More about Dice 2/6/12

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Melvin Morris
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1 More about Dice 2/6/12

2 Random Numbers As you know, random numbers have lots of uses in games, especially gambling games. Any game that is a blend of chance and skill needs a way to supply random numbers. We use the term Random Number Generator (RNG) to describe any hardware or software device that can generate random numbers.

3 Random Numbers Quality random numbers are surprisingly difficult to generate. Today s board games usually use dice to produce reasonably good random numbers but it wasn t always so.

4 Tests of Randomness Testing for true randomness uses a set of statistical techniques that we won t spend much time on. Curiously, if someone asks you to spew out a stream of random numbers, the sequence you produce will actually be highly nonrandom.

5 An Experiment I need three volunteers. One to spew random numbers One to record these numbers. One to keep track and stop everyone after 50 random numbers have been generated.

6 Human Randomness People tend to favor popular digits like 3 and 7 and will definitely tend to avoid long strings of repeating digits, or repeating patterns of digits. Yet these patterns come up in real streams of random numbers all the time. Once upon a time in a casino in Monte Carlo, red came up 26 times in a row. Everyone bet on black, thinking it just had to come up.

7 Binary Lots The oldest way to get a supply of random number for purposes of game play was though the use of binary lots. The word, lot, as in the biblical phrase, the casting of lots, is the origin of modern words like lotto and lottery.

8 Binary Lots Start with something from nature: Cowrie shells A flat piece of wood with bark on one side. Or a piece of bark, with a shiny (inner) side and a rough (outer) side. A length of dowel, flattened along its length. Later on, tetrahedral dice and coins. Tetrahedral dice were used in the original Royal Game of Ur.

9 Throwsticks We used the throwstick in History and Culture of Games: a flat or cylindrical (with flatted side) piece of wood. Ours were tongue depressors marked with X s on one side to simulated rounded or rough. But these are machine perfect, and don t behave like old-style binary lots.

10 Binary Lots Use several of these, and count how many of them meet the criterion: Cowrie shells can land open or closed side up. The wood can land bark side up or down. A dowel can land on the rounded part or the flat. Coins can land heads or tails.

1 to the throw of the lots to get a 1 to 6 outcome.

11 Binary Lots If your game needs movements from 0 to 5 spaces, you could use five binary lots. This 0 to 5 may seem just like rolling a single die (which, of course, generates 1 to 6), so just add 1 to the throw of the lots to get a 1 to 6 outcome. But the probability distributions will still differ. Thus, just changing the means of getting random numbers from binary lots to dice changes the gameplay itself.

12 With Binary Lots There are 32 possible arrangements for five binary lot objects to come up (2 5 ): Five combinations total 1 and five total 4. Ten combinations total 2 and ten total 3. Only one totals 0 and only one totals 5. This accounts for all 32 combinations. As you can see, 2 and 3 come up a lot more often than 0 or 5.

13 With a Single Die For a single honest die, each of six possibilities has an even 1 in 6 chance of coming up. Binary lots, unlike the die, have a probability distribution unevenly divided across outcomes. Most of the games that used binary lots exploited this fact: rare combinations were usually assigned especially nice or especially dire consequences.

14 Comparison All Five Binary Lots Single 6-Sided Die

15 Lopsided Throws Making matters more interesting was that the nature of the objects themselves (bark, cowrie shells) used for binary lots usually did not have true 50%-50% outcomes. These are objects from the natural world, and hardly regular in shape.

16 Lopsided Throws Is a cowrie shell more likely to land open side up or open side down? What about a flatted dowel? Only a large number of trials can tell us. So each hand-crafted game set will yield different game play because no two binary lot objects will be exactly alike.

This is because there is slightly more metal removed from the heads side than the tails.

17 Coin Tosses At least coins are 50%-50% or are they? Heads occurs very slightly more often than tails (50.2% versus 49.8% for a U.S. quarter). This is because there is slightly more metal removed from the heads side than the tails. Our flat throwsticks are reasonably honest since they are machine-made with no favored side. But you can throw them dishonestly.

18 Knucklebones As we have seen, the knucklebones (a/k/a astragals) of domesticated animals (usually sheep) were also used from the earliest times as a means of generating random numbers. Knucklebones are roughly cubical in shape, and each side looks distinctly different, making it a fairly good RNG. In practice, the two ends never come up.

19 Knucklebones They rarely land on the ends

20 Early Dice Knucklebones were known to the ancient Romans, probably through their contact with Egypt. The phrase throwing the bones to refer to the casting of dice is still in our language. Chances are, the development of dice was inspired by knucklebones. Be careful of jargon: the singular of dice is die.

