The ubiquitous six-sided die, or "d6' has been the mainstay of gaming for decades. Even the advent of role-playing in the late 1970s, with the sudden explosion of dice options, has never really unseated it. As the years passed, many people proposed abandoning the d6 altogether, usually in favor of the c110, but still the lovable cube hangs on to the number one spot. It's a bit like McDonald's: everybody knows that there are better options out there, yet somehow everybody still keeps coming back.
Geometry limits us to a handful of possibilities: the d4, d6, d8, d10, d12, and d20. I've seen others from time to time, including an absurd, almost spherical d100 that never stopped rolling. (In ancient Syracuse, Archimedes invented the "Chiliagon," a perfect 1000-sided polygon, but I've yet to see a Chili-Die.) Some games, like Piquet, try to take advantage of the whole spectrum of dice. But most game designers Choose one die and stick with it, for the simple reason that once play is underway, people would rather say: "OK, hand me the dice" than: "OK, I need a d12... No, that's a d8... No, that's a d10... hmm, where's the d12 ... hmmm.' Too many different kinds of dice clutter the table and slow things down.
But if we're going to choose one kind of die, which kind will it be?
The biggest complaint most people have with d6s is that they don't give enough options. They break down chance into six big chunks of roughly 17% each. That doesn't allow for a lot of finesse. What if you want something to have only a 5% chance of happening? True, most gamers don't want to bother with something that has only a 5% chance of happening, but there are instances when it is defensible. Officer casualties, for instance. Or the malfunction of critical systems. A single d6 roll does not allow for such fine-tuning. When using a single d6, there is a 17% chance of something happening, or a 33% chance, or a 50% chance, or a 67% chance, or an 84% chance, or a 100% chance. That's it.
Another problem game designers often encounter with d6s is their limited ability to accept die-roll modifiers. Game-designers love modifiers, which is a problem in the limited thinking of game-designers, not the geometry of dice (more on this later). Nonetheless, it can t be argued that modifiers suit d6s very well. A single +1 modifier increases the chance of something happening by a whopping 17%. And there is simply no way to add more than +4 modifiers without the die roll having become completely moot.
Let's take a look at the three traditional uses of the c16 and what is wrong with them:
The Old-Fashioned CRT
Since the days of Tactics II, game designers have created tables which cross-indexed a condition with a die roll, to achieve a result. The condition was usually numerical odds. If you have 2-1 odds, and you roll a 4, the result is "Defender Retreats." This is the simplest possible application of the die: to choose from a one-dimensional list of ossibilities. If those possibilities are arranged in order of severity, then modifiers can e used. If, for instance, a I is always the least-attractive result for the attacker, and a 6 is always the most attractive result, then every +1 modifier increases the odds of a favorable result. Amazingly, after half a century of gaming, the vast majority of results tables in wargames still look like this.
Rolling 2d6 Somewhere along the line, game designers discovered that they could roll not one but two d6s, and add the results. Eureka! If memory serves, the first big application of this was in the Squad Leader series, starting in 1981, although Dungeons and Dragons had been rolling 3d6 for character attributes since the mid-70s. A 2d6 system opened a whole new realm of possibilities, but also new problems. Now we encountered the dreaded "Bell Curve.'
| The d6 Bell curve | |||
|---|---|---|---|
| Die Roll | No. Out of possible 36 | Raw % occurring | Chances of
rolling less than or equal |
| 2 | 1 | 2.7 | 2.7 |
| 3 | 2 | 5.4 | 8.1 |
| 4 | 3 | 8.3 | 16.4 |
| 5 | 4 | 11 | 27.4 |
| 6 | 5 | 13.8 | 41.2 |
| 7 | 6 | 16.6 | 57.8 |
| 8 | 5 | 13.8 | 71.6 |
| 9 | 4 | 11 | 83 |
| 10 | 3 | 8.3 | 91 |
| 11 | 2 | 5.4 | 97 |
| 12 | 1 | 2.7 | 100 |
As you can see, the probabilities cluster up around the results from 6-8. Any time you roll two d6s, there is about a 45% chance that your roll will be from 6-8. That's a huge chunk of probability sitting smack in the center of the curve. And as Squad Leader demonstrated, there is a BIG difference between a Morale rating of 7 and a Morale rating of 6. Traditionally, games that use two dice ask you to roll "Equal to or less than' some number. If you have to check morale for your Morale-6 unit, you have only a 41% chance of passing, while your Morale-7 unit has a 58% chance.
