# Useful Notes / Dice

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Dicenote  are ubiquitous low-tech randomness generators that have been in continuous use as gaming aids for well over four thousand years. Their basic design exploits the inherent imprecision of human movement (of the hand rolling the dice) and various isohedral symmetries (most commonly, those of a cube) to produce a uniform distribution of random numbers, which, in turn, can produce near-normal (a.k.a. "bell curve") distributions by rolling many dice at once and summing up the results.

## Notation

To simplify the notation of various dice rolls, following convention has been universally adopted by gamers:

 Notation Meaning Example dX A single X-sided die ("d" stands for "die"note ) d6 NdX "Roll N X-sided dice and sum up the results" 2d10 NdX+M "Roll N X-sided dice, sum up the results, and add a flat value M" 3d6+5 Nd Used to save space in games that only use one type of dice 5d

The dice notation has long become one of the symbols of gaming and of role-playing games in particular, especially ever since Wizards of the Coast had trademarked their d20 System.

## Shapes

Counter-clockwise from the bottom:
d4, d6, d8, d10, d12, and d20 note
A fair die is one that has the same chance of landing on each face, which is naturally achieved by convex isohedrons. That said, cheap plastic dice used in most games are not particularly fair and can easily land on a particular face twice or half as often as they should due to manufacturing flaws. This simple fact is behind all popular superstitions about "lucky d20s".

The most popular die shape is the six-sided cube, followed by the rest of the Platonic solids and d10s, followed by other trapezohedrons and bipyramids, followed by the Catalan solids. The most common dice (which, incidentally, are all used in Dungeons & Dragons) are:

• d4 (tetrahedron): The only common die that cannot actually roll or land with one side up. Instead, you usually toss it in the air and after it lands, read the number written either near the vertex or near the face opposite to it, depending on design.
• d6 (cube): Easily the most common type, often synonymous with the word "dice" itself.
• d8 (octahedron)
• d10 (pentagonal trapezohedron): The only common die that isn't a Platonic solid. Often has a "0" face instead of a "10" on it, which is read as a ten in d10 rolls, but as a zero in d100 rolls (see below). For d100 rolls, dice kits often come with another d10 with the faces "00" to "90"; rolling a "0" and a "00" can be read as a zero or a hundred, depending on the game.
• d12 (dodecahedron)
• d20 (icosahedron)

Less common shapes include d2 (also known as "coin flip"; easily emulated by rolling any common die and reading odd results as 1 and even ones as 2), d3 (usually emulated by rolling 1d6, halving the result, and rounding it up), and the so-called "percentile dice" or d100 (also noted as "d%" or "d00"). While the last one is technically available as a physical non-fair die (see Zocchihedron), it is most commonly emulated by rolling 2d10 and reading one die as units and the other as tens. The (cardinally) largest perfectly numerically balanced fair die possible is the d120 or disdyakis triacontahedron.

Lastly, many games come with custom dice, although these are usually just re-skins of regular dice with non-numeric faces rather than truly unorthodox shapes. Perhaps the most iconic of these are the "FUDGE dice", denoted as dF, which are regular d6s that have plus, minus, and blank faces (two of each) instead of numbers on them.

## Basic rolls

Dice are commonly used in Board Games to generate random numbers within a certain range in order to randomize, for instance, how far a player token moves on its turn in Monopoly or which tile generates resources in Settlers of Catan. Games may also exploit non-uniform roll distributions to make certain outcomes more likely than others: when rolling 2d6, for example, a total of 7 is six times more probable than a 2 or a 12.

In Tabletop RPGs, dice are usually used to resolve whether a player's character succeeds or fails at some in-game task. Two most common methods to do so are called "roll-over" and "roll-under". Both involve a specific numeric threshold, but in roll-over, you succeed if your total roll is at least as high as that target number, while in roll-under, you have to roll below it. Historically, the target numbers for roll-over mechanics have been set by the Game Master with optional guidance from the Game System (such as in Dungeons & Dragons), while roll-under more often uses the character's own skill scores as target numbers (such as in Call of Cthulhu).

A slightly modified roll-over can be used for opposed rolls, where another player's roll replaces the target number — in other words, the player with the highest total wins. Although opposed rolls can be based on roll-under, too, they usually require additional tricks to factor in the contestants' skill levels. An opposed roll can involve multiple players (for instance, everyone at the table usually rolls for Action Initiative at the same time), and they don't even have to all roll the same (number of) dice, depending on the system. Some game systems (e.g. Sorcerer (2001)) don't have unopposed rolls at all and instead frame inanimate and even intangible obstacles as if they were hostile NPCs.

In all cases, players can usually modify either the raw roll or the target number (or both) up or down via various means. These numeric modifiers are their main means of manipulating the probability of success in most games. Others also include explicit Luck Manipulation Mechanics that allow the players to re-roll their dice by expending a limited player resource (Inspiration in D&D 5E, Hero Points in Mutants & Masterminds, etc.) or simply by GM Fiat (Advantage in 5E). Depending on the rules, the player either may take the best of two results or is forced to stick with the second one, even if it's worse.

Particularly common rolls include:

 Roll Result distribution Used in 1d20 Uniform (flat) between 1 and 20 d20 System, Dungeons & Dragons, Numenera 2d6 Bell-curve between 2 and 12, skewed towards 7 Monopoly, Settlers of Catan, Powered by the Apocalypse 3d6 Bell-curve between 3 and 18, skewed towards 10 and 11note GURPS, Dragon Age, statgen in OD&Dnote 4dF Bell-curve between −4 and +4, skewed towards 0note FUDGE, Fate, and the latter's many offshoots

Games where you roll two or three dice may also have special mechanics for rolling doubles (when two dice land with the same face up): in Monopoly, for instance, it gives you an Extra Turn; in Unknown Armies and Eclipse Phase, it scores a Critical Hit or a Critical Failure; in Dragon Age, it earns you Stunt points (see below), etc. Because of the birthday paradox, however, this mechanic loses its appeal with larger dice pools.