21 Exact Origins Dice are among the oldest gambling devices known. Dice were supposedly played by the Egyptian gods (2000 BCE and earlier). We saw that The Royal Game of Ur made use of tetrahedral dice (3000 BCE). It is thought that dice were initially used in for sacred divination ceremonies in Egypt.

22 Exact Origins Dice have been found in numerous cultures and have been made from numerous materials. The Chinese have also used dice since ancient times. It is not known whether there was commerce between them and the Middle East that brought these in trade (in either direction). The Chinese later invented the domino based on the original design for dice.

23 Materials Dice have been made of plum and peach stones; seeds; buffalo, caribou, and moose bone; deer horn; pebbles; pottery; walnut shells; and beaver and even woodchuck teeth. In later Greek and Roman times, although most dice were made of bone and ivory, others were of bronze, agate, rock crystal, onyx, jet, alabaster, marble, amber, porcelain, and other materials. Source: dicecollector.com

24 Uniform Fabrication Modern dice had to wait until it became possible to fabricate reasonably accurate cubes out of a material with reasonably uniform properties (like ivory or bone). The earliest dice were of course crudely made, and therefore would not have a true 1/6 probability per side.

Casino Dice Made out of cellulose acetate, casino dice are accurate to a tolerance of.0005 inch Casino dice are very accurately made, almost as if machined.

25 Casino Dice Made out of cellulose acetate, casino dice are accurate to a tolerance of.0005 inch Casino dice are very accurately made, almost as if machined. This is essential since casino wants to know the expected take on games of chance based on dice (usually Craps). Even a slightly inaccurate die, if detected by a skilled gambler, could work against the house.

26 To Prevent Cheating The ancient Romans used a box called a fritillus, or dice box. The dice were tossed into the top, where they hit internal baffles, and came out at the bottom after bouncing down those little stairs.

27 To Prevent Cheating At a regulation craps table, the shooter is supposed to throw the dice against the table walls, which are covered with little foam pyramids.

28 How to Cheat With Dice Easiest way is to make the cube slightly rectangular. Consider how a tossed brick lands: it lands on the face with the largest surface area most often and on the face with the smallest surface area least often. Add weights. Not possible with see-through casino dice, whose spots are only machined very slightly into the surface.

29 How to Cheat With Dice Use a precision file to add burrs to some of the edges so that can catch the green baize of the table. This favors the digits on the sides opposite the roughed-up edge. Use dice with some of the spots changed. You can only see three sides at a time, so it s possible to get away with this, at least for a little while and if the dice are opaque.

30 Multisided Dice Dice need not be cubical. Dungeons and Dragons popularized the idea of dice based on the five regular solids: Tetrahedron (4 sides) Hexahedron (a/k/a/ cube) (6 sides) Octahedron (8 sides) Dodecahedron (12 sides) Icosahedron (20 sides)

31 Multisided Dice Mathematically, the first five are the only regular solids possible in our Euclidean 3D world. Pythagoras (of Pythagorean Theorem fame) believed that these five solids (also called the Platonic solids) had mystical properties. This mystical notion meshed nicely with the ethos of Dungeons and Dragons.

33 Multisided Dice Dice have been fabricated with other numbers of sides, even up to a 100-sided one. Of course, there are also dice with things on them other than a succession of digits, for example, Poker Dice. A large icosahedron is used in the Magic 8 Ball fortune-telling toy.

34 Modern Dice Today, of course, dice are easy and cheap to make, and so are found in board games of all sorts. But it always hasn t been so.

35 How to Play Craps

36 Private Craps 2 Miss 3 Miss Shooter s Point 7 Pass Shooter s Point 11 Pass 12 Miss

37 Side Bets Flat bet - before shooter begins, two players bet equal stakes against each other that the shooter will eventually pass or miss

38 Side Bets Point bet - after shooter reaches a point number, bet whether shooter will (or won t) reach point before rolling a 7 Point Bet 4 or 10 5 or 9 6 or 8 Favorable Outcomes Unfavorable Outcomes True Odds 6 to 3 6 to 4 6 to 5 2 to 1 3 to 2 Probability of Success 1/3 2/5 5/11

39 Side Bets Hard Way bet - after shooter reaches an even point value, bet as to whether or not the point will be reached the hard way (doubles) Hard Way Bet 4 (10) 6 (8) Favorable Outcomes 1 1 Unfavorable Outcomes 8 10 True Odds 8 to 1 10 to 1 Probability of Success 1/9 1/11

40 Side Bets Come bet - Start a new game with another player at any stage in the main game Shooter s next throw becomes the first throw of your new game Equivalent to flat bets (equal stakes) Come bets continue into the next game if the shooter wins main game before a decision is reached

41 Side Bets Proposition bet - any agreed upon bet and odds at any point in the game Most popular: One Roll Bet - any bet on the outcome of the next throw - any craps is a bet that the next throw will be a 2,3 or 12 One Number Bet - a bet that a specific total will or will not be reached before a particular event

