A Look At Linear Vs. Non Linear Dice Systems
Why The d20 System May Be Misleading

Looking through the third edition Dungeons & Dragons Player's Handbook, reading on Page 58, something they said about a d20 skill roll seemed incredibly wrong, or at the very least, incredibly misleading:

"On average, Devis will roll 10 or 11 on the d20, . . ."

An anomaly? I don't think so. Reading the Dungeon's Master's Guide, Page 13 we find:

"When you figure average rolls, can the fighters hit the creature?"

Average rolls? This is a d20 we are talking about, you know?

These quotes demonstrates such a fundamental lack of understanding, I felt compelled to whip off this article to explain just why saying things like that can be so misleading.

I was particularly concerned as the entire 3rd edition system is based upon the d20, even going so far as to make great claims about the improved system and giving it a logo that reads "d20". Further still, they are apparently offering the d20 system as the basis for other people to build their games around it. So this misleading idea should be made clear, if it isn't already.

Older versions of AD&D also used a d20, but not the d20 system, but I never read such a poor characterization of the facts before. It makes me wonder if they did a lot of things the way they did them with the belief that, on average or with the greatest probability, one will roll a 10 or 11 like they seem to imply. One won't, you know. Of course I may be reading too much into this as well, but I do think it is incredibly misleading so one should have it pointed out to them, just in case they are getting the wrong idea.

Technically one may arrive at a mathematical mean of 10.5, or find the median values of 10 or 11 on 1d20, but when speaking of dice and perhaps probabilities, what one desires to know is the mode.

The Mathematical Mean or average is the sum of N data points divided by N.

The Median is the middle data point or most central data points when these points are ordered in ascending or descending order.

The Mode is the most frequently occurring data point in the sample.

The mode, therefore, is what we wish to know when we ask ourselves about probabilities. The mode of the total on 2d6, for example, is 7. Out of the 36 possible rolled combinations, of the 11 different values totaling 2 to 12, the sum of 7 occurs most frequently. For 3d6, the mode is 10 or 11. Both of these totals occur with equal frequency. Knowing the mode of the roll of some dice will help you make wise decisions - where to build your house in Monopoly as your opponent approaches your properties, or how to bet in craps, or if you are likely to succeed with this skill roll, for example.

Knowing the mode on a linear system like 1d20 will not really help us, however, since each number occurs with equal frequency. So aside from knowing your chance of rolling 1 is the same as your chance of rolling a 20, or the same as rolling any given integer in between, this isn't much help either. But you won't be mislead into perhaps believing any single skill roll will average, or tend to, or gravitate toward some central average value like 10 or 11 - like they will in 3d6, for example.

This is why I feel they are confused when they go out of their way to tell the reader the average is 10 or 11 on a 1d20 skill roll or pose questions like asking yourself if an average roll will hit or succeed. Even if the mathematical mean is 10.5 and the median is 10 or 11, neither of these things is particularly useful to know in this context. They may as well have also taken measures to point out in the skill section that the skill roll on 1d20 will give you a number between 1 and 20, inclusive, and that number will be an integer. At least that's true. But what good is knowing this, and why do they say it now? It makes one wonder why they point out this so-called average in the first place since you cannot really use that information, and particularly since it is potentially misleading, one can't help but wonder if they are confused themselves about this matter.

Linear Systems

Now a d20 is what is known as a linear system. This particular linear system generates random integers from 1 to 20 inclusive, each one as likely as the other. The probability of rolling a 1 is the same as that of rolling a 20, or any integer in between. For something like a skill roll, this is one roll and there is no opportunity to average anything like there would be with multiple dice in a single roll, such as with 3d6.

Naturally, this is rather unfortunate since most things in the real world are not well represented by linear models. Normally when we do things, we do them in average ways more often than not. That is, average for us at our own skill level, so we are not comparing our own results to some objective standard or to another person's results.

