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Post by krusader74 on Dec 4, 2017 10:20:43 GMT -6
I put together this web page to print out an N-roll binomial table. It's inspired by the binomial tables in Larry Church's article "One Roll, To Go", in Dragon issue 113 (September 1986), pages 74-75. It differs from Church's tables in a few respects: - The number of attackers is variable. You can set this to be any integer from 1 to 100. Then press the 'Calculate' button to see the table.
- There is a row for 0 hits.
- The probabilities of the cumulative distribution function for the binomial table are formatted in 100ths of a percent. You can always roll your d% a couple extra times to generate 100ths of a percent, if need be.
- Roll your d%. Then look at the column corresponding to the value needed to-hit on a d20. You want the value of the 'No. Hits' in the leftmost column corresponding to the smallest percentile that's greater than or equal to what you rolled. In other words, if your d% roll is between two consecutive rows, use the 'No. Hits' in the upper of the two rows. If that still sounds confusing, there's an illustrative example with numbers given on the page.
- The numbers agree with Church's except: (1) he rounds fractions of a percent, and (2) his table is organized so that you want the first row from top to bottom whose percentile is less than the d% you rolled.
It's a single-page web app -- all the JavaScript, CSS and HTML are together in one file -- no external libraries required. Feel free to download it and modify it to suit your needs, no permission needed. Here is a screen shot of what it looks like after I hit the 'Calculate' button... palamedes.altervista.org/images/n-roll_binomial_screenshot.jpg [http images no longer display] "N-roll binomial table screenshot"
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Post by krusader74 on Dec 3, 2017 15:41:57 GMT -6
An index of dice-related discussions(For probability-specific discussions, see also the index of prob & stat related discussions. For an index of dice-related articles in Dragon magazine, see this.)#3d20 advantage/disadvantage mechanic [1], [2]AAlternatives to dice [1]Art and dice [1]BBuying dice [1]CCarcosa dice [1]Chi-square test See Fairness, testingChits [1]Conversions: - Converting 3d6 to d% [1]
- Converting d20 to d10, d% [1]
- Converting d6s to other dice throws [1]
- Converting lots of d20 rolls with one d% roll by using the binomial distribution [1]
Dd5 [1]d6, skewed [1]d10 [1]d20 [1]d20, numbered 0-9 twice [1]d20, replacing lots of rolls with one d% roll by using the binomial distribution [1]d34 [1]d60 [1]d120 [1]dF [1]David Wesely on the introduction of polyhedral dice to RPGs [1]Diceless [1], [2]Divided die rolls, e.g., d20/d4 [1]Dragon magazine articles about dice [1]EEarthdawn dice [1]Expensive dice [1]Exploding dice [1]FFairness [1]Film and dice [1]Fudge dice [1]GGamescience [1]Gamescience, pound of dice [1]Geometry of dice [1] HHistory of dice [1]IInventor of dice [1]Inventor of dice, images [1]Isohedral dice [1]KKnucklebones [1], [2], [3], [4], [5], [6]LLong-tailed dice distributions See Divided die rollsMMaximum of 2d6 [1]Music and dice [1]NNon-isohedral dice [1]Non-transitive dice [1], [2]OOldest dice game See KnucklebonesPPalamedes See Inventor of dicePatents, d10s et al. [1]Patents, d5 [1]Poetry and dice [1]Pools [1]Predictability of dice [1]Probabilities [1]QQuirks of dice owners [1]SSkew Dice(TM) [1]Statistics for common dice rolls [1]Step die [1]Sums of two dice [1]Symmetry of dice [1]TTali See KnucklebonesTrapezohedrons, pentagonal See d10Trapezohedrons, r-gonal [1]Trapezohedrons, trigonal See d6, skewedUUnbiasedness See Fairness
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Post by krusader74 on Dec 2, 2017 9:08:41 GMT -6
And I was so proud of myself for entering that Chi square BASIC program from Dragon magazine into my Commodore Vic20 back in the day... Yeah, there were a number of great articles in Dragon about dice that left a lasting impression... Here is the Dragon magazine index of articles on dice: Subject Title Author Location System Dice: "Wild, Wild World of Dice, The" Michael J. D'Alfonsi 182(45) -- Alternatives to "What To Do When the Dog Eats Your Dice" Omar Kwalish 7(5) -- Checking for fairness "Be Thy Die Ill-Wrought?" D.G. Weeks 78(62) -- Computer "Electric Eye, The" Mark Herro 45(56) -- Divided rolls "Same Dice, Different Odds" David G. Weeks 94(18) -- Games of chance "Games of Chance" Seth Irvin Williams 346(42) D&D3 Simplifying complex rolls "One Roll, To Go" Larry Church 113(74) -- And here's my review of all these articles... 1. In "What To Do When the Dog Eats Your Dice", Dragon issue 7, pages 5-6, Omar Kwalish lists some alternatives to dice: - Percentages generated by 2d6
- Chits in a jar
- Calculators
- Cutting cards
- Numbered straws
- watch with a second hand
- Spinners
- Using six siders for higher numbers
- Coin flipping
- Phone book and blind fold
- Lazy Susan dartboard
- Classic Greco-Roman augury method
- Mouse in a maze
- Maso/macho delight
- Numbered jumping beans
2. In "The Electric Eye", Dragon issue 45, pages 56-57, Mark Herro presents - A dice rolling routine for a Hewlett-Packard HP-41C calculator.
- Quick reviews of two affordable computers:
- Sinclair's ZX-80
- Radio Shack's "Pocket" TRS-80
3. In "Be Thy Die Ill-Wrought?", Dragon issue 78, pages 62-65, D.G. Weeks describes the chi-squared test to check dice for fairness, and he provides a BASIC program to compute chi-square values. This was a great article. Its only shortcoming, like so many other articles on the subject, is that it doesn't provide an adequate answer to the question, How many times must you roll a die to test for fairness? He merely says, But what's the justification for rolling it 80 times as opposed to 8 times or 8,000 times? I posted about that here. 4. In "Same Dice, Different Odds", Dragon issue 94, pages 18-20, David G. Weeks says sometimes you want a probability roll with a "long tail" in order to model extraordinary events. So he suggests Here is his plot of this distribution using Palamedes: palamedes.altervista.org/images/divided_roll.jpgI used the command: barchart(p(floor(d20 / d4))) That rounds fractions down. If you want to round up, so that you don't get "0" as a value, use this instead: barchart(p(ceil(d20 / d4))) At the very end of the article, Weeks provides a table in the form dX/dY. For each of these "divided rolls," he gives the - Average
- Conventional roll with a similar average
- Probability of 0
- 95th percentile
This is a very interesting article, mathematically, since most "real life" distributions have long tails. 5. In "One Roll, To Go", Dragon issue 113, pages 74-75, Larry Church describes how to replace lots of d20 rolls with one d% roll by using the binomial distribution: He provides a concrete example of how his system works... And here is the table he's referring to... palamedes.altervista.org/images/10_roll_binomial_table.jpgChurch provides us with tables for - A 5-roll binomial table
- A 10-roll binomial table
- A 20-roll binomial table
He computes the values in these tables from the cumulative distribution function of the binomial distribution. I like the idea here a lot, but I have two peeves with his tables: - There's a chance that none of the 10 archers hit, but 0 is missing from the table!
- Inconsistent rounding
Despite these problems, I'd have to say this is probably the most useful Dragon article about dice!!! You could generate the "12" column in Palamedes by inputing: cdf(binom(10, (21 - 12)/20))//pct//table The output looks like this: palamedes.altervista.org/images/12_column.jpgUse the "key" if the percentage you rolled on a d% is less than or equal to its corresponding "value" but greater than the "value" above it (and assume there is a 0.00% above the first "value" in the table). These "values" are expressed in 100ths of a percent. You can always roll your percentile dice (d%) a couple extra times to generate 100ths of a percent, if need be. TODO: Maybe in the future, I'll whip up a quick JavaScript fiddle to print an entire n-roll binomial table for a given s-sided die, not just a d20. 6. In "The Wild, Wild World of Dice", Dragon issue 182, pages 45-46, Michael J. D'Alfonsi says, - Superstitions - The player who puts aside any dice that produce low rolls, because they're cursed.
