Attila's math desk, now open for business!

[dascarletm]

I've a big excel sheet that I use to calculate the impact of the malus on power attack for mean damage depending on feats, bonus, extra dice...
I'm still not satisfied with the layout but when it is I'll share with you guys.

For the question now, is it possible to find an analytic function to find the best power attack value knowing that:
- by seeing the values for the [-20;20] interval, the function isn't monotonous and may have local max (obviously other than the global max)
- the base function isn't continuous, so I can't derive it to get the mean damage

What I look for takes as input (total attack value without power attack - expected AC) and gives back the best power attack value: PA(3) = 8.21

Ah, yes I have a few of these. I made a large reference table, that automatically highlights the Highest PA value for each AC value.

[AttilaTheGeek]

Ah, yes I have a few of these. I made a large reference table, that automatically highlights the Highest PA value for each AC value.

But damage also depends on crit chance, crit confirmation chance, and crit multiplier.

[dascarletm]

But damage also depends on crit chance, crit confirmation chance, and crit multiplier.

Yes, I have one for each character that is PA dependent.

I took a simple, yet probably wrong route.

I just treated it as though all possible outcomes would happen, added the damage, then divided by the total. For each crittable # you need to calculate in 20 possibilities rolling (1-20) and divide by that. To the side of the reference table you can just have each of those crit related numbers (multiplier, etc.) referenced by the mother formula with $'s. Example A$5$ since you don't want that number to change when you pull the formula across the area.

[HurinTheCursed]

I guess it's rather similar to dascarletm's

So far, it works but it's rather dirty:
- by knowing the BAB, I get the number of attacks
- I add as input all modifiers for hitting, damaging, feats, criticals... (I added a combo box where you can pick previously entered weapons like +1 holy longsword)
- I calculate probability to hit for each couple base roll to hit / PA malus (you still have to guess the enemy's AC)
- I calculate the mean damage for one attack for that AC
- I get the full attack damage from iterating the single attack damage for a normal sequence (no snap kick, celerity...)
- The user has to find the optimal value by finding the colored cell in the line for the chosen base roll needed or by reading an histogram

It doesn't work against not critable opponents, some variable dice (holy, flaming...) when weapons don't work as usual.

[dascarletm]

I guess it's rather similar to dascarletm's

So far, it works but it's rather dirty:
- by knowing the BAB, I get the number of attacks
- I add as input all modifiers for hitting, damaging, feats, criticals... (I added a combo box where you can pick previously entered weapons like +1 holy longsword)
- I calculate probability to hit for each couple base roll to hit / PA malus (you still have to guess the enemy's AC)
- I calculate the mean damage for one attack for that AC
- I get the full attack damage from iterating the single attack damage for a normal sequence (no snap kick, celerity...)
- The user has to find the optimal value by finding the colored cell in the line for the chosen base roll needed or by reading an histogram

It doesn't work against not critable opponents, some variable dice (holy, flaming...) when weapons don't work as usual.

If you have all of your crit information available you can just set the crit multiplier to 1 to see on non-crittables. I also am thinking of putting in 2 base damage values, critable and non. You could then adjust those as needed

[AugustNights]

I have a math request, if I may.

I am struggling to ascertain which would be superior; 1d12+1d8+4, or Roll (1d12+1d8) and distinctly separate (1d12+1d8) and take the superior outcome.

[TuggyNE]

I have a math request, if I may.

I am struggling to ascertain which would be superior; 1d12+1d8+4, or Roll (1d12+1d8) and distinctly separate (1d12+1d8) and take the superior outcome.

You can implement that in AnyDice as a one-liner: `output 1d12+1d8+4 > [highest of 1d12+1d8 and 1d12+1d8]`

Or a two-liner:

output 1d12+1d8+4
output [highest of 1d12+1d8 and 1d12+1d8]

Either way, the first one is better; it has a higher average, a higher minimum, and a higher maximum. It's also tidier.

