Sums, Combinations, Permutations, and D6³

When I first pitched the dice-and-cards mechanic to Alan it was all about adding a bit of information to each die roll, and doing so without consulting a table. I didn’t realize the full implications of the system.

There’s some math here. If you’re a mathematician, you may wish to correct my terminology in places. If you’re not, this may be a bit of slog to read through.

Consider the Twenty-Sided Die

Consider a d20. You roll it, and there is a 5% chance that any particular face will come up. Any probability you want to model with a single roll of a d20 is going to have a granularity of 5% increments. You can create a 45% chance (“roll 10 or better”) but you can’t do 47%. Nor can you do 1%.

For a long time I considered 3d6 to have even less resolution. After all, there are only 16 possible sums (3 through 18) and some of them come up much more often than others.

Three Six-sided Dice: It’s More than Just Sums

If you use three six-sided dice of different colors you can easily see that rolling a 10 with 6,2, and 2 is different from rolling a 10 with 2,2, and 6.

 

Same sum, different permutations...
Same sum, different permutations…

The original mechanic I described to Alan involved using two dice of one color, and a third of a different color—the Mayhem die. If that third die was higher than both of the other two, the player drew a Mayhem card, and the card added information to the result of the roll.

MayhemDieSaysMayhem

On three six-sided dice there are 216 possible permutations (6*6*6 or 6³ is the formula here) which means that if you had a way to quickly chart all of them, your granularity of modeling is built on 00.46% chunks. This is more than ten times the number of outcomes afforded you with a single d20, and more than twice what you can get with two ten-sided dice. If you want a 1% chance of something, and are willing to settle for 0.92%, “triple-fives or triples sixes” is exactly that.

D63-FinalOur terminology makes a distinction between 3d6, which is just sums, and D6³, which takes into account the sums and a few of the permutations.

Seeing the Easy Permutations

Tracking all of the possible permutations on three six-sided dice wouldn’t make for a speedy game, but there are some permutations that are easy to see, and whose probabilities are easy to measure. Triples are just one such example, and these and other easy permutations can be used to add spikes to the smooth curve of the 3d6 sum.

It’s easiest to explain with some examples from the game.

1) The Mayhem Deck: If the odd-colored die is higher than the other two, and the sum of the three dice meets or beats the target number, draw a Mayhem card. Assuming a target of 11, there are 39 permutations  that will result in a drawn card. There are 135 permutations that will result in success.  Thus, for a target roll of 11 you have a 28% chance of drawing a card if you beat the target.

2) If you’re using a Phubar weapon and you roll triples, your weapon misfires catastrophically. There are exactly 6 permutations that meet this criteria, which means that with each roll you have a 2.77% chance of your weapon blowing up in your face. Bring a spare weapon, and maybe a spare face.

3) Autocorrect (weapon attribute): if all three dice are 3 or lower, and you do not meet or beat the target number, roll again (this does not repeat.) Assuming a target number of 11 there are 27 permutations that meet this criteria, out of 107 possible , assuming a target number of 10, so you have a 25.4% chance of getting to re-roll a miss.  That’s pretty powerful.

4) Double-Tap (weapon attribute):  if you meet or beat the target, and have rolled doubles, roll a second attack immediately. With a target of 11 you have about a 26% chance of rolling doubles, which means about one in every four attacks will generate a double-tap. Again, there’s a lot of power here. If you want more power, do “exploding Double-Tap,” and allow a re-roll to generate another re-roll.

5) Extra-Accurate (weapon attribute):  any roll of a straight (1-2-3, 2-3-4, 3-4-5, or 4-5-6) will hit the target regardless of the sum. (You can re-position the dice to spell out the straight.) There are 24 permutations that qualify. If the target number is 11, 12 of those permutations will turn non-hitting sums into hits, boosting your hit percentage by 5.5%. That doesn’t seem like much, but if you’re pot-shotting something and need to roll a 16 to hit, there are only 6 permutations where the sum alone will get you there. Extra-Accurate adds 24 permutations to that, quintupling your chances of hitting. Sure, it’s still just 13.8%, but that’s almost a fifteen percent chance if you’re feeling optimistic enough to round up. Which you probably are if you’re taking pot-shots at hard targets.

The examples above are all for combat rolls, but this permutation mechanic is not restricted to combat. It applies to skill checks with certain tools, and can even be earned as a skill attribute, providing an additional few percentage points of success, chances to re-roll failures, or doubling up on the effort—all in service of letting the dice help you tell a good story.

