Thursday, March 12, 2009

Fistfuls of Dice

Who doesn’t enjoy grabbing a handful of dice and throwing them across the game table? In Dungeons & Dragons such pleasures are normally limited to Magic Users and their spells, wands or staves spewing forth raging balls of fire or crackling bolts of lightning. Whatever the case, lots of dice spilling across the table is just plain cool. I believe that I fell in love with the old Avalon Hill game Titan for this very reason. Lots and Lots of dice. Well, that and the awesome Trampier work throughout.

In a recent post, I pondered the idea of adding some of the
neglected dice to my original D&D games. More or less I was observing how one could replicate the average scores of magic weapons that normally use a d6 with these other dice. I’m still considering these dice for aspects of OD&D, but now I am looking at ways that allow players to throw a handful of six-siders and partake in that exercise normally limited to Magic Users and Elves.

All of this harkens back to the
What Price Glory series of posts from last Summer, wherein I detailed optional rules packets for adding more depth to D&D combat. In particular, the Multiple Dice portion of the Damage packet that was offered then.

Part of that option was the idea that magic weapons all allow the wielder to throw an extra d6 when rolling damage, and take the higher result. Thus far in my games, this has been a nice additional option. But I want MORE dice. I’ve slightly modified the What Price Glory Multiple Dice option, and added more dice for magic weapons, as follows.

Multiple Damage Dice: The amount of dice used to determine melee damage is a reflection of Class, Level and Weapon. In all cases, roll the number of six-siders indicated, and take the highest result.

Standard Melee Damage: 1d6.

Fighting-Man: Level 1-3: no bonus, 4-6: +1d6, 7-9: +2d6, 10+: +3d6.
Clerics: Level 1-5: no bonus, 6+: +1d6.
Magic Users: No bonus.

Magic weapon, +1: +1d6.
Magic weapon, +2: +2d6.
Magic weapon, +3: +3d6.

For example, a Fighting-Man (including Dwarves, Elves and Hobbits) of 4th Level (+1d6), wielding a Mace, +2 (1d6+2d6), would roll 4d6 each time a hit is scored. From those 4 dice, the highest result determines the damage dealt that round. The Mace, +2 only adds to the character’s to hit roll, not damage dealt.

Thus, the maximum damage with this option does not exceed 6, unless of course you couple this option with the second part of Multiple Dice, to quote:

When rolling Multiple Dice, while only a single die is used to calculate damage, any die that rolls a 6 and is not used as the highest result will add 1 damage to the attacker’s total. For example, if a Fighting-Man throws 3d6 to determine damage, and comes up with three sixes, his base damage would actually equal 8 (6+1+1) .

This bit of Multiple Dice should have been presented as an add-on option. This is now:

The Rule of 6: The highest result is used for damage, other dice are considered bonus dice. Any bonus die which rolls a 6 adds +1 to the total resultant damage.

So, that 4th Level Fighting-Man in the above example could, potentially, deal 6+3 or 9 damage if his 4d6 came up 6-6-6-6.

A 10th Level Fighting-Man with a +3 weapon would have the luxury of throwing 7d6 anytime a hit is scored. Is this as powerful as a weapon that deals 1d6+3 each round? Perhaps not, but it is certainly more entertaining from my point of view.

The Rule of 6 in action. Someone just dealt 10 damage!

With some tinkering, Multiple Dice could be used for Cleric Cure Spells as well. For example, something like a bonus d6, use the highest result(s), at Levels 3, 6 and 9. Currently I’m somewhat liberal with Cleric Cure bonuses, allowing a 1 hit per Level of the caster bonus. But, MORE dice is always better. Most everyone has a ton of six-siders anyway. Might as well use them for more than a hall full of Goblins.

~Sham, Quixotic Referee


K. Bailey said...

I think this is something like the core mechanism of Tribe 8, which always sounded fairly slick to me.

Maybe there's a best of both worlds possibility. You roll at max 3 dice: 1 for your level, 1 for your weapon, and 1 for the enchantment.

