When you throw one die, you have an even chance of rolling all the numbers 1 through 6. But when you throw two dice, things change. The lowest number you can roll is a '2', and the highest is a '12'. The dice can only come up with ten different number totals. But there are 36 different combinations you can throw, as shown in this table.
| 1 | 2 | 3 | 4 | 5 | 6 | It's pretty easy to see that each number has a different chance or probability of coming up. A '9', for example, can come up 4 times, on the average, every 36 throws. That's 4/36 = 11% of the time. (All probabilities are rounded off to the nearest percentage point.) A number '7' is the most probable, appearing 6/36 = 1/6 = 17% of the time. The '2' or '12', as we said, is least probable, appearing only 1/36 = 3% of the time. | |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
To the right is a plot of the probability of throwing each number. You'll note that it's a lot more likely that you'll throw a number somewhere in the middle of that graph than somewhere near the ends. In fact, you've got a combined total of 14% + 17% + 14% = 45% chance of throwing one of just the middle three numbers: 6, 7, and 8. That's almost half the time! (This type of graph is called a 'bell curve', because it's got a cross-section that is bell-shaped.)
This bell curve becomes even more pronounced in the center when you're throwing 3, or 4, or even more dice. In other words, it becomes more and more likely that you will throw a number somewhere in the middle than one near the ends.
What if you want something a little more linear? What if you want a system where you have a lower chance of tossing one of the central numbers, and a higher shot at the ones on the ends? What if you'd like all the numbers to be just about equally probable?
Until now, your only choice would have been to pick up some of those funky dice with lots of sides. If you have a d20 (a die with 20 sides), for example, you can toss it and be assured that all of the numbers from 1-20 will come up, on average, equally. But we don't need no stinkin' fancy dice around here, do we? All we need is the patented (not really) Linear Dice™ system. Don't Leave Home Without It.
Here's how it works: If you need to toss one d6, throw one d6. Easy so far? Okay. If you need to toss two d6, instead throw one d6 and—here it comes!—multiply by two! So if you roll a '1', it's really a '2'. If you roll a '2', it's really a '4', etc. Keen, huh?
| 1 | 2 | 3 | 4 | 5 | 6 | This table shows the numbers you can throw by tossing just one die and using a multiplier. The number thrown is in the left-hand column, the multiplier in the top row (or vice-versa, if you think about it...). For example, if you are simulating 3 dice, look for the '3' in the top row and the numbers you can roll are in the column beneath it. | |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 |
I hear some naysayers out there whispering, "But... if you've got 2 d6, you'll never roll a '3' that way, or a '5', or...". True. But so what? You'll have a nice linear progression from 2-12 instead of a mishmosh with a big bunch in the middle. And with 3 dice, or 4, or more, you'll really smooth out that extra-steep bell curve. So what if you can't roll all the numbers? We're talking about equalling or exceeding some target number anyway. Think about it.
"But, Mark," I hear some of you whimpering, "I'd really miss my 3's, and 5's, and...". Okay. Here's what we'll do. Let's say you have to roll 4 dice. Roll one d6, like I said, but multiply by '3', one less than your total allowed. Then roll again and add. Viola! All the right numbers. And almost completely evenly distributed. Here's the 4 d6 chart to prove it. The top row is the first number, which you multiply by '3', the left side is the second number, which you add:
| 1 | 2 | 3 | 4 | 5 | 6 | Note that the first three numbers and the last three numbers each only come up once, while all the rest come up twice. Because of the addition involved, there is actually a 'bell curve' with this version of the system. The bell gets flatter the more 'dice' you simulate, and gets steeper the less dice you simulate. At 7 dice, it's totally flat. At 2 dice, it's exactly the same as not using the system at all. If you want a system that is totally flat (even though it misses some numbers), use the no-addition, totally-multiplication version of Linear Dice. If you want all the numbers and are satisfied with a little bit of curve to get them, use the multiplication-plus-addition version. No system is perfect, but this one is darn close for being so simple. | |
|---|---|---|---|---|---|---|---|
| 1 | 4 | 7 | 10 | 13 | 16 | 19 | |
| 2 | 5 | 8 | 11 | 14 | 17 | 20 | |
| 3 | 6 | 9 | 12 | 15 | 18 | 21 | |
| 4 | 7 | 10 | 13 | 16 | 19 | 22 | |
| 5 | 8 | 11 | 14 | 17 | 20 | 23 | |
| 6 | 9 | 12 | 15 | 18 | 21 | 24 |
An additional advantage of this system is that you only need one die, instead of fistfuls of dice, no matter how many 'dice' your character has in an ability. So if you can dredge up one die, you can play.
I heartily recommend this system for use with Risus Royale, my solitaire RPG that you play with a deck of cards.
Discuss The Linear Dice Sytem and games in general on the Message Board, and be sure to try all my games (including Risus Royale) here.
The original material on this site is licensed under 
and is copyright © 2005 by Mark R. Brown. 