[AugustNights]

Thanks, tuggyne.

[Phelix-Mu]

Hi Attila! Fan of the thread, here, and confirmed lifelong mathematician (if a bit rusty).

Here's my question, and it might be a bit raw, so if you need to refine it or simplify it to make the numbers easier to figure, go for it. I'm interested to see how you model it. If you already solved this in one of the pages I skipped, a link or page number would be great.

So, here's the premise:
A 20th level shaper psion and his 17th level psion(telepath) cohort.

Assume infinite wealth and unlimited experience points to spend; neither are limiting factors.

The Question:
How many iterations of psychic chirurgery, Expanded Knowledge, and psychic reformation would be needed for them to teach each other all of the powers in Expanded Psionics and Complete Psionics (excepting the 9th level powers on the other discipline lists...though there may be a trick to net those, too)?

A total cost of experience points is key here. A time amount would be great too, but once you've figured the iterations, it's not hard to get these other number. I'm obviously looking to optimize the trick, so as few iterations as possible in the ideal pattern would be great.

(If you have a quick number for the total published # of psionic powers in 3.5, use that).

Good luck. If there's a reason that I haven't sussed out why this doesn't work, please tell me.

[AttilaTheGeek]

Hi Attila! Fan of the thread, here, and confirmed lifelong mathematician (if a bit rusty).

Thanks!

The Question:
How many iterations of psychic chirurgery, Expanded Knowledge, and psychic reformation would be needed for them to teach each other all of the powers in Expanded Psionics and Complete Psionics (excepting the 9th level powers on the other discipline lists...though there may be a trick to net those, too)?

Okay, a few clarifications.

Can you link or quote all three powers?

Also, how many powers are there that are applicable?

[TuggyNE]

Can you link or quote all three powers?

Expanded Knowledge is actually a feat. The other two are in XPH.

[AttilaTheGeek]

Expanded Knowledge is actually a feat. The other two are in XPH.

Thanks

I can't crunch numbers right now because I'm at work, but I'll do it tonight probably, if not tomorrow.

[dascarletm]

I have a question.

How do you calculate in "dropping one" to finding theoretical averages or probabilities to dice rolls?

Examples:

1. I'm rolling 2d10 10 times and dropping the lowest 2d10 total.
2. I'm rolling for stats, 4d6 and dropping the lowest d6 on each roll.
3. I'm rolling 2 sets of 3d6 and taking the highest set.

[Phelix-Mu]

Thanks!



Okay, a few clarifications.

Can you link or quote all three powers?

Also, how many powers are there that are applicable?

Not sure what you mean by "applicable." The goal is just to get all psionic powers onto the powers known list of a psion.

The two psions know a combined 67 powers known as part of their class levels, 36 for the shaper, 31 for the telepath.

As I mentioned, pre-epic, it should be possible for them to learn all of the powers that qualify for Expanded Knowledge and psychic chirurgery. Basically, they miss out on only a handful of the 9th level powers on the other discipline lists (I believe this is true from the language in Expanded Knowledge). There may well be a trick to get these as well, but it's not apparent to me at the moment.

I guess the first step is that the telepath uses psychic chirurgery to teach psychic chirurgery to the shaper. Then they proceed to learn all of each others powers.

Feel free to assume that there is no overlap in their initial powers known; the shaper knows none of the telepath's powers, and vice versa.

Good luck. Post again if you need further clarification. And thanks for the assist, TuggyNE.

[TuggyNE]

I have a question.

How do you calculate in "dropping one" to finding theoretical averages or probabilities to dice rolls?

Examples:

1. I'm rolling 2d10 10 times and dropping the lowest 2d10 total.
2. I'm rolling for stats, 4d6 and dropping the lowest d6 on each roll.
3. I'm rolling 2 sets of 3d6 and taking the highest set.

