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From SCIFIPEDIA
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An open-ended roll is a special type of success dice roll used in role-playing games that allows for more extreme results (both good and bad).
For example, suppose that you are playing with dice that allow for a score of 1 to 100. This is commonly done by rolling two ten-sided dice, although actual 100-sided dice exist. It may certainly allow for you falling over trying to put your shoe on, or knocking the sword out of someone's hand. However, there is a limit and it doesn't allow for real flukes.
The game mechanic that allows a solution is to say that if you roll at the very high end (or very low end), you roll again and add the two rolls. So, you are cornered by two thieves with daggers. You attempt to knock the dagger out of one thief's hand with a kick. You roll a 97 out of 100. This would be very likely to succeed (depending on the thief's ability to resist, of course). If there is an open-ended roll for a score of 96 to 100, you roll again and add. If you roll a 90 on your second roll, this is an incredible success. You knock the dagger out of the first thief's hand, and it flies over and amazingly knocks the dagger out of the second thief's hand as well.
Some systems allow for multiple-ended rolls: if you get a very high (or very low) score) on the second roll, you might roll a third time and add and so on.
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