Occasionally we will exceed our own expectations and perform exceptionally well (for us), and occasionally we will perform exceptionally badly (for us). This will happen less frequently than performing at average levels. Mostly we hit it in the middle of the road or near it and do something quite average, pretty close to our normal proficiency. This is very realistic.

So, in the d20 system, rolling a 20 is just a likely as rolling a 1 and both are just as likely as rolling a 10 or rolling an 11. As you might imagine, it doesn't represent skills very realistically.

This is particularly sad since most dice rolls are all about luck. The bonuses and penalties are not rolls, but add or subtract to the rolls. The d20 roll is so bad for most skills, in fact, they have to invent ways around it, such as taking 10 or taking 20, for example, to take out the luck that shouldn't be there in the first place. And when the GM is already supposed to make adjustments to DC, they may as well forget about the dice altogether and just determine how strong one needs to be to do something. They either are strong enough or they aren't. No roll is required. The same is true of most skills. You either have sufficient skill for a particular DC, or you don't. No roll needed.

Alas, we like some luck involved, as long as it's reasonable, and we don't want the GM deciding our PC's fate always based on nothing more than their personal whim. They already do that enough as they prepared the scenarios ahead of time and plan for what they expect without needing to totally ban unforeseen or random factors as well. Besides, some randomness is fun and exciting. But I digress.

Now, during the course of a year, let's imagine Devis will make 1000 skill rolls. On average, he will roll about 50 ones, about 50 twos, about 50 threes, etc. etc. etc., . . . about 50 nineteens, and about 50 twenties. What the authors of that passage feel you should grasp by knowing the median - what they call the average, though it isn't - is beyond me. I mean I know what mistake they probably made, but to make it at all shows such a lack of understanding that it shakes my confidence in what they have been doing if they truly believe this.

Another big part of the problem of the 1d20 system is the confusion that rolls may be modified in 4 different ways; + to roll, - to roll, + to DC, or - to DC. This is needlessly confusing since bonuses or penalties to the skill roll, or penalties and bonuses to the DC, amount to the same thing, and only the final result matters. Since each GM already has a lot of free rein to arbitrarily set the DC, it makes little sense to adjust it later since it's those very factors they should have taken into account while setting it in the first place. Bonuses and penalties to the roll, on the other hand, frequently have to adjust rolls since the GM may not always recall what special equipment or racial bonuses a PC may have, or what penalties bad equipment or wearing armor may cause. There is no need to adjust the same roll from two different directions. Opposed checks, on the other hand, may require DC adjustments - see below.

One might also occasionally slip and confuse the term "roll" with the term "result." The roll is whatever you roll on the d20, but the result is the final number you get after ALL bonuses and penalties are applied to the roll. Statements, such as on page 62 of the PHB, "The character who rolls highest goes first" or, in general, "The character who rolls highest wins," are misleading.

Non Linear Systems

A nonlinear system, like 3d6, is often a better approximation of reality. G.U.R.P.S.. uses this as a skill system, for example, and even AD&D uses 3d6 extensively when it rolls up character statistics. Thus, one should be familiar with the bell distribution curve generated by rolling and totaling 3d6.

On 1d6, each number is as likely as the other five, but when you add simultaneous rolls for more than one dice, you will find more average results. By the way, on average your total will be a 10 or an 11 on a roll of 3d6.

This makes the 3d6 system a better method to simulate our skills as a first approximation. It is so good that using it for statistics gives some fairly decent results that more closely approximate reality. As far as skills are concerned, more often than not our characters will achieve average results, and that's good and expected and realistic.

Using 4d6 and tossing out the lowest number is just a nice method for generating "higher" than average characters, but by that they mean higher than average using just 3d6 like the normal population. This is fine. Rolling 4d6 and tossing the lowest will average about 12.25 for each stat. That's 1.75 more than 10.5, or for all 6 stats, a total of 10.5 more points, on average, above the average man or woman, in case you're interested.

