a pathfinder player*
*is confused*

[Greenish]

And then you can make a revenant blade frenzied berserker and check that for the lulz.

At that point, you'd have Shock Trooper, and the math becomes very simple: always PA for full. Always.

[Edit]: Revenant Blade isn't really relevant to the math, but Frenzied Berserker is a PrC that improves your PA ratio. Among other things.

[thethird]

A revenant blade can power attack with a double scimitar (a double weapon used for TWF) as if it was a THW. A frenzied berserker gains +4 damage for every -1 penalty to attack when power attacking.

Revenant blade is from Player's guide to Eberron pg 142. Frenzied Berserker is from Complete Warrior pg. 34