The cheap way is just to AnyDice it; the first one is considerably more complicated, and I'm not sure how to work it out without running into maximum execution time problems, but the other two are respectively "output [highest 3 of 4d6]" and "output [highest of 3d6 and 3d6]".

Edit:

And thanks for the assist, TuggyNE.

No problem!

[RFLS]

Okay, I have an absolutely massive one for you; it may in fact require its own thread. Can you do a numerical analysis that demonstrates why, exactly, the Monk is so terrible in combat, compared to, say, a Barbarian or a Paladin? Specifically, the following:

• Analysis of investment required to match heavy armor AC.
• Analysis of investment required to match damage dealt by a charger (not an Ubercharger, which verges into T.O.).
• Analysis of investment required to match the to-hit capabilities of the same charger.
• Analysis of the investment required to match the Trip/Disarm/Sunder/Grapple capabilities of a standard Barbarian.
• And, finally, one I'm sure will actually go in the Monk's favor (against the Barbarian, at least): Analysis of investment required to match saves.

For the purposes of this post, "investment" is treated as any sort of non-refundable resource (GP, skills, feats). Assume straight-classed, Core-only characters at, say, level 12; this doesn't need to be any more complicated

[unseenmage]

Would some kind math-minded person be interested in solving this little riddle for me?

Assuming each new construct is created with half the xp needed for next level in it's HD.

Assume a Recharging Staff of Minor Servitor that recharges one charge per day.

The first Minor Servitor creature is instructed to make as many as it can until it's master returns. The master never returns. As a 3.0 spell Minor Servitor can animate any non-gaseous, non-living, non-magical substance.

Assuming NPC Wealth by Level (DMG pg127), How long until they can have more magic items that they can use to cast even more instances of Minor Servitor each day?

How long until there are enough of them to amass into a Swarm?
How long before they can count as a Hivemind that acts as a level 9 Sorcerer?

How wide is the area of devastation at both the Swarm stage and the Hivemind stage if they decide to animate everything they can killing and animating anything that isn't them as well? The difference if they animate downwards effectively digging with their animations?
How fast does this radius grow after they become Hivemind and can potentially craft more Staffs of Minor Servitor?

Minor Servitor is in Savage Species page 68.
Swarm subtype is in the Monster Manual page 315.
Hivemind is in the Book of Vile Darkness page 34.

Thanks regardless.

[faircoin]

I have a question.

How do you calculate in "dropping one" to finding theoretical averages or probabilities to dice rolls?

Examples:

1. I'm rolling 2d10 10 times and dropping the lowest 2d10 total.
2. I'm rolling for stats, 4d6 and dropping the lowest d6 on each roll.
3. I'm rolling 2 sets of 3d6 and taking the highest set.

What you're asking, in math formulation, expressed in TeX is:

\[x=\frac{\sum_{i=1}^{n}(\textup{rand}_i[x,y])-\min(\textup{rand}_i[x,y])}{n-1}\]



edit

Sorry, minor mistake in the image above. The leftmost x should be some other letter, doesn't matter which, it has nothing to do with the x's on the right.

We'll start with a simple, roll four times, drop lowest scenario.

Let a be the smallest roll, and b the sum of the largest rolls.

E(x) is the expected value function. We know the expected value E(a+b)=E(a)+E(b)=14.

If p_1, p_2, all the way to p_6 are the probabilities that x >= 1, 2, all the way to 6 respectively. P(x) is the probability function. Thus, P(a=1)=p_1-p_2, P(a=2)=p_2-p_3, and all the way to P(a=6)=p_6. Hence, E(a)=\sum_{i=1}^{6}(i(p_i-p_{i+1}))=\sum(p_i), where p_7 is set to be 0. p_i is (7-i)^4/(6^4). So, sum that baby up, and you should get E(a). If you know E(a), you can find E(b) very easily.

In this case it's 12.25.

edit

Crap, I got the above number wrong. This is why you don't calculate in your head. It's actually 12.25, not 12.5.

QED.