Opposed Checks

Another problem of linear systems like the d20 is one has to avoid statistic vs. statistic rolls in many respects since they seem too unrealistic as the linear properties of 1d20 are over bearing. For example, STR vs. STR. Now, we may not roll a "height skill" roll to see who is taller, sure, but in an arm wrestling contest, STR vs. STR should be rolled. This is not to see who is stronger, but to see who will win a contest based on STR. The higher STR will not always win. He may have an off day, be hungry, perhaps tired, possibly even wounded, or the weaker man may have a burst of adrenaline. Also, the psychological factors and mind games may play a part in it. Naturally, if one were stronger they would normally win much more often than not. But using a d20 system - where most the entire range of the d20 represents luck - the stronger man may roll a 1 as easily as rolling a 20, and the weaker man may roll a 20 as easily as rolling a 1. Thus, it becomes apparent how badly the d20 linear system works to approximate certain rolls that really have little to do with luck. This is why, I feel, they just tried to avoid STR vs. STR rolls, as more there than anywhere else it clearly demonstrates just how luck shouldn't play that large a part in many rolls.

Unlike for most skill rolls, where the GM arbitrarily sets a DC for a task, opposed rolls square off against each other's result and not against any particular DC. This is fine, though I felt it could have been better explained. So Devis does not roll vs. a DC to hide, nor does a guard roll vs. a particular DC to spot hidden, but they each simply roll their opposing skills and add in their hide or spot bonuses and the higher result wins. Here, however, unlike for many skill rolls, the GM may have need to adjust rolls for DC since they won't have opportunity to do so later. For example, if the hallway is well-illuminated, Devises' hide roll should be penalized, or the guard's spot roll should receive a bonus, or both, due to situational modifiers.

As for STR v. STR or other STAT vs. STAT rolls, the book is simply wrong to liken them to "height" rolls, or state the stronger man will simply win, as mentioned above. Each pertinent STAT's modifier is added to such a roll, and some situational modifiers may apply, but those would probably not be fair contests, then.

Now, in a fair contests, the GM may decide how many times a person need win "in a row" to be declared the final victor of that contest, and set this goal to the number, N. If they feel the contest is short and sweet - like a one shot toss of a dart, then N=1, and a single roll will probably tell the tale (baring ties, of course). If it's moderate in length - like a closely matched arm wrestling contest, then N = 3. If it's a lengthy ordeal - like a marathon, then N = 5 or more. The higher N is, naturally, the more likely the "stronger" or higher STAT will finally win the whole contest.

NOTE: Since this is not the first person to win N times, but the first to win N times in a row, evenly matched contests may require many rolls and be quite exciting as reversals of fortune play back and forth. Still, the higher N is, the less "dumb luck" will prevail, while the smaller N is, the more dumb luck might win out, such as in a single throw at a dart board. That's fine, of course, since it's realistic as longer contests are frequently required to weed out luck vs. skill. Even short contests recognize this, which is why they often go for 2 out 3 or the like to determine the true champion.


Unfortunately, in most games we like to make adjustments to our rolls for various factors other than luck, and these adjustments are often +1, +2, -3, or what have you. But they are linear adjustments. We call this the "linear add." This means adding +1 to a linear roll will have an equal affect no matter what the roll's result. But adding a linear +1 to a nonlinear roll, like a 3d6, will have different results depending on where you add it or what the results were on the nonlinear roll.

Therefore, adding +1 to a d20 roll alters it by 5% (always). But adding +1 to 3d6 will add a variable percentage. On 3d6, the difference between rolling a 17 and rolling an 18 is remarkably different than the difference between rolling a 10 and rolling an 11. Both may seem like a difference of 1, but these "1's" occur at different probabilities.

On 3d6, the percentage this "+1" represents may be as high a 12.5% or as small as less than one-half of one percent. So it seems odd to add +1 to such rolls. Instead of adding the same percentage to a roll, it would almost be like adding a 1d12% roll to a 1d100% roll.