Each die can be 1d4-1d12. So a 5th level fighter with a +3 dagger might roll: level d8 + weapon d4 + magic d8.

K. Bailey said...

(though of course sometimes you roll nothing: a level 1 MU with a dagger would just roll d4)

Mike D. said...

I am a fan of fistful 'o dice too, and this seems like an elegant yet simple addition to the game. I especially like the rule of 6, adding extra +1 for each 6 not used is brilliant.

Kevin Mac said...

In my old Champions games, I loved seeing a dozen or more d6 spewing out all over the table.

But for D&D, I already feel like there's too much rolling of dice going on at my table. I have one player right now who just rolls things and rolls things and I don't even know what the hell he's rolling.

Michael S/Chgowiz said...

@Bruno - beware that player! He's discovered the secret harmonic that causes hypnosis from the sound of rolling dice. He's really lulling you asleep to steal your loot, or worse, YOUR dice. Make him roll a d4 with every roll he makes and the harmonic is broken.

Kevin Mac said...

Aw man, back in the day we kind of thought of rolling the dice as something special and had meaning. It still happened a lot, but it had some real weight. It was often plain life and death.

A couple games ago, I felt like the dice were rolling on ME, not the table. Roll Roll Roll...

The party had just had an encounter with limerick-spewing leprachauns. After the encounter, while walking with other NPC's, the bard sez "I do a limerick" and makes his appropriate performance roll. "I made it. I do another one" rolls, makes it. Two minutes later during the encounter "Ok, I do another limerick" - Roll.

ME:"You'll have an actual limerick or two later then?"
PLAYER: "I don't think it's fair to make me write an actual limerick..."
ME:"dude, you are just rolling and rolling and rolling. So your character is free-rapping out limericks and you can't, great. But why not just say you are doing limericks, and forget the roll after roll after roll for it."

Every fucking roll on the table should have weight. If you are in the finals of the limerick world's championship, then shit yeah, make a roll, but...

Anonymous said...

Something I liked about Traveller was that it was d6-based. 2D to hit. Some weapons did 4D or 5D damage.

I sure like the options the rest of the dice spectrum gives you, but I always enjoyed the piles of six-siders we flung around in Traveller.

Restless said...

Have you run numbers or actually played using this? I wonder how this runs. I can see the benefit to the electricity at the table when a lot of dice are being rolled, but also I think I can see the advantage of trying to minimize the number of dice being rolled, especially during a heavy combat.

(I am scribbling notes for house rules, and I am vacillating between class-based damage or this variant for a LL/S&W game. While I am reasonably sold on damage by class, this just seems too cool to just pass up.)

Sham aka Dave said...

I haven't crunched the numbers when using the multiple dice, select the highest, nor the Rule of 6. I'm actually not sure how to determine the odds involved. There should be an average amount which each additional d6 thrown adds to the 1d6 average of 3.5, which would be bumped ever so slightly by the Rule of 6.

I am using both rules currently, but the most any of the players throws at a time is 2d6 as the campaign is still low level and the best magic weapons thus far are +1.

I'm fairly certain my players will love both methods, especially once more dice are added to the mix. I'm also assuming that the average scores will be lower than the standard 1d6 + modifiers. I'm just not sure exactly how much lower. It would also be perfectly fine to allow magic weapons to add their plus in damage to the damage roll if you're concerned that the players might feel their damage is being penalized. You'd get the best of both worlds and I doubt it would upset game balance at all.

If you tinker with this please let me know the results. I'm always open to suggestions on how to improve an idea.

Restless said...

I'm not sure how to determine the odds either; I had a lot of math in college, but I haven't rubbed those neurons together in almost fifteen years. I am asking a guy at work who does our reporting and was a math major in college. He still does a lot with statistics, so maybe he has some ideas.

If I get some time, I might just Monte Carlo it (write a script to burn some CPU time and throw ginormous bucketfuls of virtual dice and tally results). It'd give the answers but be less pretty.

(My verification word was "dappr". While I'm flattered, I think Blogspot needs better glasses.)

