[DF] Suitable Dangerous Encounters

[Bruno]

There is a cheat involving spreadsheets which basically performs thulben's algorithm, taking advantage of a spreadsheets ability to store and manipulated data in two dimensions simply. Also makes great use of the copy-down function :D

Unfortunately I've forgotten the details, which is really irking me right now.

AHA!

OK. Instructions will be provided for Excel, but probably work in OpenOffice Calc or can be easily translated. I'll use D6s as the example, cuz this is GURPS.

Pop open a blank spreadsheet.

In A6, put the number 1.
In B7, put the foruma =SUM(A1:A6)/6
Copy B7
Paste it from B7 to the right however many dice you are interested in (so if you're interested in 5d6, copy it right to F7, so it fills an area 5 columns wide).
Paste it DOWN until your far-right column fills up with numbers and then starts becoming 0 again.

What you've just done is calculate the probability of rolling any given number on Xd6, where X is the number of columns you filled in with your formula.

You can then sum them up to find the probability of rolling less than 10 or 10, 11, or 12 or whatever you need to do.

[Nymdok]

Originally Posted by Bruno

There is a cheat involving spreadsheets which basically performs thulben's algorithm, taking advantage of a spreadsheets ability to store and manipulated data in two dimensions simply. Also makes great use of the copy-down function :D
.....

That is Damn Clever! And convenient!

Im starting to think these probability problems/solutions are worth their own FAQ.

Nymdok

[Ts_]

Originally Posted by Nymdok

P.p.s. You been writing code for this!?!?!? CHEATER!! I BEEN USING A PENCIL! Im kidding of course, but Id love to see your code. :)

Yep, some python code. A mess, of course ... And so far it seems to work nicely. There are some things missing, though, e.g., HT-checks. But I couldn't finish the simulation yet. First results in, though: Barb vs 2 Guards wins about 80% of the time (where Guards drop at -HP and Barb also at -HP). Barb vs 3 Guards: Wins maybe only 30% of the time? And vs 4 maybe 10% from the view runs I've tried.
Note that this is tougher than what I used above, because the Barb needs two average hits to drop a guard.

Gotta sleep now ...
Ts

[tjbuege]

Originally Posted by thulben

The short answer to the question of "how many ways can I come up with the sum of n on 10 dice" is (x+x^2+x^3+x^4+x^5+x^6)^10, using the coefficient of x^n in the expanded polynomial

The long answer is that you essentially use the algebra of multiplying polynomials to do the bookkeeping. A normal 6-sided die has each number 1-6 on it. We need some way to keep track of every permutation. By using the exponent of a garbage variable as the value shown on the die, algebra gives us a simple method to do this. As you can see from the polynomial above, each number 1-6 is represented once. We then multiply it by itself 10 times to see what the effect of rolling the die 10 times would be. The nice thing about this method is that you can account for "non-standard" dice (i.e. the possible sums of rolling a 6-sided die with a 1 1's, 2 3's, and 3 5's with a standard one would be (x+2x^3+3x^5)(x+x^2+x^3+x^4+x^5+x^6))

Uh...wow. It took me a bit to translate that into English...but ... wow. That's really cool. I never would have figured that out for myself.

Now, to translate this into a subroutine / function that can be reused....

Thanks!
Tim

[tjbuege]

Originally Posted by Nymdok

That is Damn Clever! And convenient!

I whole heartedly second that! Extremely convenient! I was working my way to that same solution (googled the newsgroups and found something similar), but this is a neat way to do it quickly.

Originally I was hoping for a simple equation that would give the results, but I see, even with Thulben's method, it would still require more than a simple equation, instead requiring loops or recursions. Thulben's method would be the way to go if you didn't want to store the tables, but since we seem to be focused on using Excel, no reason not to use the simple copy and paste method.

Thanks!
Tim

[Bruno]

Someone here on the forums taught it to me. I wish I knew who so I could give proper credit.

[Nymdok]

Originally Posted by Ts_

.... First results in, though: Barb vs 2 Guards wins about 80% of the time (where Guards drop at -HP and Barb also at -HP). Barb vs 3 Guards: Wins maybe only 30% of the time? And vs 4 maybe 10% from the view runs I've tried.
Note that this is tougher than what I used above, because the Barb needs two average hits to drop a guard.

Looking forward to the final results!

Nymdok

[Azinctus]

The maths is interesting and useful but:

Originally Posted by DAT

A big factor that I can't see being able to factor into a spread sheet are meta-game effects [...] e.g., cleric can cast wall of silence around the party, wizard throw a concussion blast or two into the guards, and then the minotaur acts as a reaper where his average damage from an attack puts each guard hit into the negative HP range ... and he has two attacks a second).

The Knight, Cleric and Wizard in my party might: cast Armour 5 on the Knight (to give an aggregate DR 13 or so) who would use massive deceptive attacks or whirlwind attack. The Wizard would cast Phantom (extra strong) each turn and use Fumble or another blocking spell to ensure almost complete safety. The Cleric would heal himself or summon a Divine assistant or something useful each turn.

In short they would breeze through this fight and that's without factoring in Grenades, Potions, Magic Items and what not.

[Nymdok]

Originally Posted by Azinctus

The Knight, Cleric and Wizard in my party might...
In short they would breeze through this fight and that's without factoring in Grenades, Potions, Magic Items and what not.

There are alot of things not accounted for in the model so far, thats a fair statement.

Any model put forth is probably not going to model tactical considerations and player skill (Knowledge of the rules), which is why we don't bother trying.

The general counters I would have to that are simply:

Mana doesnt last forever
A cleric that is healing himself isnt attacking.

We'll have to rely on the GM to tweak to suit, but what we're looking to do is give them a starting point that we can agree is 'about right'.

For example your knight:
Skill 20, DMG 2d+2 (just as an example, I dont know your charachter specifically)

If he Deceptive attacks down -4/-2 (ANy more the bonus starts decreasing) for 16 v 7 gives him
(Odds of hittng)(Expectation of Damage through DR)(Wounding Mod)
(83.7%) (2.28)(1.5) = 2.8 which is much WORSE than the Barbarian's 7.6.

Brute strength is a great way to get through DR and the Barbarian has it in spades :)

The knight is probably going to be better armored and have a shield though, which will contribute to his overall longevity.

Nymdok

[Bruno]

I advocate against taking into account EVERY single part of the party composition, player psychology, possible tactics, etc.

This is for a number of reasons:

1. You'll go insane trying to account for all the permutations.
2. Your players will do something you didn't plan for anyways (almost guarenteed)
3. You'll spend more time planning the fight than playing it out (I mean, if you're having fun simulating, knock yourself out, but don't stress yourself over it)
4. Perfectly and PRECISELY balancing the encounter against your players every possible move is, basically, managing to railroad your way right past the dice. If you want a scripted outcome, just be honest with your players and say "There's a few rounds of back and forth but you beat up the guards." or "The guards spank you and haul you off to jail."

The tactical planning on the players part is part of the fun they're having. If they do something brilliant and get rewarded with a smooth and easy fight, they notice that they did something brilliant and they feel good about themselves. If they do something dumb and get beaten up badly, it's a valuable learning experience, and should go towards them doing something brilliant (eventually).

GMs will have to adjust a little for their own groups, but having a baseline to adjust from is easier than nothing. I find my players tend to waltz through D&D 4e encounters "appropriately" rated for their level, so I go with the guideline, and then add an extra monster of their level. That brings the encounter to a level of difficulty that everyone seems to find "fun". In GURPS, I'd probably tweak monster defenses up by 1 instead, since an extra foe means a lot more in a game with strict facing rules.