Dice notation

• 1. Introduction

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  Dice notation is the shorthand gamers use to specify which dice to roll and how to combine them. A single line like d8+2 can condense a whole process of chance. The first edition of Dungeons & Dragons, printed in 1974, had no such shorthand. A spell called sticks to snakes simply instructed, 'From 2-16 snakes can be conjured (roll two eight-sided dice).' Who closed that gap, and what did it take for a system born in an amateur magazine to spread through an entire hobby? And how far can a single letter, d, be stretched before the notation strains to keep up?

• 2. The Basic Formula

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  Roll one 4-sided die, the rulebook says, whenever a game calls simply for d4 or 1d4. That single instruction hides the whole grammar of the system. Every expression follows the same shape: a number of dice, the letter d for die or dice, and a number of faces. Three six-sided dice becomes 3d6. Faces are numbered from 1 up to that face count, with each number treated as equally likely. The ten-sided die is the exception, printed 0 through 9, and a player can read that 0 as a 10 instead. Leave off the dice count and a single die is assumed. If a game omits the face count too, six sides is the usual default, though other defaults appear in some systems.

  1d20-10 means roll one twenty-sided die and subtract 10 from the result, one of the most common ways a modifier gets tacked onto a roll. Rolled dice are typically added together first, though some systems call for the best single result instead. Multiple dice terms can also be chained together, as in 2d6+1d8, although this form is less common. A further variation drops one die from the pool. 4d6-L means roll four six-sided dice and discard the lowest number shown. That rule bends the odds hard toward the top of the range. Under 4d6-L, a total of 3 can only happen if all four dice land on 1. A total of 18, by contrast, comes up whenever any three of the four dice show a 6. Once three or more dice are involved, the spread of possible totals starts to approximate a bell curve. That pattern follows the central limit theorem, the same principle that shapes many other kinds of statistics.

  None of that structure existed yet when early game masters had to guess whether an encounter with '30-300' orcs meant rolling 3d10x10 or 30d10 dice.

• 3. From Wargames To Fanzine

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  Miniatures wargamers began rolling dice shaped like Platonic solids in the late 1960s and early 1970s, seeking results a standard six-sided die couldn't produce. Dungeons & Dragons grew out of that same wargaming scene and became the first game with wide commercial availability to use such dice.

  A player named Ted Johnstone closed the notation gap in 1975. His article, 'Dice as Random Number Generators,' ran in the inaugural issue of the fanzine Alarums & Excursions. It laid out a standard way to write a roll. Another contributor to that same issue, Barry Gold, used the same notation, and it spread quickly through the fan community that read the zine. The notation became so embedded in that culture that Gary Gygax adopted it. He built it into the first edition of Advanced Dungeons & Dragons, published between 1977 and 1979. That bond between D&D fandom and the notation runs deep. It even shows up in the name given to the Open Game version of the D&D rules: the 'd20 System.'

  None of that explains how a roll can multiply itself, as in 1d6x5, one die whose result gets multiplied by 5.

• 4. Multipliers And Percentiles

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  3d6x10+3 reads as a chain of instructions: roll three six-sided dice, add them together, multiply that sum by 10, then add 3. The multiplication sign itself can be swapped for a division slash or the ÷ symbol. A constant number stands in for how many times the result gets multiplied. That constant defaults to 1 if none is given. A separate use of multiplication means repeating an entire roll rather than scaling it. 3x(2d6+4) means rolling two six-sided dice and adding 4, then repeating that whole roll three times and adding the three totals together.

  Percentile dice push the same idea toward a fixed target: rolling to a number out of 100, sometimes written as d%. True hundred-sided dice exist, but it's far more common to combine two ten-sided dice instead. One stands for the tens, one for the units, often told apart by color. Dice made specifically for percentile rolls usually skip printing any tens marking at all, showing two full runs of 0 through 9 instead. A roll of zero on both dice can be read as either 0 or 100, depending on the game. It is unusual, though, for the units die's zero to be read as a 10. That reading would make double zeroes equal to 10 rather than 100. A rarer relative, the d1000, uses three ten-sided dice together and turns up more often in wargames than in role-playing games.

  Before ten-sided dice became common around 1980, some twenty-sided dice were printed with two copies of each digit 0 through 9. Half of those faces were tinted a different color, marking an added ten.