In a nonlinear system, the skill you have that allows this linear add of +1 has more effect for some reason if you are performing further from the norm. It shouldn't, probably. The skill is probably consistent and should represent the same effect on your roll, and the skill certainly shouldn't be dependent on something outside of you, like an external bit of probability, since the skill is innate to you. Similarly, a bonus from something like a +1 weapon should add consistently, unless a sword can have an "off day" for example.

A similar way of thinking can be seen if you are allowed to add 1 to your stats (stats or statistics each range from 3 to 18 on 3d6). Adding 1 to your dexterity of 10 gives less desirable results than adding 1 to your dexterity of 17. Both add 1, but both don't give equal benefits. The higher a stat is, the more advantageous it is, unless the scale of advantages is perfectly linear as well.

Using strength as an example for a linear scale, if you had a strength of 10 and could lift 100 pounds, then a strength of 20 (double 10) should be able to lift 200 pounds (double 100) if the scale is linear. But it is not linear in D&D. If you add 1 to the strength of 10 you get 11 and thus can carry up to 115 pounds (15 pounds more than before the add), but if you had a strength of 17 you could carry up to 260 pounds. Add 1 to that, a strength of 18 can carry up to 300 pounds (40 pounds more than before the add). So quite simply, adding 1 will produce a greater advantage the further along or higher up you already are on a nonlinear scale. Or, in short, adding 1 doesn't mean a difference of 15 pounds no matter where you add it, but will depend on where you are on that scale.

So, the "linear add" has peculiar results when applied to a nonlinear systems like the 3d6, but excellent results when applied to a linear systems like the d20 system.

Recapping, we find the d20 is a less realistic approximation of real skills, and that 3d6 is better in that regard. But the d20 system lends itself more to realistic bonus and penalty adjustments than the 3d6, so the d20 system is better in that regard.

One might even try to use the 1d20 system rules, but substitute all 1d20 rolls for 3d6+2 rolls. Thus, 1d20+2 would become 3d6+4, for example. This 3d6+2 roll generates a nice bell curve with numbers from 5 to 20. However, rules about rolling a 1 would have to be altered so they were about rolling a 5 instead. Unfortunately, this takes a bit of work and many would not like the extra effort, even if it did help alleviate the biggest problem in the 1d20 system, that of linearity.

The 3d20 System

The 3d20 system may be employed to actually gain more average results. Whenever a 1d20 roll is called for in standard play, simply roll 3d20 instead. Then take the middle value and disregard the high and low rolls. In the event of a tie on two of the dice, take that value. For example, from the values 18, 14, 6, the roll would be 14, and from the values 18, 14, 14, the results would be 14. Discard the highest and the lowest dice and the one that remains is your result. This does generate a bell curve of sorts, so it's an improvement.

However, it unfortunately retains a great deal of luck since the bell curve is rather 'flat.' An even better method, I think, may be found in the 2d10 system.

The 2d10 System

Better yet, the suggestion of using 2d10 instead of 1d20 is wonderful. It generates numbers between 2 and 20 (very close to 1 to 20), actually does tend to roll around 11, and can easily be adapted to the d20 system. In fact, it's a remarkable match for the erroneous error they seem to consistently make - that being "average" rolls on d20 are around 10 or 11. So anytime they assumed that to be true and devised a system around it, for 2d10, it actually is true, and you will get results closer to what was intended.

Using 2d10 instead of 1d20, "freak" rolls will not play as large a part as often, and if you need a freak roll to fail, or succeed - like out at the very limit of the dice - it will happen about one in 100 times, while on a d20, it will happen about 1 in 20 times.

This would be known as the 2d10 system, and upon reflection, I actually recommend you adopt it rather than using 1d20 for your D&D games. Instead of always failing on a 1, you simply always fail on a 2 (double 1's), that happens about 1 in 100 times rather than 1 in 20 times. Similarly, you always succeed on a roll of a 20 (double 10's), that happens about 1 in 100 times, instead of 1 in 20. This assumes, of course, combat rules are in play, and not skill roll rules, where 1's and 20's aren't automatic failures or successes. So if the standard rules says a 20 always hits, then it is true for the 2d10 system as well. Anyway. . .