Obviously, there are many ways to use 2d6, but gamers tend only to think of two. The first, as I've just shown, is to roll against a target number of some kind, hoping to score less than or equal to, or greater than, or whatever. The second is to apply the 2d6 against a table, i la the old CRT system. Amazingly, game designers often neglect the Bell Curve when they do this, not realizing that the way they distribute numbers will determine the 2-3 results that will keep re-occurring on that table.
Finally, the 2d6 system has the opposite problem with regard to modifiers, than does the 1d6 system. Because of the Bell Curve, adding +1 or -1 die roll modifiers simply shifts the center of the Bell. It doesn't change the basic shape of probability. If you roll 2d6 with a +2 modifier, then your result will probably be 8-10. Remember this if you're making a table based on 2d6: a single die roll modifier will change the result you encounter in nearly half of all rolls!
Great Fistfuls of Dice (GFOD)
Many games use a system whereby a units combat strength is equated to a number of d6s it can roll. Each of these dice must then roll a "hit" by scoring a certain number(s). There are usually quite a few dice to be thrown, and thus the system has acquired the nickname GFOD. This system has the advantage of making it possible for vastly outmatched units to sometimes beat their opponents. Just because you roll GFOD doesn't mean you'll roll the numbers you need. If you roll twenty d6s and need sixes to score hits, there is always the remote possibility (about 3%) that you won't roll a single six on any of those twenty dice. Meanwhile, your puny opponent might get lucky and hit you with both of his puny dice. In a word, this is a fun way to do things, and fun counts for a lot when playing games.
I've tried a few GFOD systems that use d10s, but they never really caught on. As far as I know, all published games that use GFOD use six-sided dice. And d6s aren't really that well suited for this system if your game has any level of complexity to it. The problems come when you try to decide what the "To Hit" numbers will be. If you need sixes to hit, then how do you represent the fact that an enemy is protected by Hard Cover? How do you scale back? You could halve the number of dice rolled, but that's awkward and ruins the whole premise of GFOD. (In general, with GFOD, you don't want to mess with the number of dice rolled.) So most designers set the To Hit number at something like 4+, and then offer penalties for certain circumstances. Your unit is Disordered? Then you only hit on 5+. The enemy is behind Hard Cover? Then you only hit on 6+. But here we encounter the rigid shapes of d6 probability. If you're rolling ten dice, needing to hit on 4+, then you should score 5 hits. But if you're rolling ten dice, needing to hit on sixes, you'll score only one, maybe two hits. That's a huge disparity in performance, and makes it difficult to show gradual or slight differences.
Many game designers have inserted "Saving Throws" as a way of extending the probability curve of GFOD, but those are equally limited by the geometry of the dice, and have the added distraction of requiring another die roll. So I need a 4+ to hit, but your Soft Cover gives you a 5+ To Save? Then my net chance of hitting you is 50% X 33% = @17%. And we're back to a one out of six chance. (Any time you multiply a number divisible by six by another number divisible by six you get... a number divisible by six. What a surprise.) So why couldn't we have just done that with one roll?
So, in light of all these problems using six-sided dice, shall we just dump the venerable beast altogether and do as Wally Simon has suggested: use dlOs for everything instead? Well, sure, if you like. You'll be trading 17% increments of chance for 10% increments, and you'll have a broader, flatter Bell Curve when you roll two dice. If that's what your game needs, then by all means do it.
But before you jump ship altogether, I want to make a few suggestions. The rroblems with d6 are not just geometrical. The biggest problem is that game designers aven't been very creative in the way they use dice or construct tables. There are all sorts of interesting tricks we can perform with d6s.
1.Hexidecimal Dice
For years we've thought of d10s as "percentile dice." A colored die is the tens digit, and a white die is the ones digit, and thus a roll of 6 and 4 means: 64. But it is very rare to find a game which really needs one hundred different possibilities. Most often, percentile dice are simply used against a flat, one-dimensional table. If you roll under 40%, for instance, then your unit rallies. There might be some modifiers thrown in, but all the fancy possibilities of the percentile dice are ignored, and they're used instead for a simple Yes/No determination.
