Over the last two months, we looked at systems where you roll a single die, add a small number of identical dice or have a situation-dependent pool of dice. This month, we will look at some systems that don't fall into these categories.

Letting ability determine dice-size

The original Sovereign Stone game from Corsair Publishing (before the game was assimilated by the d20 Borgs) used an unusual dice-roll method: Attributes and skills were given as dice types. So an attribute could range from d4 to d12 (with non-human attributes of d20 or d30 possible) and skills could range from 0 (nonexistent) through d4 to d12. When attempting a task, you would roll one die for your relevant attribute and another for your relevant skill and add the results. If the sum meets or exceeds the difficulty of the task, you succeed. The more recent Serenity RPG by Margaret Weis uses a similar system, but starts from d2 instead of d4, and when you go past d12, you go to d12+d2, d12+d4, etc., instead of d20 and d30. In both systems, ratings over d12 are exceptional.

An advantage of this system is that you (until you exceed a d12 rating) only do one addition to make a roll that takes attribute and skill into account, where adding skill and attribute to a die roll requires two additions. Additionally, the numbers you add are likely to be smaller, which makes addition faster.

The disadvantage is that you need to have all types of dice around, preferably at least two of each. Additionally, the range of dice gives only five (or six) different values for attributes in the unexceptional range. This is fine for many genres, but not for all.

It gets a bit more interesting if we look at the average and spread of results as abilities increase. If you add a dm and a dn, the average is ^(m+n)/₂+1. The spread increases slightly faster than linearly in the average, which means that more skilled persons have higher spread -- even relative to their average -- than persons of lower skill. The main visible effect is that even very able persons have a high chance of failing easy tasks (by rolling low on both dice).

This observation has made someone suggest that higher abilities should equate smaller dice and low rolls be better than high. Though this makes higher skilled persons more consistent and prevents them from getting really bad results, it gives novices a fairly high chance of getting the best achievable result ("snake eyes"), which may be a problem if you want to make sure extremely able characters will always beat fumbling amateurs.

A system similar to the Sovereign Stone / Serenity system is used in Sanguine Productions' games, such as Ironclaw and Usagi Yojimbo. Here, three dice are rolled: One for attribute, one for skill and one for career. However, instead of adding the dice, each is compared against two dice the GM rolls for difficulty. If one of the player's dice is higher than the GM's highest, the player succeeds, if two are higher, the player gets an overwhelming success. If all are smaller than the GM's smallest die, the player gets an overwhelming failure (the remaining cases are normal failures). Like the Sovereign Stone / Serenity system, you get increased spread of results with higher abilities. Also, since difficulties are rolled rather than being constants, high difficulties can sometimes be quite easily overcome (if the GM rolls low). All in all, this makes results quite unpredictable, with experts sometimes failing at simple tasks and novices sometimes succeeding at complex tasks. Again, this will fit some genres, but not all.

Median of three dice

The usual way of getting bell curves is by adding several dice (or counting successes, which is more or less the same), but you can do it also by comparing dice. A simple method is to roll three dice (e.g., d20s) and throw away the largest and smallest result, i.e., pick the median (middle) result. The advantage of this method is that it requires no addition, so it is slightly faster than, say, adding 3d6. We will use the abbreviation "mid 3dn" for the median of three dn.

We can calculate the probability of getting a result of x with mid 3dn by the following observation: The median is x either if either two or three dice come up as x, or one dice is less than x, one is equal to x and one is higher than x. The probability of all three coming up as x is ¹/n³. The chance that exactly two come up as x is 3×(¹/_n²)×^(n-1)/_n (the 3 comes from the three places the non-x die can be). The chance that there is one less than x, one equal to x and one greater than x is 6×^(x-1)/_n×¹/_n×^(n-x)/_n (the 6 comes from the 6 ways of ordering the three dice). We can add this up to ^{(3n-2+6(x-1)(n-x))}/_(n³). For example, the chance of getting 7 on mid 3d10 is ^{(3×10-2+6(7-1)(10-7))}/_(10³) = ¹³⁶/₁₀₀₀.

