This article is dedicated to all the freelance RPG writers to have felt the compulsion to write their own game and have considered many types of die interpretation. I hope you find this brief treatise informative and helpful in your work. All writing and mathematics included is my own. Thank you for reading!

One-Dice Target
   The one-dice target method is by far the most used method of rolling in the gaming world. A character has a target number which they need to roll to succeed, either their own statistic or a number set by the GM. This is most often a d20 as it allows for nice, linear 5% chance increments and steady, if sometimes slow growth. The d20 can be supplemented with the occasional d100 roll, but games that have only percentile statistics and encourage character growth can be dismally boring if growth is done one digit at a time. The success odds are simple:

Where T is the target number and M is the maximum roll allowed on the die.

Opposed Checks
   Opposed checks occur when players roll a die and add it to a score or modifier set; the higher number then wins.

D = difference in base scores or modifiers
M = maximum roll possible on one die

Assuming that a tie goes the reaction's way and the action has a higher modifier than the reaction:

If the reaction has a modifier than the action:

If a tie benefits the action, then the two equations are as follows:

In addition (from Nwash):

[Ties are] complicated, since it depends on what a tie means (which is another level in my system, since my statistics have an edge that determines what happens on a tie. Sometimes it favors the attacker, sometimes the defender, and sometimes they get a 50/50 tiebreaker. Essentially, the guy with the higher edge wins on a tie, and if the edges are equal, well, then, they break the tie by a 50/50 method. With the 50/50 tiebreaker, it looks like removing the last two terms from the numerator solves that.) Also, the chance of a tie would be (M-D) / M^2 * 100% according to my data.

The graphs of both equations of D vs result, assuming M to be positive and constant, are upside-down parabolas of slightly different slope both intersecting the x-axis at (M±1)/2M, based on the equation, which represents the action's success chance of equal opponents. This is a good number to determine with various dice to see which dice you wish to use. (e.g. by d20s, the defender wins 52.5% of the time on equal footing with the tie benefit). Furthermore, the graph is fairly even over the low difference ranges, experiences sharp increases and drop-offs around the quartiles and then evens out again near the end. This is my rolling method of choice for many games since the GM can easily alter NPC and environmental difficulties to suit the story. The challenges arise when one character begins to gain attributes which are 5 and 6 points over his or her comrades; the GM would need to start producing better opponents here but they will slaughter the weaker comrades. This system is also good for games that have little character statistic alteration and a lot of character power variance, like supers games.

Rolling Under (or Over) on Multiple Dice
2 Dice:
   The chance for any sum (N) to be rolled on two dice:

This is simple for a sum up to M+1; where:

gives the chance to roll at or under sum N. If rolling sum N fails, subtract 1 from Q and recalculate. Subtracting this result from 100 gives the probability to roll over this sum.

The calculation is slightly more complex once N>M+2 since the Q-values begin to fall again back toward 1:

Substituting (M-|M+1-N|) back in for Q can be done, but it is very ugly and in my opinion, this setup is much easier.

The two-dice sum method produces a very familiar and comfortable curve. The most rapid change in odds lies in the middle range near (M+1) and the ends plateau out. A two-dice sum method, most prominently set to "roll under" a value, is good for short-term growth games, allowing characters to gain noticeably more power in a short period of time, but then be curbed by the maximum roll value, unlike the Opposed Check system where characters can get sometimes seem unrealistically powerful. It is most common seen with d6s and sometimes d8s. The downside is in its lack of staying power. After a fairly short period of time, everyone will get to the same power level and stagnate. This may not be a problem for games with a high body count or sparse character statistic growth, however.

3 or More Dice:
   I first attempted to compute this with equations, but it became extraordinarily difficult and not efficient. In the meantime, I noticed some unusual patterns and came up with a simpler, if somewhat surprising method.

It seems that our problem was almost solved hundreds of years ago by our friend Blaise Pascal. Pascal's Triangle, shown here, makes these calculations fairly quickly.

Each diagonal represents an increasing number of dice to be rolled. The diagonal 1,2,3… is two dice, 1,3,6.. is three dice and so on. For this example, I will show the odds of 13 on 3d6. Start from the top of the pyramid and move down along one side or the other once for every die over the first, and then from there, count along the diagonal until you reach the number desired. In my example, I start in the 1,3,6... diagonal and count starting with 3 until I reach 13 (66). This is the unmodified value. However, the Triangle assumes you have all numbers to work with and dice have only limited sides. In order to account for this, count backward M (the maxmimum roll of one die) along the same diagonal. Multiply that value by the number of dice rolled and subtract from the first value. (66-3*15 = 66-45 = 21) I will call this value V. The odds of rolling exactly any number on D dice are then:

Interestingly enough, the odds of rolling 13 or under on 3 dice are the same as rolling exactly 14 on four dice. Thus to determine the "at or under" number (V1), perform the above calculation on the two numbers diagonally toward the inside (in this case, down and toward the inside) of the Triangle. (at or under 13 on 3d6 is 286 - 3*35 = 181 out of 216).

Overall, a three-dice sum method is much like a drawn-out version of the two-dice sum method. It is used in games like GURPS to allow for noticeable growth over a much more extended period of time, although not as much so as the one-dice target method. It is simple to learn and understand, although it also often uses only d6s and gamers sometimes long for the odd, quirky dice that make gaming so unique.

Roll Maxing on Multiple Dice
   In recent history, many games have adopted the idea of rolling many dice and counting only the dice that show maximum values. Players will roll a set number of dice (again, generally d6s) and count the number of maxed dice to determine the outcome. This is another simple method that is intuitive and quick. Determining the odds of rolling X maxed dice is as follows:

This method will count rolling three 8s on 3d8 as three different rolls. It should be understood that there is really only one roll though it gets counted strangely in the math. I do not have much familiarity with the actual effects of this system, although from the math, I would guess that it tends to create large power gaps between characters who roll different numbers of dice and would exhibit unusually (even for gaming) wild swings in the consistency of success rolls; think for a moment of the difficulty in getting a Yahtzee and apply that to fighting off that 6-dice Drow over there...

I hope this document has been useful and entertaining. Big thanks to Nwash for editing, checking my math, and publishing! Please email me with any feedback.