But d6s can be percentile dice, too. And this really starts to get interesting, because now we have not 100, but rather 36 possible outcomes. Do you need to break down chance into increments of roughly 3%? Try one colored d6 and one white one, and rolling them just like percentile dice. A 2 and a 5 equals 25. The span of possible numbers looks like this:
11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
It might be a bit weird at first to think that the next number after 16 is 21, but I'll be damned if it doesn't work! Do you want to show a roughly 40% chance to rally? No problem: the unit needs to roll 25 or less on two dice.
This system does not lend itself to the traditional system of modifiers (but I hate the traditional system of modifiers, anyway - more on that later.) It's hard to add a +1 modifier to a roll of 16 on hexidecimal dice. In your mind, you want to make it 17. The best rules and tables have factored in all possibilities up-front, and don't use modifiers at all. But if you need modifiers, here's another possibility....
2. Dual/ Linear Dice
Keep the hexidecimal model in your head for a moment, but consider a new use for it. Let's say that you need a table that shows how many hits a unit inflicts when it fires. You could specify a number that must be rolled, given the circumstances. Remember - try to get away from using modifiers. Incorporate all the factors urfront into this 'To Hit" number. Now we'll say that the goal is to roll higher than eac digit on the two digit dice.
For instance, if the number was "23" then you'd have to roll higher than 2 on the blue die and higher than 3 on the white die. For each digit over on each die, a hit results. Thus, if the number is 65, there's only one chance out of 36 of scoring a single hit: you'd have to roll a "6" on the white die. If the number is 55, then there's two chances for one hit, and one chance for two hits. Let's say the"To Hit" number is: 43:
Chance of no hits: 67%
1 hit: 2 chances (roll a 53 or 44): 6%
2 hits: 4 chances: 12%
3 hits: 3 chances: 9%
4 hits: 2 chances: 6%
5 hits: you'd have to roll boxcars, thus: 3%
Do you see the weird new probability curve we've created? The most likely result is "No Hits," but if you do hit, then the median result is actually two hits, not one. You're twice as likely to score two hits (12%) as to score one (6%). Play with this system and experiment, and I think you'll see that it has all kinds of useful and unexpected applications.
You can manipulate this system with modifiers, too, if you must. Perhaps one type of modifier only applies to the blue die, and another type only to the white die. Or you can throw in various monkey wrenches that simulate malfunctions or disasters, without having to make the player roll a second time. For instance, if both dice roll below their respective To Hit numbers, then something has gone wrong.
3. Dice as Tables
Here we abandon the old notion that the die's function is simply to tell which possibility we hit on a table of pre-determined possibilities. Again, the idea is to factor up-front all the contingencies you'll encounter and want to model. Let's say you're designing a table to show the results of artillery fire in the horse-and-musket era. What factors must you include? Crew quality? Distance to target? Ground conditions? Number of guns firing? Size of guns firing? Type of target? Difference in elevation? The target's cover?
Step one is to break these factors into relative numerical values. Which conditions affect each other most directly? You might decide that crew quality, and the number and size of guns can all be considered related in terms of sheer firepower. Perhaps you include distance to target in that, perhaps not. Meanwhile, ground conditions, elevation, and the target type and cover are a different issue - they determine how effective that firepower will be when it reaches the target.
So now you have an X value and a Y value. There are a zillion different ways to organize them, but you should decide whether or not you want an "average" result. If so, then you'll want to use a Bell Curve. If not, then you'll want either a single die, or decimal dice - an even distribution of chance. Most designers want an average result: a grand battery of Old Guard 12-pounders ought to be able to blast the average target to smithereens. So down the vertical spine of the table you use a standard 2d6 - the firepower has an "average" value. And across the horizontal you might want a straight percentage distribution representing the target's odds of surviving the fire (that's the stuff like cover, ground condition, etc.) So in the end you'll be rolling three dice: two white ones and a blue one, and instantly looking at the result.
This all sounds more complicated than it is, because we're looking from a gamedesigner's viewpoint. The average player simply looks for his numbers, rolls his dice, and sees the result. No modifiers needed - a clean roll of three dice.
4. The Dice as Benchmark
Instead of using a table to arrive at a kind of "to kill" number, and then rolling against that, we can shorten the process without losing any of the factors. Let's say you've established your To Kill (TK) number, and now you want a randomizer to see how well the unit actually performed. In this case the TK number serves as a minimum - the bottom of the average that you're hoping to simulate. You then roll against it. Since a single d6 is too small for much variation, let's roll 2d6 against a TK number. Let's say that the TK number is 7, and anything above that scores a number of hits equal to the difference. In other words, if you roll a 10, you've scored 3 hits.