## Dice pools

The key distinguishing feature of dice pools is that the players can manipulate their "pool size" — i.e. how many dice (usually but not always d6s or d10s) they roll at once, — as opposed to applying modifiers to standardized rolls. Depending on the Game System used, the results of dice pool rolls can become extremely non-linear and difficult to predict, which frustrates some players (who prefer clear probabilities), while energizing others (who like not knowing what happens next). Dice pool mechanics come in three basic categories:

• Add-up-and-compare: Sum up all dice and compare the result to a target number, like in roll-over (usually, bigger pools mean better success chances). This method was invented — along with the dice pool mechanic itself — by the Ghostbusters RPG and popularized by Star Wars d6.
• Success threshold: Count the number of dice that rolled above a fixed threshold (e.g. 7 for d10 in the Storytelling System) and compare it to the target number. Variable thresholds are a common modification, where the threshold value is also variable, representing the difficulty of the roll; this approach was pioneered by Shadowrun.
• Set building: After rolling the pool, the player must group the dice into sets, e.g. by the face up number (such as in the One-Roll Engine) or so that they sum up to 10+ (the Raises in the 7th Sea reboot). The number and, optionally, the size of these sets is then used for situation resolution.

Dice pool mechanics often feature one or more common modifications:

• Special die: One die in a pool is colored/styled differently from the rest and has special effects attached to it. As old as dice pools themselves, this mod goes all the way back to Ghostbusters' Ghost Die, which induced a Critical Failure whenever the ghost face on it came up.
• Exploding dice: All dice rolled above a certain threshold (which can range from the basic success threshold to the highest possible roll, like in the Storytelling System) count towards the end result and are rerolled immediately, effectively adding extra temporary dice to the pool. This can lead to a single die "exploding", if the player keeps rolling high longer than statistically probable, hence the name. Exploding dice are sometimes noted with an exclamation mark: "NdX!".
• Roll-and-keep: After rolling the dice, the players can only use some of them (usually the highest) for determining the action outcome. This is denoted with "NdX(kY)" ("Roll NdX and keep Y of them") or simply "NkY" if the game only uses one type of dice. If Y equals 1, only the highest roll in the pool counts towards resolution.
• Complex faces: Instead of regular dice with numbered faces, some games use custom dice with at least two types of symbols on them that are tallied up independently of each other, such as Successes/Failures and Advantages/Threats in FFG's Star Wars: Roleplaying Game or the Swords, Bows, Shields, and Cheese in Mice and Mystics.

## Dice game design

Gerald "Linnaeus" Cameron has outlined four general principles of game design to keep dice-rolling engaging and fun:

1. "Downtime is the Enemy". Watching other players roll dice if their rolls have little to no impact on your own position is boring. For this reason, game designers should minimize the time needed to resolve any one player's move/turn and, ideally, give other players something to do in the meantime.
2. "No More Than One Roll Per Turn". A corollary of the above: the more consecutive rolls it takes to resolve a single player turn, the longer the downtime for other players. Rerolls are particularly dangerous in this regard if not capped in some way.
3. "Give Players a Chance to React to the Dice". No one enjoys putting up with the Random Number God's whims, so a dice game should have mechanics for both before and after the roll — if only to let players mitigate particularly bad luck (for a price, of course).
4. "Low Rolls Should Not Suck, High Rolls Should Not Rule". Ideally, players should be able to use any roll to their advantage, but the minimum requirement is to prevent an Unstable Equilibrium from occurring simply as a result of dumb luck.

## Twists

Some interesting twists that use the physical or mathematical properties of the dice to introduce unusual game mechanics:

• Dungeons & Dragons 5th Edition has streamlined the older editions' plethora of situational 1d20 modifiers into the Advantage/Disadvantage mechanic, where the player rolls 2d20 and keeps the highest or the lowest result, respectively. Statistically, this is the equivalent of shifting the mean by ±3.325note  and the median, to 6 and 15, respectively.note
• In addition to the standard dice set of a d4, d6, d8, d10, percentile die, d12 and d20, Dungeon Crawl Classics handles enhanced rolls with "Zocchi dice", a set of non-standard dice originally created for use in wargaming. DCC's official set alone adds to the standard set with a d3, d4, d5, d7, d14, d16, d24 and a d30.
• Stunts in Dragon Age: A basic attack roll is 3d6+modifiers, but if you roll high enough to land a hit and roll doubles, you get a number of "stunt points" to spend immediately on additional effects, such as a Knock Back or extra damage. The number of stunt points you get is determined by one die colored differently from the other two.
• Unknown Armies often represents mystical shenanigans by allowing or forcing a player to "invert" their d% roll (by swapping the tens and the ones), which has the effect of either saving or screwing a roll, with probability nearly independent to that of making the roll in the first place (basically, its a reroll that doesn't require actually rerolling the dice).
• The D63 mechanic of Planet Mercenary uses not only the total of the 3d6 roll, but also has a "Mayhem die" which causes interesting things to happen when an action succeeds and the Mayhem exceeds the other two, as well as weapons qualities where interesting things happen on rolling doubles, triples, or straights.
• The German RPG New Hong Kong Story uses roll-under for opposed rolls in a clever way: when a character is hit in combat (but not critically), they can roll to evade or to parry (only in melee). Rolling under your own skill lets you half the incoming damage, while rolling under the attacker's roll lets you avoid it completely. Finally, if you crit on a parry roll, you evade all incoming damage and inflict a small but unblockable amount of damage back on the attacker.