Here's some common dice rulings.
- Chance Rollsd20 x>10, 20++ - This ruling is a simple d20 chance roll. A 10 or higher is a success, a 20 is a critical success. Critical success typically implies that no matter what happens to the target of the action, some effect will be achieved. In the case of trying to create an object or solve a problem, or throw a dart... a critical success implies a moment of brilliance. Perfect solution just suddenly pops into your head, the way to build whatever you're trying to make just becomes apparent to you, the dart hits the bullseye despite your 'friend' kicking you in the back of the leg. Some actions are more difficult, and the required rating for a success would change accordingly.
Saving Rollsd20 +/-B, x>y - Saving rolls are a form of chance roll, using a d20 as the base and adding or subtracting a modifier based on a character's stats. An example would be a Reflex Saving Roll, where a character that just stepped on a trap door is trying to grab the edge. They have a dexterity of 17 and an intelligence of 15 for an average of 16, which divided by eight (element of surprise lessens modifier) results in +2. The roll is a 13, and the DM has decided a success in this case should be at least 15. The +2 gives our character a complete success, but not particularly spectacular. The actual action could be anywhere from catching the edge to leaping back at the last second... but it would've failed had the saving roll been a miss.
Limited Max Rollsdx+y mz - Limited max rolls involve an 'mz' where the M stands for maximum and the z is the upper limit, anything higher is rounded down. In the example of d10+1 m10 the minimum roll is 2, and the 'm10' means the maximum result is a 10. You could apply this to a d20 roll as well, or add multiple dice together (3d5 m12 for example) but the general idea of setting a 'maximum' is to restrict the roll. Maximum values are typically reserved for duration rolls (d10+1 m10 turns/seconds/whatever)
Accuracy Rollsdx acc - A die is rolled for accuracy with a bow (or thrown weapon), in the case of a d6 6 would be a direct hit and 1 a miss. Say your target is three feet wide at fifty yards with a bow - 6 is bullseye (assuming that's your target), 3 is in the white ring (about half the radius out), 2 is grazing the edge and a 1 is a complete miss. (thank you .deadbatteries.)
Damage rollsxdy +/-z - A damage roll can be done with any combination of dice, but only really matters in the case of numeric or percentile health/stamina/durability statistics.
Statistic rollsxdy - Again only really applicable in the range of numeric or percentile statistics, a statistic roll would be done for the sake of base stat, temporary modifiers, or raise/lower from training. Typical basic stat roll is 3d6 +racial/class bonuses (taken from most recent D&D rulebook).
As a general note, a basic d20 chance roll should really be done for every major action taken in a fight. You can treat it however you like... 20 is a garaunteed hit, 10 is a hit unless the target rolls a higher success on their evade, below 10 the opponent can evade or block without any need for a defensive chance roll for example.
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Updated! - Rulings reclassified as formula rather than specific examples.