The Bell Curve does interesting thincis with this s . ystem. Yes, it's true that the lower the TK number, the more hits youl score, but its not a linear progression.
| (rounded) Percent Chance of getting... | ||||||
|---|---|---|---|---|---|---|
| TK# | No hits | 1 hit | 2 hits | 3 hits | 4 hits | 5 hits |
| 4 | 16 | 11 | 14 | 17 | 14 | 11 |
| 5 | 27 | 14 | 17 | 14 | 11 | 8 |
| 6 | 42 | 17 | 14 | 11 | 8 | 5 |
| 7 | 59 | 14 | 11 | 8 | 5 | 3 |
| 8 | 73 | 11 | 8 | 5 | 3 | - |
| 9 | 84 | 8 | 5 | 3 | - | - |
| 10 | 92 | 5 | 3 | - | - | - |
It's interesting to watch the probabilities shift on this scale. Notice how there's never more than a 17% chance of any one type of hit result. With a very low TK number, you're more likely to score three hits than to score one. This type of probability curve would be very useful for a limited number of outcomes; you're trapped by the 2d6 scale. But if you can work within those confines, this system has the great advantage of being very fast, and not requiring any tables at all.
5. Get the Dice out of the Margins
Thus far we've only looked at the traditional one-dimensional arrangement of using dice as determining factors along the frames of the tables. Don't forget that you can reverse this arrangement and stick the dice in the middle of the tables, with the results at the edges. This allows for a more two-dimensional look at results. Here's an example from my latest game project. This game gives units "Strength Points" (SP) which are lost in combat and by other ways, but may be regained under certain circumstances. When you try to regain them, you roll one d6 and add it to a factor based on that unit's type. A Guard unit, for instance, might give you a 5, whereas a militia unit might give you only a 2. So you'd take that number, add it to your die, and then look at the table to see how you've done:
| SP Recovery Table | |||||
|---|---|---|---|---|---|
| SPs Recovered | SPs to Recover | ||||
| 2 | 3 | 4 | 5 | 6+ | |
| None | 0-6 | 0-4 | 0-3 | 0-2 | 0-1 |
| One | 7+ | 5-7 | 4-6 | 3-5 | 2-4 |
| Two | - | 8+ | 7-8 | 6-7 | 5-6 |
| Three | - | - | 9+ | 8-9 | 7-8 |
| Four | - | - | - | 10+ | 9-10 |
| Five | - | - | - | - | 11+ |
Along the top are the number of SPs you're trying to recover. Down the left spine are the number that you DO actually recover. Inside the table itself are your dice results. So let's say yciu're trying to recover 5 SPs, and your roll totals "7." From the 5 column, go down until you hit the "6-7." Then go left to find out how you did. Not very well, it seems; you only recovered two.
6. Avoid multiple rolls when one roll will do
Some games allow for minute chances by using multiple rolls of d6s. Do you want to simulate a rare possibility? Roll a d6, and if you get a 1, roll again. On another 1, your general is dead. The possibility of the event is now whittled down to .17 X .17, which equals .0289, or roughly 3%. The Warhammer games take this to extremes in their combat system. In general, this is a bad idea. Whenever possible in a game, things should be decided by a single die roll. Time is everything in a game, especially a game with multiple players. You want the results to be as quick, clear, and decisive as possible.
7. Get rid of modifiers, as much as you can
How many times have you seen a game with a wonderfully simple-looking Combat Results Table, and then a whole bloody page of modifiers. Sometimes there are so many modifiers that they have to be broken down into categories: Modifiers for formation, Modifiers for terrain, Modifiers for numerical odds.... If a game is using this many modifiers, then the designer has failed to design a qood table in the first place.
A good results table should be able to encompass all the results which are possible. It should have factored them all in from the outset. Whatever modifiers are still needed should reflect a minor adjustment here or there, and only because the modifier represents something oddball that doesn't deserve to be incorporated into the table design. For instance, let's say that all artillery crews have the same basic effectiveness / rate of fire / ammunition supply / whatever, except the French Old Guard, who are somewhat better. This is a prime example of a justifiable modifier: give the Old Guard a +1. They're the only exception to a whole category.
