I’ll start by answering my own question.

No, it will never replace the d20, if only because “roll a d20″ is far quicker to say than “roll 4d6 minus 3.” Of course, there are also some good math reasons, as well as player satisfaction reasons that the d20 is here to stay. That’s not to say that 4d6-3 doesn’t have a place in your game, however. I believe there are instances where it can be a viable alternative to a d20 – more on that in a moment. But first, let’s look at the math involved.

After doing some quick math in your head, you should conclude that 4d6-3 will yield a range of numbers from 1 to 21. And obviously, a d20 yields a range of numbers from 1 to 20. However, the mathematical difficulties of 4d6-3 do not stem from that extra number, but rather the probabilities involved with rolling multiple dice versus rolling one die. Below you will see two bar graphs. The one on the top shows the probability of rolling any given number on a d20. As you may already know, the probability of rolling any number is the same – 5%. The graph on the bottom shows the probabilities for 4d6-3. You can see that it is a curve, with its peak at 11. This is a classic bell curve.

Probability for a d20

Probabilities for 4d6-3

###### (Special thanks to anydice.com for the graphs)

The fact that the d6’s create a curve (as opposed to the d20’s flat line) is both its selling point and weakness. The d6’s produce less “swingy” results, instead generally giving more “average” results – most of the time you’d roll something between 9 and 13, and VERY rarely a 1 or 21. The weakness shows itself when you add a bonus to the roll. A +1 bonus to a d20 roll increases the chance of success by a flat 5%, every time. With 4d6-3, because you’re moving up (or down) a curve instead of along a line, you’re not getting the same flat increase in probability that you would with a d20. The increase in probability of success would depend upon the target number you’re shooting for, as well as your base bonus to the roll. I’m not going to throw a bunch of math at you, just trust me when I say it’s a headache, and not intuitive, and you’d need a spreadsheet to figure it out. And if THAC0 taught us anything, it’s that no one likes to use a chart when they play D&D. The bottom line is that, with 4d6, a flat +1 can have a different impact on different players. That might be ok in a system that is designed for multiple d6 rolls, but D&D was designed with the d20 mechanic.

On top of the mathematical problems of introducing a bell curve mechanic to a game that was designed for a linear mechanic, we have the issue of player satisfaction. Yes, when we use 4d6-3, rolling tends to be more even. That’s good at first blush, but when we look at the facts, it’s more bad than good. Consider this: with a d20, you’re going to crit (over the long term) 5% of the time. With 4d6-3, you’re going to crit .8% of the time. Even if you open up “crits on a 20 or 21,” your crit percentage is still far below 5%, at only .39% – less than 1% of the time! And let’s face it, critting is FUN. So, yes, you’re not going to miss as often, but you’re also giving up the rush you get when you roll a natural 20. That’s a tradeoff I don’t think a lot of people are willing to make.

Now, I’ve given this a lot of thought, and I feel that there can be a place for the 4d6-3 mechanic at your table. I absolutely do not believe that it should be used in combat. The bonuses and penalties thrown around like candy in your average combat situation would wreak havoc. Plus, you’d almost never crit. So, when should you use it? For skill checks, of course! Here are four ways you could try using 4d6-3 in place of your d20.

**To replace the “Take 10″ mechanic**

You could have your players roll 4d6-3 when they would normally take 10 on a skill check. This would yield roughly the same result as taking 10 (most of the time you’d roll pretty close to a 10), but sometimes there would be a great success, or terrible failure. Think of it as a way to add an element of chance to taking 10, but not the same wild chance that you get from rolling a d20.

**For trained skill checks only**

Think about it: two characters could have the exact same skill bonus, but one’s bonus could be from training in that particular skill, while another’s could be from innate ability (ability score + racial bonus). From a purely “simulationist” standpoint, it doesn’t stand to reason that someone who is trained in something has the same chance of success as someone who just kind of has a “knack.” You would think that a character trained in a skill would perform around the average most of the time due to their training, with the occasional spectacular success or failure. On the other hand, one might expect an even amount of success and failure from a character who was performing a skill out of innate ability – there’s been no training in the finer points of the skill, or pitfalls to watch out for. Let those who have training in a skill roll 4d6-3, and have untrained checks be rolled with a d20 to reflect the difference between someone who is trained at something and someone who is not.

**As a DM reward**

Instead of handing out +2 to a skill check, why not occasionally give players the option of rolling 4d6-3 instead? This gives the players a decision point (+2 or 4d6-3?), and also gives you something else to hand out besides a +2. While this is like comparing apples to oranges (“+2 to a d20″ is nothing like “+0 bonus to 4d6-3″), it does give the DM another way to spice up what may be “just another skill check.”

**For any skill check**

You could give players the choice of rolling 4d6-3 for any skill check. Be careful with this one though; as stated before, the game (including skill checks) was built with the d20 in mind. Most players, especially those with decent skill bonuses, would always choose to average 9-13 on every skill check. If you do go this route, it would be wise make the choice more meaningful – the easy solution is to incentivize using the d20. For example, you could add a “crit” mechanic to skill checks for natural 20’s – a character’s “amazingly inspiring, perfect execution” of a skill could give another character +2 or even +4 to their similar skill check, but only if that second character also rolls a d20.

I hope that I have gotten you to consider using this mechanic in your game. It presents a lot to think about, and could really add some spice to the normal d20 roll. It won’t, after all, ever replace the d20, but it’s nice to know there is more than one way to skin the proverbial cat.

Would you ever consider introducing this mechanic to skill checks? Which option would you choose?