Sham aka Dave said...

I'd be very interested in the results or outcomes of your research. I fairly handy at figuring the odds for most dice combinations, but these two options are a bit different than anything I've used before.

Anonymous said...

The distribution for picking the highest die is fairly easy. I think I've calculated it once and could do again.

The methodology is as follows:
1. The probability of the total outcome being 1 is (1/6)^n, where n is the number of dice being rolled; each die must be 1.

2. The probability for the final outcome being 1 or 2 is (2/6)^n, as each die must show 1 or 2.

3. Now that the chance of rolling 1 or 2 is known and the chance of rolling 1 is known, simple subtraction gives the chance of rolling a two.

4. Probability of the final result being one, two or three is (1/3)^n.

5. Subtract the probability of the result being one and subtract the probability of the result being two. You've got the probability of having 3 as the highest die.

Continue in similar way. A nice pattern may manifest out of that if someone crunches the numbers; I think there was one but the details elude me.

The case where the number of sixes rolled matters is more complicated. It might be possible to write down the combinations or to use conditional probabilities.

Or maybe use binomial distribution with parameters p = 1/6, n = number of dice rolled. It gives the probabilities of rolling k sixes. If k is at least 1, then the final outcome will be 5 + k (= 6 + (k-1), but 5+k is simpler).

So use the first method to determine probabilities of the highest roll being anything between 1 and 5. Use the second method to determine the probability of rolling 6+.

If I have not explained these clearly enough, do ask for clarifications.

Sham aka Dave said...

Hey Thaunir, thanks for making this understandable. I think I get it now.

Let's say we want to determine the chance of a 6 being the highest result of 3d6. We have a 1/6 chance of a 6 on a single die. So you are saying that the equation would be (1/6) times 3?

When I have the time I'll consider this further, but off the cuff I'd say that the odds are 50% if the equation works. This is actually a simple example, as using 3 dice provides an even 50/50 chance of any single number appearing. Right?

The Rule of 6 does provide an entirely different problem.

Anyway, thanks for the comment. I wish I had the time right now to play with these numbers!

Anonymous said...

Hello Sham.

It seems my explanation was not good enough, as this is wrong: "Let's say we want to determine the chance of a 6 being the highest result of 3d6. We have a 1/6 chance of a 6 on a single die. So you are saying that the equation would be (1/6) times 3?" With that logic, were you to roll six dice, you would always get at least one of each result, which is clearly not true.

I suggest trying to follow the steps I outlined there; they are not very fast, but they get the job done, at least if I did not make any mistakes. Are they intelligible?

Restless said...

As threatened, I burned up an unreasonable amount of CPU time and compiled the results here. Most likely, you'll be interested in the d6 results here. It's not pretty, but it works, and if you want to do anything with the data it's all there.

I added an additional rule I thought of while writing the scripts, similar to the rule of six. I call it the rule of max, which would go as so:

The rule of max: The highest result is used for damage, other dice are considered bonus dice. Any bonus die which equals the highest die roll adds +1 to the total resultant damage. So, if the 4th Level Fighting-Man in the example above rolled a 2, 4, 5 and 5, then he would score a total of six damage, or five plus one bonus die. He could potentially score up to nine points, or 6 + 3 if his 4d6 came up 6-6-6-6.It's like the rule of six, except that bonus dice are equal to your maximum roll add one to the result. This has a few interesting features, which I will summarize below.

Here are the interesting generalizations I could come up with regarding the results I gathered:

* The fistfuls of dice approach causes large peaks towards the higher end of the die, but stops abruptly at the max for the die, of course. The rule of six basically clips off the peak at the maximum die value and spreads that out over the end of the graph for the bonus amounts available. Rule of max does the same, but it tends to stick closer to the values for the fistfuls of dice method in that it has more of a peak near the top of the die scale, but also it shifts more of the bulk of the chances near that peak (or even slightly higher).

* Overall, due to the larger number of potentially matching situations, the results of the rule of max tend to be slightly higher than the rule of six.