Recall, using 2d10 is great insofar as those suggestive comments in the PHB and DMG of rolling an average around 10.5 are now correct inasmuch as 11 is around 10.5, and it is actually an 'average' roll now. Do it, do it, do it! I don't think you'll regret it.

Some have said they feared using 2d10 instead of 1d20 would reduce the number of critical hits, even making some weapons less deadly. This is true. It would. Adopting the 2d10 system will strike a slightly different balance than before, but this is fair since all are in the same boat. And quite frankly, since there are always more NPCs than PCs, they will have far greater opportunity to critical hit your PC to death than you have to do the same to them. Does this mean critical hits are bad? No. I just feel they should occur less frequently.

If a 'natural' 20 was required before, the 2d10 system will require at least one of the 1d10's be a natural 10. One will probably already have to do this in order to achieve a total value of 20, but with enough bonuses, perhaps that won't be required. So, just as a total of 20 or more is insufficient to threaten a critical hit - and only a natural 20 will do - so too will one of the d10's be required to be a natural 10.

For those weapons that required an 18 or 19, we also insist at least one of the 1d10's is a natural 10. This is similar to the original requirement of rolling a natural 18 or higher.

Critical fumbles may occur when one rolls a value of 2, or double 1's. The GM may employ special rules for such instances. Here are some you may enjoy including in your game.

Critical Hits And Fumbles (Optional Extras For 3e Combat.)

So, with critical hits, it's still possible a farmer may kill a high level adventurer with a lucky shot; it just won't happen as often in the 2d10 system as it might in the d20 system. If you prefer skill and ability to dumb luck, adopting the 2d10 system seems a good idea.

On the other hand, if you feel you'd sorely miss it in combat, one could still employ a d20 for combat, but use 2d10 for all other skill rolls. I wouldn't recommend keeping d20 just for combat, but you're welcome to do this too if you'd miss the high frequency of critical hits and fumbles.

And yes, it might take a touch more work to use two dice and add them rather than one dice, but this is not really that hard, and still easier than GURPS or other systems that use 3d6. Most added realism comes at a price. The question is, is it worth it? I think it is in this case, and I hope you do too.

Buy The 2d10 System Today!

Please send $0.00 to God, or his/her nearest representative, and it's yours. You can't lose. Try it for thirty days and if not completely satisfied, you will receive a full refund, no questions asked ;-)

Of course, which system you prefer is up to you. Many games use more than one system, depending on what they are trying to do at the time. 3rd edition D&D has striven for consistency and tried to make the d20 system more universal as having only one system to learn is often easier. Even so, they still use more than one. For example, different dice for damage and 3d6 for stats as well as the 1d20 system, and even (gasp) d% rolls where you want to roll low (gosh, rolling high is not always the goal in 3e after all. So 1d100 or d% rolls are part of it too, like on page 129 of the PHB where you have to roll low to recover, or other places where d% is called for). But that's ok.

But I will say, however, that you shouldn't be mislead into thinking the d20 system has the primary advantage the 3d6 system or the 2d10 system has, since, in fact, it doesn't. Devis will not roll, on average, a 10 or an 11 on his roll of ONE d20. So if you think probability suggests he will roll a 10 or 11 more than any other number, you'd be wrong. Shame on anyone for suggesting this. But more importantly, after you have read this article, shame on you if you still think this.

But is that what they really implied when they said that? Well, read it yourself and you be the judge. But ask yourself, when they tell you this so-called average, what are you going to do with that knowledge, and why are they telling the reader that? I can't imagine why, but maybe you can. At least in a game like Monopoly when they tell you on average the toss of 2d6 will most likely be a 7, that's information you can use.

© April of 2001
James L.R. Beach
Waterville, MN 56096