Dice notation

• 1. Standard Notation Basics

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  A player rolls three six-sided dice and adds the results to a character's skill score. The notation 3d6 appears in rulebooks from the mid-1970s onward as a compact way to describe that action. The letter d stands for die or dice, while numbers before it indicate quantity and numbers after it define sides. A single digit like 4 means roll one four-sided die, but most games default to six-sided cubes when no number precedes the d. Some systems label the ten-sided die with digits 0 through 9 instead of 1 through 10, treating zero as either nothing or ten depending on house rules. Modifiers follow the dice expression with plus or minus signs, allowing calculations like 1d20 minus 10 to subtract ten points from a twenty-sided result. Chaining expressions together, such as 2d6+1d8, remains rare outside specific game communities. Dropping lowest results uses an L suffix, so 4d6 minus L removes the smallest value from four six-sided dice. This adjustment shifts probability curves toward higher totals because rolling a three now requires every die showing one.

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• 2. Historical Origins And Adoption

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  Miniatures wargamers began using Platonic solid shapes in the late 1960s to generate outcomes impossible on standard cubes. Dungeons & Dragons emerged in this environment as the first commercially available game to embrace polyhedral dice widely. Early editions lacked standardized notation, sometimes offering verbal instructions or implying rolls through result ranges like 30, 300 orcs. Ted Johnstone introduced formal dice notation in an article titled Dice as Random Number Generators published in Alarums & Excursions during 1975. Barry Gold used similar conventions in that same inaugural issue before the method spread throughout fan circles. Gary Gygax eventually adopted these notations for Advanced Dungeons & Dragons between 1977 and 1979 after seeing their utility among players. The d20 System name reflects how deeply embedded standard dice notation became within D&D fandom. Before ten-sided dice appeared around 1980, twenty-sided dice often carried duplicate digits zero through nine for percentile calculations. Half of those faces received distinct coloring to indicate addition of ten when randomizing numbers from one to twenty.

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• 3. Mathematical Probability Distributions

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  Rolling three or more dice produces a probability distribution approximating Gaussian curves according to the central limit theorem. A roll of 4d6 minus L demonstrates this shift clearly since obtaining a total of three requires all four dice showing ones. That specific outcome carries a probability of one in twelve hundred ninety-six combinations. Conversely achieving eighteen happens whenever any three dice display sixes, yielding a probability of one in two hundred sixteen. Standard deviations vary across different die types; a single six-sided cube shows about 0.816 while combined pools increase variance significantly. The average result of rolling two six-sided dice multiplied by ten plus another six equals thirty-eight point five with standard deviation near seventeen point one six. Non-linear distributions emerge when mechanics like d10x combine multiplication with dice rolls, concentrating most results at lower ranges. Mean values for such systems reach thirty point two five with standard deviations approaching twenty-three point eight two. These mathematical properties allow designers to balance randomness against predictability within game mechanics.

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• 4. Advanced Multipliers And Modifiers

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  Some games expand notation using multipliers where AdX×C means multiplying dice results by constants like five or ten. Expressions such as 3d6×10+3 instruct players to sum three six-sided dice, multiply that total by ten, then add three points. Division symbols sometimes replace multiplication signs, allowing operations like C÷dX to split outcomes evenly among participants. Repeating throws appear through lowercase x instead of multiplication symbols, so 3x(2d6+4) repeats the inner expression three times while accumulating sums. Selective strategies discard specific results using k suffixes indicating kept dice counts from larger pools. Players might keep highest values via kh or lowest via kl depending on system rules. Combining these modifiers allows complex expressions like AXdYkHlC where multiple conditions apply simultaneously. Alternative notations use L and H letters directly instead of numbers to drop extremes automatically. When used with plus signs, these letters double lowest or highest values rather than removing them entirely. Games like 7th Sea omit side counts when relying solely on ten-sided dice, streamlining notation for experienced groups.

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• 5. Percentile And Specialized Dice Systems

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  Ten-sided dice labeled zero through nine often pair together as percentile dice representing units and tens separately. One die shows multiples of ten while another displays single digits, creating combinations ranging from one to one hundred. Rolling zeros on both dice may equal either nothing or exactly one hundred depending on game-specific house rules. Three ten-sided dice occasionally combine into d1000 rolls common in wargames but rare within role-playing contexts. Custom dice marked with success and failure symbols exist for systems requiring binary outcomes without numerical ranges. Fudge dice feature six sides split evenly between minus signs, plus signs, and blank spaces representing negative one, positive one, and zero respectively. Four fudge dice generate results spanning negative four to positive four equivalent to rolling four three-sided dice minus eight points. Variants include dF.1 cubes containing four blanks alongside single plus and minus faces for narrower outcome spreads. Base-six variants like D66 rolls originate from Traveller games where two six-sided dice form numbers from eleven to sixty-six. These combinations average thirty-eight point five with standard deviations near seventeen point one six. Blood Bowl introduces block dice using Xdb shorthand where defenders choose which result counts when negative values appear.

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• 6. Dice Pool Mechanics And Variants

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  Games including Ghostbusters and Star Wars Roleplaying Games count successes by tracking how many dice meet fixed thresholds within pools. Vampire: The Requiem asks players to roll ten-sided dice noting those showing eight or higher as successful actions. Companies produce custom dice marked directly with success and failure icons for streamlined gameplay without counting individual pips. Open-ended mechanics allow rerolling highest values repeatedly until non-maximum results appear, creating exploding dice effects. Feng Shui and Savage Worlds employ such systems where each successive roll adds to previous totals before stopping. Notation suffixes like exclamation points indicate explosion rules while asterisks or capital X letters serve alternative purposes. Hackmaster implements dice penetration subtracting one from rerolled totals after initial explosions complete. Diana Warrior Princess explodes all successes simultaneously rather than stopping at first failure condition. Anydice software includes functions named explode to simulate these behaviors programmatically. Three-dice variations called d666 combine success determination with degree assessment using separate dice for different outcomes. Planet Mercenary tracks Mayhem die comparisons alongside lowest numbers rolled and matching pairs across three cubes. These specialized systems prioritize narrative tension over pure mathematical predictability while maintaining structured randomness.

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