- Specific dice - The player who always uses one d20 to roll high, and another to roll low.
- Strange habits - The player who sleeps with their dice bag under under their pillow.
- Dice bags - The player who uses a bottle of Crown Royal as a dice bag.
- Too many dice - The player with 200 dice.
- Cheater, cheater! - The player who uses loaded dice.
- A few last thoughts.
7. The blurb for "Games of Chance" by Seth Irvin Williams in Dragon issue 346 reads: ... Has anyone else used any of the dice advice in these articles? Let me know if I missed any dice articles from Dragon, and I'll add it to my review!
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Post by krusader74 on Dec 1, 2017 9:29:06 GMT -6
Quoted from the article: McLean, K. (1990). "Dungeons, dragons and dice." The Mathematical Gazette, Volume 74, issue 469, pages 243-256. D&D and its polyhedral dice got me interested in math as a kid, and keeps me interested in math as an adult. Here is a brief outline for this article: - The definition of an isohedron: "Any two faces are congruent and any face may be mapped onto the other by some rotation or reflection of the die onto itself."
- Euler's formula, V - E + F = 2, and the proof that an isohedral die cannot have an odd number of faces.
- A complete table of convex isohedra:
- 5 platonic solids
- 13 Archimedian solids
- r-gonal dipyramids (an infinite sequence)
- r-gonal trapezohedrons (another infinite sequence)
- Non-isohedral dice, like the d5, a truncated prism. Can they be unbiased?
- Non-transitive dice
And here are some comments and criticisms about this article... RE: The definition of an isohedron and the complete table of convex isohedraI would have liked to have seen the article talk about the algebra of isohedrons, not just their geometry. Isohedrons are defined by their rotational and reflective symmetries. These symmetries are captured in the idea of a Coxeter group, and may be expressed very concisely in a Dynkin diagram. One could use these diagrams to enumerate all uniform polyhedra. ... RE: Euler's formula, V - E + F = 2Euler discovered this relationship studying planar graphs. And each polyhedron is associated with a planar graph. I discussed this in the Appendix: Learn You Some Graph Theory to the answer I wrote to the question posed by the thread Non-Linear Randomly-Generated Dungeon?... RE: r-gonal trapezohedronsAn r-gonal trapezohedron is a solid whose faces are quadrilaterals. Applying Euler's formula, it has F=2r faces, E=4r edges and V=2r+2 vertices. H.P. Lovecraft's 1935 short story The Haunter of the Dark features a "crazily angled stone" of extraterrestrial origin called the "Shining Trapezohedron." r=3: You can buy fair, non-cubical 6-sided dice with trapezoidal faces, called Skew Dice(TM). (A trapezoid is a quadrilateral with two sides parallel.) I wrote about Skew Dice in the post on Weird But Fair Dice. r=5: There's a recent discussion of the pentagonal trapezohedron in the thread: When did 10-sided dice start being used in D&D? Its 10 faces are congruent kites. (A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.) You can make your own origami pentagonal trapezohedron by following the 56 steps on pages 105-109 of the Dover book, A Plethora of Polyhedra in Origami (2002) by John Montroll: If you prefer to "cheat" -- by cutting and gluing instead of just folding -- the instructions are here. ... RE: Non-isohedral dice, like the d5In 2003, a patent was filed for five-sided dice. In 2005, the USPTO granted patent US 6,926,275: Theoretically, the probability of landing on a rectangular face is a continuous function of the ratio of the lengths of a rectangular side to the triangular side. In the limit, as the prism gets longer, the probability of landing on a rectangular side goes to 100%. And in the limit, as the length of the rectangular side shrinks to zero, the probability of landing on it goes to 0%. So somewhere in between, it must be unbiased. Hence, fairness depends on thickness: The problem with this argument is that the world isn't continuous. In practice, the fairness of such a die depends on the conditions in which it is thrown. You want: - High initial angular velocity.
- Rough, perfectly inelastic surface.
McLean's paper references other studies where these practical dependencies were demonstrated. ... RE: Non-transitive diceI posted about non-transitive dice in detail, in the thread on The Most Powerful Dice.
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Post by krusader74 on Nov 27, 2017 19:08:35 GMT -6
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Post by krusader74 on Nov 24, 2017 5:53:13 GMT -6
While US 809,293 patents the pentagonal trapezohedron -shaped d10 shown here... ..there is an older patent for a ten-sided die: US 614,524, filed in 1897 and granted in 1898. As you can see from its image, this decahedron is not a pentagonal trapezohedron: There are in fact 32,300 topologically distinct decahedra!Gamescience's d10, protected by design patent US D267,569 filed in 1981 and granted in 1983, differs slightly in shape from US 809,293 (1906). The white one Greg. Blue d12, white d10, greend8, pink (ick, ick, ick,) d6. yellow d4 (sharp points). The original d10 were red and black w/o filled numbers. I used correction fluid to fill them. Three of them actually still exist. Bought in England about 1967. Dave Arneson "Dark Lord of Game Design" THE Polyhedral Dice Patent, US 3,208,754, was filed in 1963 and granted in 1965. So it makes sense that Dave Arneson was able to buy a set in 1967. Here is the image: There is a "Master List" of dice patents here.
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Post by krusader74 on Nov 24, 2017 1:25:14 GMT -6
I wanted to start a thread where folks can suggest realistic scifi that might become an “Appendix N” for Traveller. Stuff that when you look at it you think “yeah, that’s Traveller!” There is a book-length treatment of this subject... The Science Fiction In Traveller: A Reader’s Guide to Traveller Role-Playing Fiction (2016) by Shannon Appelcline Availability:Print Length: 197 pages Publisher: Far Future Enterprises Publication Date: March 24, 2016 Opening words:Table of Contents:Blurb:Appelcline gives each book (or series of books) a summary and an explanation of how it relates to the Traveller RPG.
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Post by krusader74 on Nov 13, 2017 17:16:52 GMT -6
Marc issued the following health update yesterday on the CotI mailing list. It's great to hear that his condition is improving!