My point in giving the proof is to show that the solution isn't completely trivial, so you probably don't want to calculate it out every single time. Use that link, AnyDice.

The cheap way is just to AnyDice

Gives you a Poisson distribution of your expectations.

[Douglas]

This question isn't strictly speaking for D&D, but rather DDO, the D&D-based MMO. I have two weapons, one of which has far fewer base bonuses but gets a lot more benefit from strength, power attack, etc., and I'm trying to figure out at which point they balance out. I may need to explain some DDO-specific mechanics first.

Chance to hit

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A player's chance to hit his target is (Player's attack bonus + 10.5) / (Target's Armor Class * 2) + 25%, rounded to the nearest 5%.

So, for example, attack bonus of 10 vs AC of 20 would look like this:
(10+10.5) / (20 * 2) = 21.5/40 = .5375 = 53.75%
53.75% + 25% = 78.75%
Rounded to nearest multiple of 5% is 80% chance to hit.

Assume, for the purpose of the math, that my attack bonus before adding weapon properties and situational modifiers is +55, and my target's AC is 60.

Glancing blows

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3 of every 4 attacks is accompanied by a free "glancing blow" attack. Glancing blow attacks with my modifiers have a +5 attack bonus, do 60% normal damage, and have a 15% chance of triggering on-hit effects (which includes things like Holy and Flaming).

Attack sequence

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The DDO version of iteratives goes +0/+0/+5/+10 instead of +0/-5/-10/-15. The glancing blow attacks come on attacks 1, 3, and 4. My attack progression is thus:
Normal attack +55
Glancing blow +60
Normal attack +55
Normal attack +60
Glancing blow +65
Normal attack +65
Glancing blow +70

Plus the weapon's enhancement bonus on each attack, of course.

The weapons:
Weapon 1: Green Steel Lightning II Greataxe

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+5 enhancement bonus to attack and damage rolls.
Base damage dice of 1d12 * 1.5.
Crit profile of, after accounting for my feats and other bonuses, 19-20/x5.
Holy (2d6 damage vs evil)
Shocking Burst (1d6 electric on all hits, 1d10 times (multiplier - 1) more on crits)
Shocking Blast (just the crit part of Shocking Burst, plus an additional 4d6 damage on a confirmed natural 20)
Lightning Strike (1.5% chance on any hit to add 20d20+400 electric damage)

Glancing blows have only a 15% chance to apply the Holy, Shocking Burst, Shocking Blast, and Lightning Strike properties, and these properties are not multiplied by crits except as specifically noted.

Weapon 2: Sword of Shadow

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+5 enhancement bonus to attack and damage rolls.
Base damage dice of 2d6
Crit profile of, after accounting for my feats and other bonuses, 15-18/x3 and 19-20/x5 (I have abilities that increase crit multiplier by 2 but only apply to natural 19 and 20).

Pretty much all the power of the Sword of Shadow comes from its expanded crit range, which multiplies static bonuses. The question is, how much total bonus from strength, power attack, etc., do I need for the Sword of Shadow's average damage to catch up? How does the answer change against enemies with 25%, 50%, or 75% crit immunity? Does the target's AC significantly affect the answer, presumably through crit confirmation chance? Assume, for the purpose of calculation, that the enemies are evil and have no electricity resistance.

Bonus question:
Weapon 3: Epic Sword of Shadow

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+10 enhancement bonus to attack and damage rolls.
Base damage of 2d6 * 2.5
Crit profile of 15-18/x3 and 19-20/x5

I'm pretty sure this upgrade blows the green steel weapon away with even a moderate strength bonus against enemies with no fortification, but what about enemies that have 25%, 50%, or 75% crit immunity? How close is it against 100% crit immune enemies, keeping in mind that the extra base and enhancement damage apply (at 60%) to glancing blows?

[TuggyNE]

Weapon 3: Epic Sword of Shadow

I might work on the math later but do you

do you actually have an eSoS??