A BAD use of modifiers is when you use a whole list of them for something that should have factored into the basic design, like numerical odds, for instance. There are too many combat tables that tell you to roll against each other for results, but then add +1 if you have 3:2 odds in your favor, +2 for 2:1, +3 for 3:1, and on and on, not to mention modifiers for formations, weather, and so forth. The solution here is to use a more sophisticated table design, not to stick with the old one-dimensional model and heap on the modifiers for all those things you wanted to include in the table but didn't.
8. Don't Forget the Law of Accumulating Probabilities...
I said earlier that most gamers don't want to bother with something that has only a 5% chance of occurring. These things are usually reserved for exceptional events, such as officer casualties or system malfuntions. Bear in mind, however, how many times your game will require a player to check for something like that every turn. Do you require the player to check his big gun every time it fires? If so, then even a puny 5% chance begins to add up pretty quickly. Let's say you only want a unit to hit only on a roll of "1." It seems remote that this unit will hit anyone. Yet if you try four times, the cumulative chance that you succeed is 52%.
There are three things you should consider when writing rules that cover remote possibilities. First, is the possibility so remote, and the consequences so negligible, that it's even worth it to simulate it? For instance, if an officer casualty simply results in another officer taking over, then why bother to clutter the game with this rule at all? If everybody's gun malfunctions roughly the same number of times, and has roughly the same chance of being repaired, then is it really worthwhile to model it?
Second, how many times will a player have to check during the course of a turn or a whole game? If the cumulative probabilities are even modestly high, then perhaps your base chance is too high. For instance, if a game lasts twelve turns, and the average officer in the Union army has to check for a casualty once, on average, per turn, then there is a very strong probability that by the time the game ends every single Union officer will be dead! That's a big pile of dead Yankees. (For some of my friends here in Charleston, that translates to 'Good Game Design.")
And third, will there be any modifiers added to the base probability? In one of my games, for instance, some officers get a (V) bonus for Valor. They're brave and often lead from the front, giving their units an advantage in combat. That, however, also gives them a "bonus" when checking to see if they got killed, too. One must be careful with this. When dealing with accumulating probabilities, even a small increase in the base chance results in a massive increase in the chances after repeated attempts. What looks like a tame difference can have the result that a fire-eater like Marshal Ney has about a 90% chance of getting himself shot by the end of the battle, whereas a "normal" general has only a 30% chance. Is that really the probability spread you were looking for? Then kindly explain how Ney and all those other fire-eaters managed to survive so many battles!
And Finally: Use as few Tables as you possibly can In most game designs there are quite a number of things which call for the application of random chance. That does not mean, however, that they all need tables. In fact, if you are looking for an Either/Or or Yes/No result, then you most definitely do not need a table. This may seem obvious, but consider how many games are cluttered with charts and tables to tell us something we could have figured out with one die roll and our own heads.
Empire is an infamous culprit in this regard, though there are many others. On one table where players must roll to see if their artillery has re-located or not (a Yes/No answer), French horse artillery is given a 130% base chance of success, which then might be whittled down in increments of 10% given fatigue and casualties. In other words, for there even to be a possibility that French horse artillery will fail, the unit in question needs to be badly shot up and exhausted. Is this really worth modeling with a table? Can we not simply state an either/or rule that covers such a contingency?
Over the years my games have gotten more and more randomized, and nowadays I make players roll to do things that most games (including my own, older designs) used to allow them to do automatically. In fact, I am now gravitating toward forcing players to roll to do things, whether they want to or not, with the idea that the commander's role is to try to keep a chaotic situation under control as best he can. Most games, I think, make the commander a kind of orchestra conductor, whose worst problems involve one or two instruments getting a bit lost here and there. I try to simulate the exact opposite - a total cacophony, with the commander trying desperately to make some kind of tune out of it. So my games tend to use a lot of dice.
That means, of course, that there are many different contingencies which need to be modeled in many different ways. Some things need to sit on an average, while others can be a flat spread of probabilities, with any one result no more and no less likely than any other. These are the kinds of things you need to consider when designing game rules. I don't think that the d6 is past its prime. I don't have anything against using different kinds of dice, but I think gamers would be well advised to get a bit more creative in the ways they use probabilities, and they would probably find that the shape of the dice aren't nearty as crucial as the way the game is designed.
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© Copyright 2001 Hal Thinglum
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