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Post by krusader74 on Nov 10, 2017 17:00:59 GMT -6
In a recent thread, there's a discussion of neutrality. I thought I would post this "addition" to provide some historical background. It is excerpted from The Strategic Review #6 (Vol. II, No. 1) February 1976, pages 3-5.THE MEANING OF LAW AND CHAOS IN DUNGEONS & DRAGONS AND THEIR RELATIONSHIPS TO GOOD AND EVILby Gary Gygax Many questions continue to arise regarding what constitutes a "lawful" act, what sort of behavior is "chaotic", what constituted an "evil" deed, and how certain behavior is "good". There is considerable confusion in that most dungeonmasters construe the terms "chaotic" and "evil" to mean the same thing, just as they define "lawful" and "good" to mean the same. This is scarcely surprising considering the wording of the three original volumes of DUNGEONS & DRAGONS. When that was written they meant just about the same thing in my mind -- notice I do not say they were synonymous in my thinking at, that time. The wording in the GREYHAWK supplement added a bit more confusion, for by the time that booklet was written some substantial differences had been determined. In fact, had I the opportunity to do D&D over I would have made the whole business very much clearer by differentiating the four categories, and many chaotic creatures would be good, while many lawful creatures would be evil. Before going into the definitions of these four terms, a graphic representation of their relative positions will help the reader to follow the further discourse. (See #I) Notice first that the area of neutrality lies squarely athwart the intersection of the lines which divide the four behavioral distinctions, and it is a very small area when compared with the rest of the graph. This refers to true neutrality, not to neutrality regarding certain interactions at specific times, i.e., a war which will tend to weaken a stronger player or game element regardless of the "neutral" party's actions can hardly be used as a measure of neutrality if it will benefit the party's interest to have the weakening come about. Also note that movement upon this graph is quite possible with regard to campaign participants, and the dungeonmaster should, in fact, make this a standard consideration in play. This will be discussed hereafter. Now consider the term "Law" as opposed to "Chaos". While they are nothing if not opposites, they are neither good nor evil in their definitions. A highly regimented society is typically governed by strict law, i.e., a dictatorship, while societies which allow more individual freedom tend to be more chaotic. The following lists of words describing the two terms point this out. I have listed the words describing the concepts in increasing order of magnitude (more or less) as far as the comparison with the meanings of the two terms in D&D is concerned: LAW | CHAOS |
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Reliability | Unruly | Propriety | Confusion | Principled | Turmoil | Righteous | Unrestrained | Regularity | Random | Regulation | Irregular | Methodical | Unmethodical | Uniform | Unpredictable | Predictable | Disordered | Prescribed Rules | Lawless | Order | Anarchy |
Basically, then, "Law" is strict order and "Chaos" is complete anarchy, but of course they grade towards each other along the scale from left to right on the graph. Now consider the terms "Good" and "Evil" expressed in the same manner: GOOD | EVIL |
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Harmless | Unfit | Friendly | Mischievous | Kind | Unpleasant | Honest | Dishonest | Sincere | Bad | Helpful | Injurious | Beneficial | Wicked | Pure | Corrupt |
The terms "Law" and "Evil" are by no means mutually exclusive. There is no reason that there cannot be prescribed and strictly enforced rules which are unpleasant, injurious or even corrupt. Likewise "Chaos" and "Good" do not form a dichotomy. Chaos can be harmless, friendly, honest, sincere, beneficial, or pure, for that matter. This all indicates that there are actually five, rather than three, alignments, namely: | | |
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LAWFUL/GOOD | | LAWFUL/EVIL | CHAOTIC/GOOD | | CHAOTIC/EVIL | | NEUTRAL | |
The lawful/good classification is typified by the paladin, the chaotic/good alignment is typified by elves, lawful/evil is typified by the vampire, and the demon is the epitome of chaotic/evil. Elementals are neutral. The general reclassification of various creatures is shown on Illustration II. Placement of characters upon a graph similar to that in Illustration I is necessary if the dungeonmaster is to maintain a record of player-character alignment. Initially, each character should be placed squarely on the center point of his alignment, i.e., lawful/good, lawful/evil, etc. The actions of each game week will then be taken into account when determining the current position of each character. Adjustment is perforce often subjective, but as a guide the referee can consider the actions of a given player in light of those characteristics which typify his alignment, and opposed actions can further be weighed with regard to intensity. For example, reliability does not reflect as intense a lawfulness as does principled, as does righteous. Unruly does not indicate as chaotic a state as does disordered, as does lawless. Similarly, harmless, friendly, and beneficial all reflect increasing degrees of good; while unpleasant, injurious, and wicked convey progressively greater evil. Alignment does not preclude actions which typify a different alignment, but such actions will necessarily affect the position of the character performing them, and the class or the alignment of the character in question can change due to such actions, unless counter-deeds are performed to balance things. The player-character who continually follows any alignment (save neutrality) to the absolute letter of its definition must eventually move off the chart (Illustration I) and into another plane of existence as indicated. Note that self-seeking is neither lawful nor chaotic, good nor evil, except in relation to other sapient creatures. Also, law and chaos are not subject to interpretation in their ultimate meanings of order and disorder respectively, but good and evil are not absolutes but must be judged from a frame of reference, some ethos. The placement of creatures on the chart of Illustration II. reflects the ethos of this writer to some extent. Considering mythical and mythos gods in light of this system, most of the benign ones will tend towards the chaotic/good, and chaotic/evil will typify those gods which were inimical towards humanity. Some few would be completely chaotic, having no predisposition towards either good or evil -- REH's Crom perhaps falls into this category. What then about interaction between different alignments? This question is tricky and must be given careful consideration. Diametric opposition exists between lawful/good and chaotic/evil and between chaotic/good and lawful/evil in this ethos. Both good and evil can serve lawful ends, and conversely they may both serve chaotic ends. If we presuppose that the universal contest is between law and chaos we must assume that in any final struggle the minions of each division would be represented by both good and evil beings. This may seem strange at first, but if the major premise is accepted it is quite rational. Barring such a showdown, however, it is far more plausible that those creatures predisposed to good actions will tend to ally themselves against any threat of evil, while creatures of evil will likewise make (uneasy) alliance in order to gain some mutually beneficial end -- whether at the actual expense of the enemy or simply to prevent extinction by the enemy. Evil creatures can be bound to service by masters predisposed towards good actions, but a lawful/good character would fain make use of some chaotic/evil creature without severely affecting his lawful (not necessarily good) standing. This brings us to the subject of those character roles which are not subject to as much latitude of action as the others. The neutral alignment is self-explanatory, and the area of true neutrality is shown on Illustration I. Note that paladins, Patriarchs, and Evil High Priests, however, have positive boundaries. The area in which a paladin may move without loss of his status is shown in Illustration III. Should he cause his character to move from this area he must immediately seek a divine quest upon which to set forth in order to gain his status once again, or be granted divine intervention; in those cases where this is not complied with the status is forever lost. Clerics of either good or evil predisposition must likewise remain completely good or totally evil, although lateral movement might be allowed by the dungeonmaster, with or without divine retribution. Those top-level clerics who fail to maintain their goodness or evilness must make some form of immediate atonement. If they fail to do so they simply drop back to seventh level. The atonement, as well as how immediate it must be, is subject to interpretation by the referee. Druids serve only themselves and nature, they occasionally make human sacrifice, but on the other hand they aid the folk in agriculture and animal husbandry. Druids are, therefore, neutral -- although slightly predisposed towards evil actions. As a final note, most of humanity falls into the lawful category, and most of lawful humanity lies near the line between good and evil. With proper leadership the majority will be prone towards lawful/good. Few humans are chaotic, and very few are chaotic and evil. Attachments: - Law-Chaos.md (10.64 KB) - article in Markdown plain text
- - article in PDF
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Post by krusader74 on Nov 8, 2017 16:02:59 GMT -6
I just want to add a few late (relatively speaking) but relevant (for OD&D) sources to Jon Petersen's excellent list: 1. The article "Critical Hits" by Lew Pulispher in White Dwarf #8 (Aug/Sep 1978) on page 12. Pulispher's (1978) mechanics work like this: - If you roll a "natural 20" on your attack, then roll again.
- If the second roll is high enough to hit, then it is a "critical hit." Roll the d20 a third time for a special effect.
- Normal damage is inflicted, unless modified by the special effect.
Pulispher's "critical hits" only work against humans, humanoids and human-sized creatures, not non-human monsters. In the table of special effects, MDR=minimum damage roll: d20 | Special effect |
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1 | Shield arm unusable d6 turns | 2 | Shield arm unusable d3 days, MDR=2 | 3 | Shield arm unusable 2d6 days, MDR=3 | 4 | Weapon arm unusable d6 turns | 5 | Weapon arm unusable d3 days, MDR=2 | 6 | Weapon arm unusable 2d6 days, MDR=3 | 7 | 20%: Shield arm, 80% Weapon arm, unusable d4 weeks, MDR=4 | 8 | Leg limp d6 turns | 9 | Leg limp d3 days, MDR=2 | 10 | Leg maimed d6 turns, MDR=2 | 11 | Leg maimed d3 days, MDR=3 | 12 | Leg maimed d4 weeks, MDR=4 | 13-14 | Head hit, stunned -- no attack d6 rounds but may defend | 15 | Head hit, knocked out d6 rounds (MDR=2 unless no helmet) | 16 | Head hit, concussed, knocked out as above, can't walk without aid when wake up, can't fight/cast spells. MDR=3 unless no helmet; concussion lasts 2d12 days. | 17-20 | Body hit, x2 damage |
2. The article "Good Hits & Bad Misses" in Dragon #39 (July 1980) on page 34, where Carl Parlagreco describes mechanics for both "critical hits" and "critical fumbles". He also acknowledges that mechanics like this are a lightning rod for criticism: 3. In the "RPGA INTERVIEW with... E. Gary Gygax" in Polyhedron #1 (Summer 1981) on page 5, Gary Gygax clearly distinguishes between the terms - "double damage on a natural 20", and
- "critical hits"
The difference is that "critical hits" have side effects including a hit location. Jon noted that Gary dislikes and rejects "critical hits." In this Polyhedron interview, Gygax lumps "critical hits" together with a lot of other unbalanced mechanics that "pervert", "destroy", and "spoil" the game. Here is an excerpt where he talks about this: So why did these destructive "mutations" come about? He offers several explanations: - The game was originally intended for hard-core miniatures wargamers (who would never use such unbalanced mechanics). But then the game went to the mass market, who did not understand miniatures wargaming.