[Hyde]

Given an even distribution of monsters in the Beastiary (We'll say one of each, because anything else is going to be an extension of that), what is the best possible +8 Weapon that could be crafted to fight them?

Assume 16th level fighter or barbarian with a 26 STR and 22 Dex.

Ignore Abominations.

[Douglas]

I might work on the math later but do you

do you actually have an eSoS??

No.

Working on it, and I do have the base item, but no.

Given an even distribution of monsters in the Beastiary (We'll say one of each, because anything else is going to be an extension of that), what is the best possible +8 Weapon that could be crafted to fight them?

Assume 16th level fighter or barbarian with a 26 STR and 22 Dex.

Ignore Abominations.

This question is specifically beyond the scope of what the OP offered. If you want to present some specific options, fine, but "what in the entire game is best" is far too broad and has too high a chance of something being overlooked.

Someone might try to answer anyway, but I wouldn't expect it.

[lsfreak]

Given an even distribution of monsters in the Beastiary (We'll say one of each, because anything else is going to be an extension of that), what is the best possible +8 Weapon that could be crafted to fight them?

Assume 16th level fighter or barbarian with a 26 STR and 22 Dex.

Ignore Abominations.

While it's outside the thread, I'll throw out several points.
- Depends on your level of optimization. Overcoming DR is going to be much more important if you've just got Power Attack than if you've got Shock Trooper + Valorous Weapon + Headlong Rush, which if it's going to be mathed out needs to be known.
- Depends on your build. While Valorous is going to be a must-have for a charger (unless it surpasses your Op threshold), it's much less useful on Jack B. Quick.
- Depends on your party and your other gear. If you reliably have a bard dropping DFI, the elemental damage stuff isn't as bad as it normally is because it won't be completely knocked out by resist 10. If you can't get haste from another player or from items, speed goes from a huge waste to your #1 enhancement.
- Having said that, +1 collision (+2, MIC) shadow striking (+3, ToM) with sudden stunning (+2000gp, DMGII) is probably a good starting point that would be decent for any build. Wrathful healing, transmuting, speed, magebane, vampiric and wounding are some other that are good irrespective of specific build, and of course there's others that are only good for specific builds.

[Douglas]

Well, I finally went and made up a spreadsheet to answer my own question, and the results are in.

AC is mostly irrelevant for the comparison, even doubling it to 120 had almost no effect on which one was better (though absolute damage did drop a lot, just about the same for everything).

Against enemies with no fortification, it takes about 70 static damage bonus for the normal Sword of Shadow to match a greensteel lightning 2 weapon. Against enemies completely immune to crits, even 1000 leaves the SoS still slightly behind. For in between partially crit-immune enemies, the needed bonus is less extravagant but still not feasible.

Epic Sword of Shadow blows it away, with a huge margin even with 0 static bonus against fully crit-able foes, and only slightly behind (4 damage out of ~150 vs greensteel greatsword, less than 1 vs greataxe, per full attack) even against 100% crit immune. Give it any significant static bonus, and it wins even with no crits at all.

Extra bonus answer, Breach, The Dividing Blade, is a bit behind Epic Sword of Shadow for full crits, but close enough that only a moderate crit resistance makes it even and full crit immunity puts Breach well ahead.

[Hyde]

Even though it was outside the scope, I apreciate the answer all the same. Thank you.

[killem2]

My DM and a fellow Psion player and I were in a three way debate on using Empower Power on a crystal shard.

She's only level 5.

At what point does this little Petal Psion start seeing a true benefit of using empower power over using nothing at all?

[lsfreak]

At what point does this little Petal Psion start seeing a true benefit of using empower power over using nothing at all?

Unless I screwed something up, a six-PP crystal shard is equal between spending the points directly and using empower, and higher PP expenditure favors empower. Or put a different way, 6d6 and 4d6 plus empower are equal, which after writing it that way instead of how I worked it out seems painfully obvious