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| D&D Combinatorics |
 Title text: Look, you can't complain about this after giving us so many scenarios involving N locked chests and M unlabeled keys. |
Explanation
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This explanation may be incomplete or incorrect: Created by a BOT THAT GRABBED A CURSED ARROW - Please change this comment when editing this page. Do NOT delete this tag too soon. |
Dungeons and Dragons (D&D) is a role-playing game that usually has a "Dungeon Master" (narrator) that takes a team of players through scenarios where they attack monsters and go on quests.
Often, there will be semi-random events: e.g., when attacking a monster, often a player will roll a die and deal damage based on the result. D&D uses a variety of dice, from regular d6 (6-sided, cubic dice) to other polyhedral dice, with the number of faces denoted by dX (e.g., d10 is a 10-sided die, with numbers from 1 to 10 on it). Common sets include: d4, d6, d8, d10, d12, d20, and occasionally d100 (typically not, however, the d65536).[citation needed]
With these, you can simulate events with a wide variety of denominators. In this case, Cueball gives a combinatorial problem:
- There are 10 arrows.
- 5 arrows are cursed.
- You randomly take two.
- What are the odds that neither of them are cursed?
Calculating using binomial coefficients, there are “10 choose 2” (45) ways to choose two arrows, of which there are “5 choose 2” (10) ways to choose 2 arrows that are non-cursed. As a result, the odds of taking all non-cursed arrows is 10/45, which simplifies to 2/9.
The Dungeon Master (DM) in this case has to map that probability into rolling multiple dice, whose sums are also not evenly distributed: i.e. if rolling 3d6 (3 six-sided dice) and a d4 (1 four-sided die), the sums can range from 4 to 22. It's pretty hard to do this in one's head, but it does happen that the odds of rolling 16 or more with this combination is 2/9, matching the probability that we want to simulate. The caption elaborates that the DM has a degree in the relevant field, and is unable to resist applying this to the D&D game when the opportunity arises - opportunities that Cueball eagerly provides for this very reason.
There is a much easier way of implementing the operation: literally present 10 similar-looking arrows, or other objects that are taken to represent arrows (cards, for example), with the assigned information of whether each one is cursed hidden away from Cueball, and then let Cueball pick any two. It may be inferred that as the DM's mind is too occupied with advanced, high-level knowledge, she is no longer capable of considering straightforward solutions to the problem. Or that the DM actually enjoys calculating probabilities and combinatorics in a geeky way.
The title text claims that Randall only started doing this to the DM after she herself insisted on forcing another combinatorial puzzle on the players several times, involving a bunch of locked treasure chests and a multitude of keys to unlock them with. This might be a reference to an M-of-N encryption system, where a system has n valid passwords (instead of just one) but requires m of those passwords to be given before it will open; it is assumed m is greater than 1 but less than n. While this is easy enough to implement in a computer system, it would be extremely cumbersome to build for a physical lock with keys, and spreading the mechanism across multiple separate treasure chests would be impossible without literal magic (luckily, magic is in plentiful supply in a typical Dungeons and Dragons game).
Transcript
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This transcript is incomplete. Please help editing it! Thanks. |
- [Cueball, Megan, Ponytail, White Hat, and Knit Cap are sitting at a table. Everyone is looking at Cueball. Ponytail is facepalming. The table is covered in sheets of paper and assorted dice.]
- Cueball: I grab 2 of the 10 arrows without looking and fire them, hoping I didn't grab one of the 5 cursed ones. Did I?
- Ponytail: Sigh. Umm. Okay.
- Ponytail: Roll... Uh... Hang on...
- Ponytail: Roll 3d6 and a d4. You need... 16 or better to avoid the cursed arrows.
- [Caption below the panel:]
- I got way more annoying to play D&D with once I learned that our DM has a combinatorics degree and can't resist puzzles.
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