- Referees like to spoil (or in his words: "baby") their players: Gary chuckles, "who can say 'nay' to someone who's having a good time with the game?"
Gary previously warned players against employing unbalanced game mechanics in White Dwarf #7 (Jun/Jul 1978), on page 22. There he writes: - "Rewards must be in proportion to risk, possibly less and never greater."
- "Things must not be too easy or there is no sense of accomplishment gained from playing the game."
- "Conversely, a campaign must never be so difficult that players become discouraged with the hopelessness of never being able to have a character to survive long enough to actively associate with."
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Post by krusader74 on Oct 30, 2017 8:10:01 GMT -6
[You could get a binomial distribution using d6 (or d10) dice pools. That not any different than rolling six attacks. You just rolling all six dice at once and counting the successes. There are a number of practical and aesthetic differences, though I realize not everyone will appreciate them: - In a standard set of polyhedral dice, the d6s are 16mm and the d20 is 23mm. So it's easier to fit a bunch of d6s in my fist. (Even more so, because I use 30mm d20s, which are easier on my eyes and have a nicer heft to them than the 23mm). And for really big dice pools, there are tiny 5mm d6s.
- It's faster to roll a fistful of dice concurrently than consecutively, so it saves time.
- I have tons of d6s, but only a few d20s.
- Buying d6s in bulk is usually cheaper than other dice.
- It's faster to count successes on a d6 than a d20, because there are fewer faces.
- I am not sacrificing as much mathematical consistency with the ACS, like I am if I use a discrete uniform distribution by simply rolling one d6 to count kills.
- I do, however, sacrifice some accuracy, since a d6 is grainier than a d20: 16.67% granules versus 5% granules.
- Some of us simply enjoy rolling big giant fistfuls of dice.
To approximate ACS probabilities with a d6 dice pool, I wouldn't over think it. A rough approximation for the success range might be [edited nov. 3]: AC + Level | success range on a d6 | P(success) |
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3 - 4 | 6+ | 16.7% | 5 - 8 | 5+ | 33.3% | 9 - 12 | 4+ | 50.0% | 13 - 17 | 3+ | 67.7% | 18 - 21 | 2+ (anything but "1") | 83.3% | 22+ | 1+ (automatically hits) | 100.00% |
The method is... Pre-conditions: Attacker is a Level 2+ fighting man, and defenders are one-hit-die mooks - Determine defenders' AC* and attacker's Level and sum (AC + Level)
- Use the table to lookup the success range
- Roll a number of d6s equal to Level
- Count successes: Each success kills one mook
* If the defenders' ACs vary, you could use a weighted sum, and kill off mooks with the highest AC first. That means re-determining AC+Level each round. Example 1. A Level 6 fighting man encounters 6 orcs, AC 6. Level + AC = 12. So he rolls 6 d6s and counts each 4+ as a kill. Example 2. A Level 8 fighting man encounters seven duergar, who fight as 1 HD dwarfs with AC 4. So Level + AC = 12 also. He rolls 8 d6s (even though there are only 7 targets) and counts each 4+ as a kill.
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Post by krusader74 on Oct 29, 2017 14:54:43 GMT -6
Based on my work trying to recreate the math behind the Battlesystem resolution table the actual probabilities are a bell curve. So instead of a single die I would look for a 2 dice combo to replicate the odds. For example for a 6th level fighter I would roll 2d3 instead of 1d6. For the odd levels I would add +1. For those interested in the math, the odds of having X successes in Y attempt at Z odds is a binomial distribution. What the Battlesystem designers did was to take that and combine it with armor class, THACO, and damage dice to produce a chart that with one roll told you how many hit dice of damage a group of X guys inflicted on another troop. Quite elegant and while designed for AD&D works with any edition of D&D. You could get a binomial distribution using d6 (or d10) dice pools. For example, a 6th level fighting man faces off against 6 orcs. The orcs are AC 6. so the FM hits an orc on a roll of 11 or better on a d20. Probability( d20 >= 11 ) = 50%. So instead of rolling a d20 six times, one at a time, and looking for scores in the range 11-20, you could instead roll six d6s all at once, and count up how many score in the range 4-6. (Skip damage rolls, of course. A "hit" automatically kills.) Or you could roll six d10s and count how many hit in the range 6-10. It's still six rolls any way, whether you use the ACS with a d20 or this proposed dice pool method, but I find it easier to roll a fistful of d6s or d10s at once rather than d20s one-at-a-time. This is more like the CHAINMAIL mass combat system than Battlesystem. You'd need to translate the ACS table into a table of hit-ranges, after settling on using a d6 or a d10.
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Post by krusader74 on Oct 28, 2017 14:30:11 GMT -6
d10 random links1. The Uncanny Resurrection of Dungeons & Dragons - a New Yorker article by Neima Jahromi, October 24, 2017. (The remaining 9 links are to things mentioned in this article.) 2. Rise of the Dungeon Master: Gary Gygax and the Creation of D&D - a graphic novel by David Kushner (Author) and Koren Shadmi (Illustrator), 144 pages, May 9, 2017. Amazon, B&N, Google books #NotAnEndorsement 3. Critical Role - voice actor Matthew Mercer leads other voice actors on epic Dungeons & Dragons campaigns. Live Thursdays at 7PM PT on Twitch. 4. The Adventure Zone - a bi-weekly comedy and adventure podcast based on Dungeons & Dragons. 5. HarmonQuest - a role-playing television show with celebrity guests, created by Dan Harmon who plays Fondue Zubag, a half-orc ranger. 6. This tweet from Ta-Nehisi Coates, at 6:27 AM on September 7, 2017. Coates is a journalist and currently writes the Black Panther series for Marvel Comics. The tweet says: 7. Brooklyn Strategist - a club and café where you can pay $10 to play D&D 5E on Wednesdays in Brooklyn, NY. 8. Guard Up! Wizards & Warriors Summer Camp for teens in Burlington, MA. 9. Orcs! Orcs! Orcs!, which is a self-described Portland, Oregon. $40/ticket. 10. Learning through role-playing games: an approach for active learning and teaching (2013) by Marco Antonio Ferreira Randi and Hernandes Faustino de Carvalho. The abstract to this research paper says: (I recently wrote about another teacher's success using D&D in the classroom to help teach writing to 7 year olds.) Feel free to append your own list of D&D-related d10 random links!
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Post by krusader74 on Oct 27, 2017 15:08:45 GMT -6
Roast unicorn, cannabis bread and other culinary delights of the middle agesHere are some old recipes to add flavor to your Medieval fantasy role-playing... On honest indulgence and good healthIn 1999, the world's oldest printed cookbook was auctioned for about £6,000. The book, entitled De Honesta Voluptate et Valetudine, was written by Italian physician Batholomaeus Platina circa 1480. It's in Latin. It's 93 pages. It was bound in calf leather on wooden boards with two brass clasps. It contains 300 recipes and their medicinal use. About 20 copies survived. Recipes include: - Medieval cannabis bread, to ward off the plague
- Cooked bear
- Left-over-hog
Here's an example, a recipe for Exicium ex Pulpa (meat sausage): - Finely slice a veal haunch and add lard
- Combine and grind marjoram and parsely
- Combine and beat egg yolks and grated cheese
- Mix everything together
- Use the mixture to stuff pork (or veal) casing in egg-sized lumps
- Cook the sausages on a spit over a slow fire
Roast UnicornFor April Fools Day in 2012, the British Library reported that they found a 14th c. "unicorn cookbook" complete with pictures in the margin. The pictures are hilarious: (More recipes and pictures on their website.) Even though this book is a joke, you could use the idea and pictures in a medieval fantasy campaign setting. ZinzibarIn 2013, the oldest known Medieval cooking manuscript was found. It was written in the Durham Cathedral's monastery in 1140. Like De Honesta Voluptate et Valetudine, it is written in Latin and it concerns itself with health and the medicinal value of the food. Recipes include: "hen in winter" and "tiny little fish." Sauces were important -- they typically feature parsley, sage, pepper, garlic, mustard and coriander. This book was translated into English, and published in 2017: Zinziber - Sauces from Poitou: Twelfth-Century Culinary Recipes from Sidney Sussex College, Cambridge, MS 51 (Google books link) #NotAnEndorsement (Zinziber is Latin for "Ginger.") Baking the LawIn 1266, England enacted the Assize of Bread and Ale, the first law in England to regulate food and drink. It regulated the price, weight and quality of beer and bread. This law was discussed by comedian Sue Perkins on The Great British Bake Off, series 3, episode 1, about 20 minutes into the show. (I'm not providing a link, but you can probably still watch it on dailymotion.) A court met tri-weekly in baker's hall to try lawbreakers. The court beadle was in charge of tracking down lawbreakers and sending them to trial. The position still exists today, but it is purely ceremonial. Here is the current (as of 2012) beadle, Neil Fletcher, talking with Sue Perkins: Your punishments for producing illicit beer or bread1st Offense: You are tied to a gurney by your wrists and ankles and dragged by a horse through the shi filth on the street.The punishment might seem harsh, but people were baking sand and cobwebs and other illicit substances into the bread that could kill someone. 2nd Offense: You are pilloried at the royal exchange, where people spend the day throwing rotten fruit in your face. (At least you hope its only rotten fruit that they're throwing in your face...) 3rd Offense: The beadle and a bailiff go to your bakery and smash your ovens so you can no longer operate in the city. Tragically, the law was repealed in the 19th century, and so the beadle doesn't get to smash ovens anymore.