* In rule of max, it's very good to get doubles, triples or more matching dice. In fact, this may seem counterintuitive, but it's better to get a lot of ones than a couple twos in a roll, because then you get bonuses on your ones. It makes rolling doubles more significant.

* In the rule of max, your lowest possible roll on multiple dice is two, since rolling two ones gives a two (one plus a bonus of one). I sort of like that. Some others might not.

* I did some other dice to consider the possibilties of giving players other dice to roll on rather than making everything d6. For instance, fighters and dwarves might get d8, while semi-skilled combatants (clerics, thieves, elves, halflings) get d6 and magic users get d4. Maybe some magic items promote you to the next die type (d8 rather than d6). Thus, I wanted to look at the curves for those. The flaw with that, as I can see it, is that you're less likely to bring the rule of six or rule of max into play, because there are more potential results on the dice.

* If you wonder why I stopped at lower numbers of dice rather on some, it's because these charts were computed exhaustively; 7d20 is 1280000000 potential dice combinations, which takes a while to compile the results.

So which one would I choose? I really don't know. I like fast combat, but I don't know if I like rule of max because it seems to skew high which leads to a lot of attrition if a combat goes south on the players (which I didn't expect, honestly). Fistfuls of dice is good, but it's got a hard stop at the die size which is a little jarring to me. Overall, I think the rule of six is a good compromise.

Anonymous said...

Sham posting anon

thaunir: Actually, right after I wrote that response we got on the road for our trip. I realized while driving that using that assumption I had so quickly reached meant that 6d6 would yield a 100% chance of a 6, and that was not the case at all.

Somehow I glossed over the ^ bit in my haste.

Anyway, I think I've got it sorted out all the way up to 8d6, BUT I just came back to reread your steps and have a question.

4. Probability of the final result being one, two or three is (1/3)^n.Shouldn't this read (3/6) OR (1/2)^n?

In otherwords, let's say I am figuring the odds for a 3 being the highest result on 3d6. (.5)^3 = .125. I then subtract the odds from the previous results when figuring 1 and 2 in the calculation steps...(.125 minus .037) which equals .088. According to my round numbers the chance is therefore 8.8% for a result of 3 as the highest of the 3d6 roll.

Using this logic I came up with the following numbers for each result. Maybe I'm doing it wrong, BUT the odds add up to 100 so I think I got it:

1: .46%
2: 3.24%
3: 8.8%
4: 17.12%
5: 28.24%
6: 42.14%

I was just confused by step 4 reading 1/3.

Restless: I think the Rule of Max is very interesting! That's a great idea.

When I get home and have more time to consider all of this further I will take a long look at those numbers you compiled.

~Sham (its raining at the beach right now)

Anonymous said...

Hello Sham.

Yes, it should be 3/6 or 1/2. My apologies. I should really have proofread that.

The numbers also look fairly realistic. A good mnemonic: With four dice, the probability of rolling at least one six is (very close to) half.

Anonymous said...

Sham posting anon again

Hitting the road soon and heading home from the beach.

Restless: those numbers you ran are invaluable, and also confirm that I used Thaunir's equations properly. I'll be consulting those tables extensively as I continue to work with these two options (in particular the Rule of 6 results). Thanks for posting those charts!!

Thaunir: Thanks again for sharing the equation you spelled out for me. I really appreciate it!

Here's what I came up with using the process. You'll notice it is very close to the results Restless posted. When I get home and have a proper calculator I'll redo these combinations and make up some tables.

Chance for a 6 when rolling:

2d6: 30.56
3d6: 42.14
4d6: 51.79
5d6: 59.83
6d6: 66.53
7d6: 72.11
8d6: 76.76

Under the proposed rules, 7d6 would be the highest number of d6 thrown, but I included 8d6 and will take the table out to 10d6 for future expansion.

I've already devised an alternative Combat and Saving Throw system using six-siders which will likewise employ this convention (which I'm tentatively calling the Multi-Roll). Other name suggestions for the "Roll X, use the Highest" mechanic are welcome!