Summary

Players will enjoy the added flavor of doing more than just adding up the dice they’ve rolled, and since we’ve used the easy-to-spot permutations (doubles, triples, straights, and the color of the highest number) game-play won’t stop while players stare at the dice.  They will, however, covet items with these attributes, and they’ll roll those dice with a little more glee.

(Note: We used 11 for the target number in the above examples because the math is easier. Target numbers in actual game play can vary widely depending on the encounters crafted by the Game Chief. We’ll do another post on the science of character stats, skills, and target numbers at a later date.)

16 thoughts on “Sums, Combinations, Permutations, and D6³”

  1. This sounds very interesting in concept and i will have to wait until i have the book to try it.

    I have a few questions though.
    1. How does the double tap combine with the Mayhem deck? for instance what if i get a hit with 4,4, 6 and 6 is the off color and the mayhem card suggests the weapon jams. do i still get the second roll? or if both rolls get a mayhem card (by your chance this happens 7,84% of the time with a target of 10.
    2. With the extra accurate does Mayhem also happen if it would not be a hit without the extra accurate?
    3. Can you give us a few concrete examples of Mayhem cards?

    Can’t wait to play full court jungle-ball in the game. Now i just have to find money in the budget to upgrade to the signed version.

    1. Hey Marc! I can answer those!

      1.) Ask your Game Chief. His call. We leave the adjudication of the Mayhem cards to the GC because every situation is different. It’s easier to let the person on the ground make that call.
      2.) Ask your Game Chief. His call.
      3.) Ask your Game Chief. His…oh. I mean, yeah. That’s a separate post.

      1. Let me tell you how I’d play it.

        1) You roll a second time, and then the misfire becomes the story of that second shot. Instant RiPP to the player who screams “HE WAS ALREADY DOWN WHY DID YOU PUSH YOUR LUCK?” in character.

        2) It depends on the Mayhem card. If it’s something cool like “This is How it’s Done” (which awards skill points to a grunt who has survived Ablative Meat, or lesser points to a PC) then I’d absolutely play the card, role-playing the NPC grunt in question (or giving a RiPP to the player who does it before I can get to it.) If it’s something terrible like “That’s Coming Out of Our Pay,” I’d gauge the state of the party and encounter and then either spend RiPPs to discard the card, or invite the player to spend a RiPP. If however, the player is ready to jump on the card and role-play the disaster, I will not stand in their way.

        3) That’s a separate post. For now let’s just say that they can be categorized by their attributes: Beneficial temporary, beneficial permanent, detrimental temporary, detrimental permanent, and wild.

        Developing an aversion to certain foods after a critical hit is a good example of “detrimental permanent,” because that becomes a player attribute. It’s not really detrimental, though, because it’s an opportunity for the player to earn RiPPs in the future.

        “A Bit Too Inspiring, Perhaps” is a good example of detrimental temporary. That’s where your success gets upgraded to critical success, and the grunts rally and charge, even if that would put them in a bad tactical position. Again, it’s an opportunity to role-play. If a player can convince me (and the other players) that their four-word St. Crispin’s Day speech is good enough, the grunts may return to their positions.

        Both of these cards would be interpreted differently depending on the skill check that drew the card. An engineering check that modifies a weapon might inspire grunts to attempt the same thing, resulting in hilarity. The development of an aversion during a bit of comm-system work might mean you saw something on the hypernet that you cannot un-see, and now you’ll see it everywhere.

        If this sounds challenging, that’s because it is. The ultimate reward is a fantastic story, jointly told. Even the worst of failures, including the dreaded Total Party Kill (which isn’t going to happen unless you and the Game Chief know that’s the kind of adventure this is) can turn out a fantastic story. The success of your gaming session is independent of the success of the party in meeting or exceeding the mission objectives.

        1. Thank you both for the reply.

          And can’t wait for the extra post on mayham cards.

          And more insight on creating your charter and/or ship if that is an option.

  2. Thanks for the breakdown on how some of the rules and cards work. I’m really liking what I’m hearing so far. It sounds like it moves pretty quick, is easy to learn, and will make for some great stories. Thanks for the peek behind the curtain– liking what I see so far.

  3. Clever! Thank you for sharing. I’m starting a campaign via Roll20.net’s virtual tabletop, and this seems like an awesome way to get better granularity… especially when the math on the back end can be automated via macro.