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Post by krusader74 on Oct 25, 2017 14:13:12 GMT -6
... Unless you train your muscles to hold shields like that, you just can't maintain it. Yes, and the Medieval "Fighting Man" did precisely that. He spent most of his life -- age 7 to 21 -- training his muscles to use armor and weapons in combat. I think we agree on that. Quoting from Training a Knight:
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Post by krusader74 on Oct 24, 2017 13:32:00 GMT -6
But the type asked about by the original poster has a limited range of sizes. I agree 100% with your point that size and weight constrain how someone wears a shield. But I disagree that the constraints in the OP put much limitation on the size or weight of the shield. The only constraints in the OP seemed to be: - time: Medieval (inferred),
- place: Europe (inferred),
- shape: "flat top, slowly tapering sides to a pointy bottom," which I call "kite-shaped" for short,
- decoration: "coat of arms,"
but not size or weight, and there was still quite a large variation in these qualities, even given those constraints. Surveying a whole bunch of resources, not just Ashdown, I found: Shield Type | Weight (lbs.) |
---|
Medieval Kite-Shaped | 5-10 | Small Buckler | 2 | Ancient Greek | 15 | Ancient Roman | 22 | Modern Non-Ballistic Shield | 6 | Modern Ballistic Shield ("Body Bunker") | 16 or 20 |
(I'm including ancient and modern shields in this list to provide broader historical context -- it shows that the large weight range for shields hasn't really changed much in 3,000 years.) There's going to be a significant difference in how someone wears a 5 lb. shield versus a 10 lb. shield. I don't have any shields to experiment with. But I do have weight plates (2.5#, 5#, and 10#, 4 ea.), and there is quite a difference when you do the same exercise after doubling the weight! I remember reading about another SCA member who conducted an experiment to determine how long a shield would last in combat. He made a 6 lb. Medieval kite-shaped shield, attached it to a post, then bashed it with a 9 lb. sword. The shield lasted 2 hours. One problem with these experiments is that they only furnish one data point. And so we need to invoke something like the mediocrity principle to draw a conclusion. But what if the same experiment were repeated a dozen times, would we get the same result? And how long would the shield last if we vary the parameters -- 10 lb. shield vs. 5 lb. sword? Can't really say based on the one data point we have. Incidentally, replicas of Medieval kite-shaped shields can be purchased on Amazon and they're typically 6 lbs. and range in price from US$50-100. #NotAnEndorsement On an entirely different note: I have rehosted the public domain images from my last post, in order to get rid of the annoying error messages. Sorry about that. Hopefully, everyone can see those images now.
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Post by krusader74 on Oct 23, 2017 15:56:16 GMT -6
There were exceptions. It all depends on the time period and culture in question. For example, regarding the late 10th century Saxons, Ashdown says, Similarly, while we generally think of shields being large, some were actually small. Depends on when, where, and how it was being used. For example, regarding a French knight going on a pilgrimage circa 1370 AD, Ashdown says "the shield is small, notched in the right-hand corner for the lance rest," like so: How did men really wear those classic looking shields If you were a noble (10th level lord?), you'd have your shield bearer carry it for you. Ashdown talks about "the battle of the three kings," saying: Here is the (c. 1000 AD) illustration: And when you're not using it, you'd carry it like this:
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Post by krusader74 on Oct 23, 2017 8:29:17 GMT -6
There is a video on YouTube dated Oct 7, 2017 called Engaging students in writing | Using D&D in the classroom. It's about a teacher, Bec West, who uses D&D in her writing class in order to motivate her kids to use their imaginations and write interesting stories. There are 18 kids in her class. They're about 7 years old. The kids really enjoyed it! This video might interest some of you - if you are a teacher
- if you have young kids (the kids in her class are only 7)
- if you need to run a game for a large group (she has 18 kids in her class)
- if you want a simplified game mechanic that encourages collaborative storytelling
The video is somewhat long -- 36 and a half minutes -- so I thought I would write up a short summary that you can skim and decide whether to watch. Character Creation. She had the kids create "group characters", i.e., 3 or 4 kids control 1 character as a group. First they must cooperate to develop a character concept. They went through the following steps (approximate video time-stamp in parentheses): - Prewriting activities: The groups brainstorm what kind of character they want, considering characteristics like name, age, class, race, tools, traits, and most importantly how s/he became a hero. Then they organize this information in a mind map. (12:33)
- Backstory rough draft: In the Poetics, Aristotle wrote that "A story that is whole has a beginning, middle and an end." The groups now translate their prewriting into a one-page handwritten character backstory. They fold one sheet of paper into thirds (13:44) and write the beginning in the top third, the middle in the middle third, and the end in the bottom third. The story describes how the character became a hero.
- Typing: Next they use a computer to type-up their character backstory. (15:22)
- Picture: They use the computer to find a picture suitable for their character. (16:00)
- Editing and Layout: They proofread their stories and lay-it-out on the computer along with the picture they chose for the character. (16:30)
Mechanics. Rather than the usual six abilities, the characters have a small set of skills (fighting, sneaking, thinking, talking). I'm not sure, but I think that they roll a six sided die to determine these skills. The task resolution mechanic is pretty simple: Roll a d20 (and add any applicable skill modifier). A roll of 20 is success. A natural 1 is failure. In-between rolls are up to the teacher/DM. But the important thing is: The kids get to make up their own story describing how the character succeeded or failed at the task. Keep in mind: This is a writing class, and the goal is to get the kids to use their imaginations and tell interesting stories! Inventory. The teacher gave the kids a list of 30 items. She had them pick 5 items from the list. The idea was that they had a knapsack that could fit 5 things inside. The kids know they need to eat and camp out, so all the groups chose food rations, flint or magnifying glass to start fires, and tents or bedrolls for shelter. Character Sheet. The kids combined the main characteristics with the skills and inventories to create character sheets. Here's a plain text schematic of one character sheet: +---------------------------------------------------------------+ | N A M E | +---------------------------------------------------------------+ | | | | P I C T U R E | Name: Loradiand Shieldheart | | | Age: 20 years old | | | Class: Bard | | | Race: Half elf | | | Tool: Custom guitar | | | Traits: Quiet joker | | | Hero Origin: Killed a dragon | +---------------------------------------------------------------+ | | | | Skills | Inventory | | +6 = talking | Small sack | | +4 = sneaking | Magnifying glass | | +3 = fighting | Smallknife | | +2 = thinking | 3x Food rations | | | 1x Bedroll | +---------------------------------------------------------------+
And here's a screen capture (22:00) of another: Adventure Scenario.- Invitation: The characters find a scroll challenging them to find the "puzzle room" hidden under a temple in the city. If they solve all the puzzles, they'll receive a great reward. (22:49)
- City Map: The kids need to find the temple within the city. (24:01)
- Temple: They need to find the general location of the secret door in the temple (24:30). A skill roll was used.