• 5. Keeping And Dropping Dice

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  Cortex Plus games hand a player dice of several different sizes at once and ask them to keep only two results, normally the best two. A player can spend a Plot Point to keep an extra die, and some character abilities let a third die be kept automatically. Other systems spell the same idea out in notation rather than table rules. Extending the basic formula with a 'k' lets a game say how many dice to keep from a larger roll. That k can turn into 'kh', 'kl', or 'kc'. Each version specifies keeping the highest, the lowest, or dice of the player's own choosing, and these can even combine. A parallel version replaces 'k' with 'd' to mark dice dropped instead of kept. Writers have to be careful with this form, since it produces two d's in a row. It also means the die's side count can no longer be safely left out. 4d6, with the highest and lowest results dropped, is one instance of this drop notation. It's considered clearer than the keep version, since 'keep' notation has no way to describe holding onto only the middle results. 7th Sea and Legend of the Five Rings only ever use ten-sided dice, so both games drop the side count from their notation entirely. Their shorthand for eight dice with the top six kept and summed needs only the dice count and the keep count to make sense.

  One online die roller pushes the idea further still. On OpenRoleplaying.org, adding L or H instead of subtracting it doesn't drop a die at all. Instead it doubles the lowest or highest result, counting that die twice.

• 6. Dice Pools And Thresholds

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  Vampire: The Requiem has players roll a pool of ten-sided dice and count how many come up 8 or higher. That count of 'successes' becomes the result of the roll. The same basic approach runs through the original Ghostbusters role-playing game, the Storyteller system, and Fantasy Flight Games' Star Wars Roleplaying Games. In each, a set number of dice gets rolled, and only the dice that clear some fixed threshold count toward the outcome. Some publishers even manufacture custom dice for this, printed directly with success and failure symbols instead of numbers.

  The Fudge system takes a different approach to the same pool idea. Its dice carry a minus sign, a plus sign, or a blank face instead of numbers, standing for -1, +1, and 0. The standard mix is one third of each symbol. That's usually built from a six-sided die with two faces of each, a die known as dF.2 or simply dF. Rolling four of them together, 4dF, produces a result from -4 to +4, the same range as 4d3-8 written the standard way. A rarer variant, dF.1, replaces two of those faces, leaving four blanks alongside just one plus and one minus.

  None of these pool systems needed a special name for themselves. Elsewhere, dice tricks got stranger nicknames entirely, built around block dice, beast numbers, and lopsided multipliers.

• 7. Named And Unusual Variants

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  Traveller, published by the Game Designers' Workshop, first rolled two six-sided dice to generate two-digit results for its encounter and event tables. It never wrote that practice down as formal notation, though. Games Workshop later formalized that same trick as the D66, used in Necromunda and Mordheim. The formula d6x10+d6 gives one die the tens place and the other the ones place. Rolling a 1 and then a 5 makes 15; rolling a 3 and then a 6 makes 36. Across all 36 possible results, from 11 to 66, the average outcome is 38.5, with a standard deviation of about 17.16.

  Blood Bowl, another Games Workshop title, uses a block die with the shorthand Xdb. Rolling it means throwing the absolute value of X in six-sided dice and keeping just one result. A positive X means the attacker picks which die to keep, while a negative X hands that choice to the defender. Since -1 and 1 work out the same, only 1 is ever written. Common values are 3, 2, 1, -2, or -3. Some players write 'for' or 'against' after the number instead, using those words in place of a plus or minus sign.

  Steve Jackson Games' In Nomine calls its own three-dice roll a d666, a nod to the number of the beast. Underneath that ominous name it's really just 2d6 for success or failure, combined with a separate 1d6 that measures the degree of that result. Planet Mercenary takes a similar three-die roll, called d6³, and layers extra meaning onto it. Alongside the usual 3d6 success check, the game compares the individual dice results. It notes whether a marked Mayhem die rolled highest, what the lowest number showed, and whether any two dice matched.

  The plainest of these named rolls is the d3, just a six-sided die whose result gets divided by 2. It averages 2, with a standard deviation of 0.816. Gary Gygax's Cyborg Commando pulls in the opposite direction with d10x, equivalent to multiplying two ten-sided rolls together, d10 times d10. That produces a lopsided spread weighted toward the low end, with a mean of 30.25 and a standard deviation near 23.82.

  None of these named rolls explain what happens when a die refuses to settle after just one throw.

• 8. Exploding And Penetrating Dice

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  Feng Shui and Savage Worlds both let a die reroll itself, usually when it lands on its own highest face. Each new roll adds onto the running total. The practice goes by several names: open-ended rolling, exploding dice, or penetration rolls, depending on which game is describing it. On the probability tool Anydice, the function that makes a die explode on its top value is simply named explode. Writers mark an exploding roll a few different ways. One is an exclamation point after the expression, as in AdX! or 6d6!. Another is an asterisk instead, as in AdX*, or simply doubling the letter X, as in AdXX.

  The Storyteller system folds exploding dice into its own pool mechanic. It layers the reroll rule on top of a threshold and a target number the pool has to clear. Diana: Warrior Princess pushes the idea further, exploding every success in a pool rather than just one die. Hackmaster takes the opposite tack with a variant called dice penetration. It subtracts 1 from the total of any rerolled die, a small tweak that keeps totals from climbing as high as pure exploding dice allow.