  4. The only downside I see for this is that it makes online play a bit more complicated, because my electronic dice roller isn’t really designed to distinguish between dice. That said, the one I usually use has a bunch of options (so it might have something like this set up, possibly for In Nomine or something) and, if worst comes to worst, we can always roll 1d6 separately, so it’s not a huge deal and the added value seems worth it.

    …Come to think of it, the cards might make online play a bit complicated too.

    Any suggestions on how to deal with online play, either in the book or via the blog?

    1. We’re developing the very best tabletop RPG that we can, and we’re leveraging lots of things only available with physical presence. That’s where our expertise lies.

      Clever players and Game Chiefs with mad programming skills will be much better equipped to solve online play problems than we are, and we look forward to seeing what they come up with. That said, coding a die roller for D6³ permutation tracking is straight math, and whiteboarding tools could be used for Ablative Meat.

      1. With Roll20, it shouldn’t be too hard. When you roll more than one dice (for instance, 3d6,) it gives you the dice in a specific order, instead of sorting them by roll value. So your group can just know that the third die is the wild die. (Incidentally, the first roll I made to test this was 5, 3, and 6, so I expect Shennanigans would have ensued.)

        Roll20 also has a card system already in place. I haven’t played with it much yet, but it lets you deal cards to players (including the GM,) who then drag the card out to the map to lay it out. It comes pre-loaded with the standard 52 card playing deck, but it would be basically a matter of a little electronic elbow grease on the GC’s part to screenshot the cards out of the PM:RPG PDF and build it into a deck. (Will the PDF of the cards also feature an image of the card backs?) Then they just need to deal some cards to themselves and drag them to the table when appropriate.

        The only thing that I don’t think Roll20 would handle very well by itself is the list of ablative grunt-shields, but that can be easily solved by the use of a collaborative Google document or Etherpad.

        Hitting the “Delete” key may not have quite as much impact as tearing physical paper to physical shreds, but it does have two advantages in that you can delete a grunt that was just made by someone in Hong Kong while you’re in Omaha, and it saves you a small sum on paper supplies.

          1. It solves most of it for Roll20, at least unless the people behind Roll20 make some catastrophic decision to sort the dice by result instead of by the order in which they’re rolled.

            Barring that, though, it’s easy to pick out whether your dice rolls are triples, straights, or if the wild die is high. It may even be easier done than said.

            I’m expecting you still draw a Mayhem card if you get a straight and the wild die (is there an official name for it beyond the different-colored d6?) is high.

            Do straights and triples have any effects under ordinary generic circumstances, or only when special abilities/equipment traits/what-have-you call them out?

  5. I’m trying to understand how ‘extra accurate’ works it we do not have 3 distinctly colored dice to create a canonical ordering. How do I distinguish a 1,2,3 roll from a 3,1,2? If I cannot, then there are 24 permutations, not just 12. If I insist that the color die is in a specific position (high, low or middle) then I get 3 sets of 8, but don’t see how to combine to get exactly 12 valid results.

  6. Good question. Canonical ordering is not required, which means I had the math wrong. That section has been corrected.

    Also, I’m going to add pictures. I saw your earlier confusion, and realized pictures solve everything.

    (Except doing the math wrong.)

  7. I have to say, this all sounds really neat, but I was wondering how the heavy weapons like Schlock’s plasgun or multicannons or the AP-229 or some form of grenade/rocket launcher where you aren’t so much targeting an enemy as a general direction or room? Stuff like Schlock filling the area with low intensity plasma like he did on Oisri seems very different from a roleplay/narrative perspective than toasting an attorney drone with said plasgun, so I was wondering if that would also use the d6 cubed pattern or if there would be some other method of figuring out results?

    1. Heavy weapons, ship weapons, and WMDs will still use the D6³ Mayhem system, but some effects just won’t apply. For instance, if you hose the room down with the BH-209, the TN is probably 5, and accuracy boosts and re-rolls won’t come into play. If, however, you’re aiming at a target two city blocks away, you might need to roll a twelve or higher to hit. Having Auto-Correct will be handy.

      Mayhem rolls and card draws will always apply, however, and when you’re firing the BH-209 (or the jury-rigged main gun on your Harmadillo-class gun boat) Mayhem is probably appropriate. Keep RiPPs handy to save yourself from pesky things like friendly fire and new nick-names.

  8. I’m really excited how the d6^3 mechanic is a lot more than just rolling 3d6 and adding up the results. This is the post that got me to actually pledge for the kickstarter today. I look forward to downloading my copy of the PDF and talking my gamer buddies into playing it.

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