- Secret Door: The kids get a picture and need to find the door hidden in it. (25:42)
- Puzzle Door: After entering the secret door, they're in a chamber with 4 more doors. An inscription says: One door is the entrance to the puzzle room, and the other 3 open to certain doom. The kids get a picture showing all 4 doors. Each door has unique features. There are also shared features. For example, 3 of the doors have handles, but 1 doesn't. One door has tiles, but the others don't. And so on. DM only: It doesn't matter what door they pick -- whichever door they choose leads to the puzzle room -- what's really important is their justification for choosing. They need to talk it out and articulate reasons for their choice. (26:50)
- Word Puzzles: Inside the puzzle room, the kids get pictures of word puzzles that they must solve. For example, one puzzle (28:46) has blanks _ _ _ _ _ with a picture under each blank. The first letter in the word represented by the picture is the letter they need to use to fill in the blank. More difficult puzzles follow, including math puzzles.
- Caged goblin puzzle: In the final puzzle, the puzzle master shows the kids a goblin in a cage (31:48). He is going to release the goblin. The kids must get the goblin back in the cage. But the kids are not allowed to harm the goblin in any way! A skill roll is used. One character tried to push the goblin back in the cage, but he rolled a 1. Loradiand Shieldheart, the Bard (shown above), used her guitar to sing the goblin a soothing lullaby, in order to put it to sleep. She rolled a 20, the goblin fell asleep, and the characters put it back in the cage, winning the final challenge!
- Certificate of Achievement: The teacher printed up certificates of achievement for each character (33:31), congratulating them for solving all the puzzles.
Future plans. The experiment was a huge success -- the kids were very excited to play D&D, and it really helped them to write more. Next semester, the teacher plans to let each of her 18 kids run their own character instead of using "group characters". Commentary/Criticism. One might argue that this isn't "really" D&D, but I think it captures the spirit of D&D, and that it's a good adaptation of the game for younger kids. In the beginning of the video, the teacher makes a faux pas and talks about "0" (instead of "1") as the low roll on a d20, but corrects herself later in the video. No a big deal; "chalk it up" to nerves. I wish this teacher the best of luck in continuing her experiment using D&D in the classroom! I'd like to see her method adopted at other schools, and expanded beyond writing class to math class also.
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Post by krusader74 on Oct 22, 2017 14:55:28 GMT -6
Have a look at British and Foreign Arms and Armour by Charles Henry Ashdown. At the bottom of the document is the index. There's an index entry for Shields: Assyrian, Bronze Age, etc. For each given era, there's a link to a description of shield construction, how it was used, and pictures. Hope this helps!
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Post by krusader74 on Oct 15, 2017 14:16:33 GMT -6
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Post by krusader74 on Oct 12, 2017 14:40:38 GMT -6
Dante and Milton have quite different maps of Hell. If you Google, you can find lots of images depicting these conceptions. Just to follow up on this point, I found the following paper online: The Structure of Milton's Universe: The Shape and Unity of the World in Paradise Lost (2012) by Gábor IttzésIn it, the author discusses four different maps/diagrams of Milton's Universe. Three of these are from old, public domain sources. The four are - David Masson's 1874 diagram. On Google Books, you can find an 1892 edition of Paradise Lost with an introduction by David Masson that reproduces this diagram on page 22.
- Thomas Orchard's 1913 diagram. (NB. The date on the figure below says 1977, b/c it was excerpted from a more recent work.)
- William Warren's 1915 revised version of Professor Hime's diagram.
- Walter Curry's 1957 diagram. The date indicates it is not public domain, but I'm excerpting it below, since it is only a tiny part of a larger work.
And here are the four diagrams: There is also an 1885 edition of Paradise Lost that reproduces David Masson's diagram on page xix. It contains additional diagrams by Professor Hime, including his diagram of Milton's Hell (p. xxiv): Wikipedia has a public domain image of Dante's Universe by an "unknown artist, renaissance to modern". You can see that our world is a sphere, divided into 2 hemispheres: (1) Earth and (2) Water (the Divine comedy was written in the 1300s, before Europe discovered the New World). Hell is beneath Jerusalem, which is located at the center of the Earth. Satan is at the center of the sphere. And the Mountain of Purgatory is antipodal to Jerusalem, in the center of the hemisphere of Water: EDIT: I should throw Gary Gygax's diagram of the "Planes" into the mix. For an explanation, see this post:
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Post by krusader74 on Oct 6, 2017 15:17:36 GMT -6
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Post by krusader74 on Oct 4, 2017 17:05:26 GMT -6
Thanks for the additional literary references! There is also a 1929 short, silent film inspired by the Mallarmé poem you cited, called Les Mystères du Château de Dé (English: The Mysteries of the Chateau of Dice) by Man Ray. It starts by quoting the (French) title of the poem: "Un coup de dés jamais n'abolira le hasard." I don't comprehend French (or Surrealist cinema!) well enough to understand what's going on, but the basic plot seems to be: Some guys throw a pair of oversized six-sided dice in order to decide what to do. You can watch it on YouTube:
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Post by krusader74 on Oct 3, 2017 16:41:58 GMT -6
Nice article! Inspirational material with Halloween coming up at the end of the month. I noticed a few important "hot spots" missing from the list though... 1. Boca do Inferno (English: "Mouth of Hell"). Quoting from another page on atlasobscura.com it is On yet another atlasobscura.com, there's the story about how Alesiter Crowley supposedly went there to commit suicide, after a fight with his mistress. It was just a publicity stunt. He published the following suicide note (in Portuguese) in the newpaper. It is written to his mistress: 2. Cleveland, OH. In the Buffyverse, there's a Hellmouth in Sunnydale, CA, and another one in Cleveland, Ohio. 3. Under the Temple Mount in Jerusalem In Dante's Inferno, the author finds the gate to Hell in a cave outside a dark wood. Geographically, this gate is underneath where the altar of burnt offerings stood in the Temple on Mount Moriah in Jerusalem. Dante and his peers believed this location was the center of the world. The gate of Hell has a stone with the following famous inscription on it. I'm using John Ciardi's translation, since he sticks with the original terza rima rhyme scheme: Hell is described as a vast pit in the shape of a funnel (or inverted cone) having nine ledges (concentric "circles") where sinners get tortured forever. The apex is at the centre of the earth. There, at the center of the earth, Satan is frozen in ice. Beneath Satan is a tunnel, leading to the Mountain of Purgatory, on the opposite side of the earth. The Mountain of Purgatory is like a giant pyramid with nine ledges. At the apex was the Garden of Eden. Above that are the nine heavens of Paradise. You could use the Antipodes Map to find the exact point on the surface of the earth opposite to Jerusalem. You find that point (-31°46'S, 144°47'W) right smack in the middle of the south Pacific, roughly east of Brisbane Australia and west of La Serena Chile... If there is a Mountain of Purgatory there, it is probably an underwater volcano. 4. Monster mouths. In Old English prose and poetry, a "Hellmouth" was quite literally the mouth of some giant monster that would swallow you, and you'd go to Hell. Some examples: Note that these creatures don't "bite" sinners -- they swallow them whole. Hell is in the bowels of these creatures. No wonder it smells like sulfur. 5. The Gates of Hell in Milton. Book Two of Milton's Paradise Lost describes the gates of Hell. His descriptions are adapted from Virgil's descriptions of the gates of Tartarus. There are "three thrice-fold" gates: The inner is brass, the middle gate is iron, and the outermost stone. After Satan's daughter, Sin, throws open these gates for him, Satan flies out of Hell into Chaos, a gap between the creative power that makes Hell a place of punishment and Heaven a place of bliss. Chaos is totally absent of divine Law and order. It is "governed" by three beings: Chaos, Chance and Night. As such, Chaos (the place, not the person) is filled with confusion, uncertainty and darkness. These three beings are personified, and they have a throne in the middle of Chaos (the place). At the end of Chaos (the place) is the "Pavillion of Chaos," equated with Pluto. This in turn leads to Limbo and the World. Dante and Milton have quite different maps of Hell. If you Google, you can find lots of images depicting these conceptions. The number three is very important to the architecture of both conceptions--- In Dante: Three main locations (Inferno, Purgatory, and Paradise), each with nine (= 3 * 3) circles. Terzi rima (three-line stanzas). Satan has 3 faces, devouring a different arch-sinner in each of his three mouths. In Milton: Three unearthly locations (Hell, Chaos, Heaven). There are three thrice-fold gates of Hell. An infernal Trinity: Satan, Sin (his daughter), and Death (his grandson). And a chaotic Trinity: Chaos - Chance - Night. Milton's Satan has 12 (=4*3) disciples (like Jesus's 12 disciples), who are drawn from 3 ancient religious pantheons: Greek - Egyptian - Semitic. And so on... Three s all over.