42 The Odds Somewhat more involved Bottom line, shooter has a 49.29% chance to pass (valid for Center, Flat, and Come Bets) Slight disadvantage, but most people consider it negligible for the fun of being in the captain s seat (such as it is)

43 One Number Bets Hard Way Note! New! One Roll Bets Sucker!

44 Playing the Field The Field bet is a one-shot bet that the next roll will be a 2, 3, 4, 9, 10, 11 or 12. Doing the math: = 16 (out of 36) So 4/9ths of the time the player wins. This is 11% for the house, a tidy profit. Moral: Never Play the Field (at least in Craps)

45 Play the Boring Bets! Pass, don t pass, come, don t come - casino gets cut of ~ 1.4% One Roll Bet 7 Craps Favorable Unfavorable True Odds 30 to 6 32 to 4 34 to 2 35 to 1 34 to 2 35 to 1 5 to 1 8 to 1 17 to 1 17 to 1 Payout 4 to 1 7 to 1 15 to 1 30 to 1 15 to 1 to 1 Banker s Cut 1/6 1/9 1/9 5/36 1/9 5/ % 11.11% 11.11% 13.89% 11.11% 13.89%

46 Category Games

49 Catego Each turn, player rolls 2d6 and enters the number in any unused column in their row. Every player must choose a different column in round 1. Play 11 rounds. The player with the highest number in a column at the end scores the value of that column. Ties do not score. Player 1 Player 2 Player 3 Player Scores

50 Catego Scores Player Player Player Every player must choose a different column in round 1.

51 Criss-Cross

52 Mechanics 2d6 will be used to generate single numbers, 0-9 (10 = 0, 11 = 1, 12 = 2) Everybody rolls for their individual center square number Play proceeds clockwise, with each player rolling the dice, announcing the result to the group, and passing dice to the left After each throw, each player places the number in their grid where they see fit

53 After 4 Rounds

54 Scoring The idea is to align the numbers in valuable rows and columns to score maximum points Five equal digits: n points Four equal digits: m points Three equal digits: p points Two equal digits: q points Sequence of 5 consecutive digits: s points 0s can be low or high Sequences can loop (e.g = s) It is possible to get more than one score in a line (e.g. full house = p + q points)

55 Sample Score (assumes 5/k = 6 pts, 4/k = 4 pts, 3/k = 2 pts, 2/k = 1 pt, sequence = 4 pts

56 Exercise Make the scoring fair 5 of kind=? points 4 of kind =? points 3 of kind =? points 2 of kind =? points Sequence =? points Other scoring rules? Count diagonals?

57 Variants Criss and Cross You may only score the lower of the row and column scores Criss vs. Cross Two players, one score sheet, alternate turns One player is scoring the rows, the other the columns! (madness!)

58 Homework Longer term

59 Dice Project Design and develop a game featuring the use of one or more dice. The game may have other components, but the dice should be center stage Your game must have significant intrinsic value, meaning it will not rely on wagering of real money to make it interesting. Demonstrates triangularity (risk/reward structures) via probability You should evaluate your game based on how long it holds the players attention. Things to consider: Should your game have a theme? Who is your audience? Is your game understandable? Is it fun? Why is it fun?

60 Dice Project For Thursday, 2/15: A working prototype that has been through at least two iterations of testing and revision. Bring it to class for group playtesting

61 Dice Project For Thursday, 2/23: A finished game prototype (use a Ziploc bag or other secure container for pieces). Notes from your brainstorming, or whatever process you used to get to your initial idea. The initial set of rules. Written analysis of each of the playtests you conducted (note dates and times, and who played even if it is just you), and the revisions you made. A final set of rules. An estimate of what this game would sell for in retail, if it were mass produced. Explain what you base this on. Anything else you feel is relevant.

62 Homework Extra Credit Prove the dealer s cut on a Miss or a Don t Come bet is ~1.4% (Remember - boxcars don t count on first roll!)

63 Back to Criss-Cross What Reiner Knizia came up with

64 As published Five equal digits: 10 points Four equal digits: 6 points Two equal digits: 1 point Diagonals count Three equal digits: 3 points Sequence of 5 consecutive digits: 5 points

65 A B C D E F RK 5/k /k /k /k Seq / Diag? Y N N N Y N Y

66 G1 G2 G3 G4 G5 G6 RK 5/k 10 4/k 6 3/k 3 2/k 1 Seq. 5 Diag? Y

67 A A (alt) B C RK 5/k /k /k /k Seq Diag? N N (FH=5) 4seq=3 N Y

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as:

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as: E n ( Y) y f( ) µ i i y i The sum is taken over all values