Since the curve of mid 3dn is a parabola, it is arguable whether it can be called a bell-curve (it lacks the flattening at the ends), but it is a better approximation to a bell-curve than adding 2dn. You can get closer to a "real" bell curve by taking the median of 5 dice instead of 3.

For a given range of values, the curve obtained by taking the median of three dice is somewhat flatter than what you get by adding three dice, while the one you get by taking the median of five dice is slightly steeper than that of adding three dice. For example, all of the rolls below have identical ranges from 1 to 10 and identical averages of 5.5, but their spreads differ:

The value of the median roll is typically used in the same way as the value of a single die or the sum of a few dice.

If the value of the median roll is compared to ability (i.e., you must roll under your ability), the method is a special case of the method where you count how many of three dice meet or exceed the difficulty rating, except that you don't distinguish degrees of success and failure: If at least half of the individual dice meet the target, it is a success, otherwise a failure.

A variant is to combine median rolls with the idea of letting abilities equal dice types. So you would roll one dice for your attribute and another for your skill, but what about the third? You can add a third trait (like the career in ``Ironclaw''), you can let the third die always be the same (e.g., always a d10), or you can duplicate the skill die, so you roll two dice for your skill and one for your attribute. This makes skills more significant than attributes and limits the maximum result to the level of the skill. Regardless, you still have the effect of spread increasing with ability that you get when the dice-sizes increase with ability.

Adding a subset of the rolled dice

In d20, it is common to generate attributes by rolling four d6 and then add only the largest three results. This generates an asymmetric bell curve but retains the same range of results as just adding 3d6:

 sum 3d6:
 
    Value   2.16 bars per %
       3 :  |
       4 :  |||
       5 :  ||||||
       6 :  ||||||||||
       7 :  |||||||||||||||
       8 :  |||||||||||||||||||||
       9 :  |||||||||||||||||||||||||
      10 :  |||||||||||||||||||||||||||
      11 :  |||||||||||||||||||||||||||
      12 :  |||||||||||||||||||||||||
      13 :  |||||||||||||||||||||
      14 :  |||||||||||||||
      15 :  ||||||||||
      16 :  ||||||
      17 :  |||
      18 :  |
 
 Average = 10.5
 Spread = 2.95803989155
 Mean deviation = 2.41666666667
 

 sum largest 3 4d6:
 
    Value   2.16 bars per %
       3 :  
       4 :  |
       5 :  ||
       6 :  ||||
       7 :  ||||||
       8 :  ||||||||||
       9 :  |||||||||||||||
      10 :  ||||||||||||||||||||
      11 :  |||||||||||||||||||||||||
      12 :  ||||||||||||||||||||||||||||
      13 :  |||||||||||||||||||||||||||||
      14 :  |||||||||||||||||||||||||||
      15 :  ||||||||||||||||||||||
      16 :  ||||||||||||||||
      17 :  |||||||||
      18 :  |||
 
 Average = 12.2445987654
 Spread = 2.84684444531
 Mean deviation = 2.31853947569
 

Note that the graphs (which were made with Troll) are turned sideways compared to the usual presentation. The main difference between the two distributions is that the very high values are much more common largest-3-of-4 roll than it is with an unmodified 3d6 roll. Rolling 18 is, for example, 3.5 times as likely.

You can generalize this method to rolling n+m dice, select n of these and add them. The result will have the same range of values as just adding n dice, but you can modify the distribution. Selecting the lowest dice will slant the distribution towards low values and selecting the highest dice will slant it towards the high end of the range. You don't have to select the largest or smallest dice, you can, for example, add the smallest and the two largest of 4d6 or discard the smallest and highest and add the rest (which shows that this is a generalisation of the median-of-three method described earlier).

Odds and ends

I'm sure that there are interesting dice-roll methods that I haven't covered. If any of you want a particular system discussed, drop me a line and I will look into it.

Next month, I will look at how scale can be integrated in dice-roll systems and some alternatives to using dice. I will also tell you some of my personal dislikes, so I will probably need to don my asbestos suit. :-)