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Post by krusader74 on Sept 27, 2017 18:19:36 GMT -6
I wanted to create a list of references to dice in art, music and literature. Here's what I got so far: Dicing with... DeathIn the Rime of the Ancient Mariner by Samuel Taylor Coleridge, the mariner's ship is driven south by a storm and eventually reaches Antarctic waters. An albatross (a symbol of Christ) appears and leads them out of the ice jam where they are stuck. The mariner shoots the albatross with a crossbow, ending Part I. In Part II, the ship gets lost in uncharted waters near the equator. The angry crew makes the mariner wear the albatross around his neck (like a crucifix, as a reminder for his sin). The crew starts to die of thirst ("Water, water, every where, / Nor any drop to drink."). In Part III, the mariner's ship encounters a ghost ship (the "hulk"). On board the hulk, Death plays a game of dice with his wife, the Nightmare Life-in-Death, who wins the game ("The game is done! I've won! I've won!"). The outcome of this dice game determines the mariner's fate: He will endure a fate worse than death as punishment for his killing of the albatross... I won't spoil the ending. I just wanted to point out the image of the dice game with Death. For the 1876 edition of the poem, artist Gustave Doré did a series of engravings. Here is his illustration of the dice game with Death: Shakespeare plays... at diceHere's a list of quotes from Shakespeare's plays about dice, dice games, fortunate/unfortunate rolls, and cheating with loaded dice: - "If Hercules and Lichas play at dice..." (Merchant of Venice II.i.32)
- "Be these the wretches that we play'd at dice for?" (Henry V IV.v.8)
- "Proud of their numbers and secure in soul, The confident and over lusty French Do the low-rated English play at dice" (Henry V Act IV, Prologue, line 19)
- "fullam" (Merry Wives of Windsor I.iii.94): a kind of false dice loaded at the corner
- "...were it good To set the exact wealth of all our states All at one cast? to set so rich a main On the nice hazard of one doubtful hour?" (Henry IV, Part I IV.i.45-48)
- "...false As dice are to be wished by one that fixes no bourn 'twixt his and mine" (Winter's Tale I.ii.132)
- "die" (Winter's Tale IV.iii.27)
- "...the very dice obey him" (Antony and Cleopatra II.iii.35)
- "cogging" (Othello IV.ii.132): cheating with loaded dice
- "Slave, I have set my life upon a cast, And I will stand the hazard of the die" (Richard III V.iv.10)
- "You know how much the grosse summe of deuce-ace amounts to... which the base vulgar call three" (Love's Labour's Lost I.ii.48): deuce-ace: roll 2+1=3 on 2d6
- "... well run, dice!" (Love's Labour's Lost I.ii.232)
- "This is the ape of form, monsieur the nice, That, when he plays at tables, chides the dice In honourable terms" (Love's Labour's Lost V.ii.326): tables=backgammon
- "Abate throw at novum, and the whole world again / Cannot pick out five such, take each one in his vein." (Love's Labors Lost V.ii.545): novum=novem quinque, a dice game where the principal throws are the 5 and 9
- "Your lordship is the most patient man in losse, the most coldest that ever tum'd up ace" (Cymbeline II.iii.2)
- "I had rather be in this choice than throw ames-ace for my life" (All's Well That Ends Well II.iii.85): ames-ace=roll 1+1=2 on 2d6; aka "snake eyes"
Dicing with... JesusIn 1480, Franco-Flemish composer Josquin des Prez wrote Missa Di dadi, the "Dice Mass." Sacred renaissance music used polyphonic textures. The foundation, or cantus firmus, was a Gregorian chant, sung by the tenor. On top of this traditional melody, the composer added other vocal parts. In this case, Josquin adds 3 voices, for a total of 4. But here's the interesting part about this mass, Josquin determines the speed-ratio between the tenor cantus firmus and the other voice parts by rolling dice. And in the musical score, he draws pictures of the dice rolls, as shown here: Josquin wrote the mass in Milan where dice gambling was rampant. An ordinary mass has 5 divisions. Here are the divisions, along with the dice rolls/speed ratios: - Kyrie 2:1, i.e., the the note-lengths of the cantus firmus is doubled to fit with other vocal parts
- Gloria 4:1
- Credo 6:1
- Sanctus-Benedictus 5:1
- Agnus Dei
These ratios represent an imaginary game of dice, which goes through several turns until the Sanctus's roll of 5+1=6 ends the game in a victory, a metaphor for the victory of Christ over the sinful gamblers' dice. You can listen to this beautiful, polyphonic mass on YouTubeWhat other references to dice can you find in classic art, music and literature?
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Post by krusader74 on Sept 27, 2017 18:18:49 GMT -6
Are dice throws predictable?Dice throws are deterministic mechanical systems. But determinism does not imply predictability: They may be chaotic, in which case, a small, immeasurable perturbation in the initial conditions would lead to drastically different, unpredictable outcomes. To answer this question, you first write out the equations of motion for dice being thrown and bouncing on a surface. I won't bore you with the equations... you can see them in the references at the end. These equations merely encapsulate the physical laws governing the system: - Conservation of mass
- Conservation of energy (potential + kinetic)
- Conservation of momentum (translational and angular)
Then you check for sensitive dependence on initial conditions: Do trajectories with different initial conditions separate exponentially? In the case of dice, it turns out the answer to this question is NO, as shown in the following two papers: 1. Dice Throw Dynamics Including Bouncing (XXIV Symposium Vibrations in Physical Systems, 2010) by Juliusz Grabski, et al. Quote: 2. Iterated-map approach to die tossing ( Physical Review A, Volume 42, No 8, 1990) by R. Feldberg, et al. Quote:
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Post by krusader74 on Sept 13, 2017 8:15:47 GMT -6
I'm thinking about a 3d6 sum roll-under mechanic where bonuses and penalties are applied with additional dice (usually one die, but sometimes two, and rarely three). In a bonus situation, (3+x)d6 are rolled, and the lowest three die are totaled. In a penalty situation, (3+x)d6 are rolled, and the highest three die are totaled. Hopefully that's clear. Now my question is, what would be the probability distribution of the 3-18 results in each case? Really great question, foxroe !! sixdemonbag already posted a very practical answer, using anydice.com. So let me try to provide additional theory and some computational alternatives, since I'm sure others will ask questions about this same distribution in the future. The probability mass function f(x) of the sum of the k-highest of n s-sided dice is given by an extremely unwieldy combinatoric formula: where and The formula was derived by a user called "techmologist" on Physics Forums. To see the step-by-step derivation of this formula along with further explanation and analysis, please read the linked 2-page article: But this only answers part of your question, because - you also want the distribution for rolling n s-sided dice and keeping the lowest k
- you really want the cdf, not the pmf, since you're thinking about a "roll-under mechanic"
To answer these two concerns succinctly: First: The pmf for rolling n s-sided dice and keeping the lowest k looks like the mirror image of the pmf for rolling n s-sided dice and keeping the highest k. For example, the pmf g for rolling 4 6-sided dice and keeping the lowest 3 looks like the pmf f for rolling 4 6-sided dice and keeping the highest 3 after you swap the 18 with the 3, swap the 17 with the 4, and so on. In other words if π is the permutation: then g = f ∘ π Note that π is an involution, i.e., π∘π= 1 or equivalently π=π -1, so that f=g∘π too. Second: If f(x) is the pmf of the discrete random variable X, then its cdf is F(x) = P(X ≤ x) = ∑ t≤x f(t) If you were a masochist, you could write out the cdf F(x) for 4d6k3 in full to see what it looks like. It looks ugly. Very, very ugly. You'd see (among other things), five nested summation symbols. You really want a computer to manipulate these functions for you... I tried looking into your Palamedes program, but I couldn't figure out what code to input to simulate the conditions. Not your fault! Palamedes is highly idiosyncratic. For example, you can't use the letter "x" as a variable name, because it is the syntax for an exploding die. Furthermore, Palamedes isn't finished or fully debugged yet. Nevertheless, you can get the Prob & Stat data you need from it, because it implements techmologist's formula. Here are some input examples: 4d6k3 /* generate a random variate */ cdf(4d6k3) ∘ 18 /* evaluate cdf at 18, returns 1 */ p(4d6k3 ≤ 18) /* same as above, only more pleasing notation */ table(p(4d6k3)) /* table for the pmf */ table(cdf(4d6k3)) /* table for the cdf */ table(QQ(cdf(4d6k3))) /* table for the cdf as fractions, not decimals */ table(stats(p(4d6k3))) /* mean, variance, etc. */ barchart(p(4d6k3)) /* plot of the pmf */ barchart(cdf(4d6k3)) /* plot of the cdf */ perm <- [3..18] -> [18..3] /* the permutation shown above */ roll_4d6_keep_lowest_3 <- p(4d6k3) ∘ perm /* explained above */ p(roll_4d6_keep_lowest_3 ≤ 18) /* cdf, returns 1 */ p(roll_4d6_keep_lowest_3 ≤ 3) p(roll_4d6_keep_lowest_3 < 0) /* returns 0 */ table(roll_4d6_keep_lowest_3) table(cdf(roll_4d6_keep_lowest_3)) table(stats(roll_4d6_keep_lowest_3)) barchart(roll_4d6_keep_lowest_3) barchart(cdf(roll_4d6_keep_lowest_3)) And here is the (really long...) output from running this script: You can PM me with follow-up questions. Anyway, I hope this helps!
