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.
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.
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.
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.
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.)