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Post by krusader74 on Sept 12, 2017 17:58:03 GMT -6
How many times must you roll a die to test for fairness?I see this question raised a lot. But so far, I haven't seen a thoroughly satisfying answer. So I decided to try and write one... Effect sizeProblem: Too many statistical studies fetishize an arbitrary 0.05 significance level (alpha) such that a p-value of 0.04999 is a publishable finding but p=0.05001 is not. The big problem with this is that even the most trivial effect will eventually become statistically significant if you test enough (because: the law of large numbers). Solution: The solution to this problem is a system for deciding precisely how large the effects in our data really are. In statistics, the effect size is used to describe a difference between two groups. Calculating effect sizeEach type of statistical test has its own way of calculating effect size. When we test dice for fairness, we typically use Pearson's chi-squared (χ²) test, a type of Goodness of Fit test (of which there are many). The measure of effect size used for chi square tests is Cohen's w, defined as: Here the p 0i refer to the probabilities under the Null Hypothesis H 0, and the p 1i refer to the probabilities under the Alternative Hypothesis H 1. Example: You throw a six-sided die a bunch of times to test for unbiasedness. The Null Hypothesis H 0 says that p 0i=1/6 for i=1 to 6. The Alternative Hypothesis H 1 says that the p 1i ≠ p 0i for some of the i's. Remark 1: to actually compute w, we need concrete values for the p 1i, e.g., the die is loaded and the probability of throwing a six is 1/4 and the other sides are equiprobable. Remark 2: The null hypothesis states what's commonly assumed to be true. When we test a die for fairness, we generally assume all sides are equally probable. So if you're testing a d20 for faireness, then H 0: p 0,i=1/20 for i=1 to 20. However, if you happen to manufacture loaded d20s such that the "20" appears 10% of the time and the "1" never appears, and you want to test your loaded dice for quality control, then your null hypothsis won't be equiprobability. Instead it will be H 0: p 0,1 = 0, p 0,i = 1/20 for i in 2 to 19, and p 0,20 = 1/10. What is meant by "small" and "large" effect sizes?In Cohen's terminology, a "small" effect size is one in which there is a real effect --- something is really happening in the world --- but which you can only see through careful study. A "large" effect size is an effect which is big enough (or consistent enough) that you may be able to see it "with the naked eye." The following table summarizes what is meant by "small" and "large" in terms of Cohen's w: Type of effect | Effect Size (w) |
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trivial effect | < 0.10 | small effect | ≈ 0.10 | moderate effect | ≈ 0.30 | large effect | > 0.50 |
Example: You throw a six-sided die a bunch of times to test for unbiasedness. If the die is really unfair and the percentage of times each face appears is given by the following table, then it has the effect size shown in the last column: Type of effect | % of 1s | % of 2s | %of 3s | % of 4s | % of 5s | % of 6s | Effect Size (w) |
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trivial effect | 14.27 | 16.67 | 16.67 | 19.07 | 16.67 | 16.67 | 0.083138 | small effect | 18.60 | 14.40 | 16.80 | 18.00 | 14.20 | 18.00 | 0.105451 | moderate effect | 14.70 | 20.70 | 16.50 | 7.10 | 21.00 | 20.00 | 0.291452 | large effect | 0.00 | 23.60 | 23.60 | 11.80 | 21.00 | 20.00 | 0.506454 | (very!) large effect | 3.30 | 0.00 | 93.30 | 0.00 | 0.00 | 3.40 | 2.058253 |
PowerPower and sample size estimations are used to determine how many observations are needed to test the null hypothesis against an alternative. There are two types of errors we can make in a statistical study: - A type I error is said to have occurred when we reject the null hypothesis incorrectly. The significance level (or "alpha") is the probability of making a type I error, i.e., α = P(reject H0 | H0 is true). Note: alpha is a conditional probability, and there are many common fallacies concerning them.
- A type II error is said to occur when we fail to reject the null hypothesis incorrectly. The chance ("beta") of making a type II error is β = P(fail to reject H0 | H1 is true).
The power of a test is the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. Mathematically, power is defined as 1 - β = P(reject H 0 | H 1 is true). It is the complement of beta. Note: Each type of statisical test has its own method for computing power (and/or beta). In Pearson's chi-squared (χ²) test, the formula for power (and/or beta) is (**)where - F is the is the cumulative distribution function (cdf) for the noncentral chi-square distribution
- nc is the noncentrality parameter of the noncentral chi distribution. It is defined as nc = n * w2, where n is the sample size and w is the effect size.
- df are the degrees of freedom for the test. E.g., when you throw a six-sided die under a given hypothesis which defines the probability distribution, then df=5 because knowing 5 of these probabilities determines the sixth.
- xcrit is the critical value for the given value of alpha, i.e.,
Sample SizeTo determine the sample size required to test for an unbiased die, follow these steps: - Decide the significance level α. By convention, this is usually chosen to be 0.05, 0.01 or 0.001.
- Decide the power 1 - β. By convention, this is usually chosen to be 0.80 or 0.90.
- Decide the effect size w you want your test to detect.
- Solve the above equation (**) for n.
The last step would be extremely difficult to compute by hand, given the complexity of the noncentral chi distribution. There are many solvers freely available online. I've written my own free, public domain Python module (chisq.py) to do the math. It computes: - Effect size w, given Alternative Hypothesis (and optional Null Hypothesis, assumed to be equiprobable outcomes by default)
- Power 1 - β, given w, n, df (and optional alpha, assumed to be 0.05 by default)
- Sample size n, given w, df (and optional alpha (0.05 by default) and power (0.80 by default))
The script requires the scipy.stats and scipy.optimize libraries. There's actually very little code in the script, since scipy does all the heavy lifting. If I find a decent JavaScript library that computes the noncentral chi distribution (or write my own someday...), then I will write a JavaScript version of these functions and put them online as a jsfiddle.
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Post by krusader74 on Sept 1, 2017 15:01:51 GMT -6
I was the person who told David about the forums, and talked him into posting there. Thank you Robert, we all owe you a great debt of gratitude! These posts are a true treasure trove!
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Post by krusader74 on Aug 30, 2017 14:22:21 GMT -6
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