|Ivars Peterson's MathTrek|
December 22, 1997
In Tricky Dice, I described a remarkable set of four, specially numbered dice. In a two-person game, no matter which one of these dice your opponent would pick, you could select one of the remaining dice and, with a high probability, come out the victor in a game of at least 10 throws, in which the higher number wins each throw.
Nathaniel Hellerstein of San Francisco pointed out a set of three dice that has the same nontransitive property. The dice have faces numbered as follows:
Red beats yellow, which beats blue, which beats red, each with a probability of 5/9.
Hellerstein calls them magic dice because their numbers can be found in the following magic square, in which the rows, columns, and diagonals all add up to 15 (see More than Magic Squares):
Other magic squares also yield nontransitive dice sets, Hellerstein says.
Any set of nontransitive dice can be used to play a diabolical game called "Black Magic." Players roll one die each, with the loser of the last round being the first to choose a die. "This way, losers tend to keep losing, and winners tend to keep winning," Hellerstein remarks.
In Monopoly Dollars and Sense, I discussed recent efforts to calculate the probability of landing on different squares in the board game Monopoly. However, those calculations typically required making some simplifications, such as leaving out the consequences of certain Chance and Community Chest cards.
In his analysis of Monopoly, software engineer Truman Collins not only included the Chance and Community Chest cards but also took into account such subtle factors as the differing probabilities of having already rolled two doubles when sitting on a given square. He also examined the probabilities of landing on each square when the player's strategy is to get out of jail immediately by paying $50 instead of staying in until doubles are rolled.
Here's the resulting table of probabilities (top 12), expressed as percentages and ranked in order (from highest to lowest) for the get-out-of-jail-as-soon-as-possible strategy (listed first) versus the prolonged-jail-stay strategy (listed second). "Visiting jail" and "in jail" are considered separately.
|1 In Jail||3.9499||1||9.4569|
|4 New York||3.0852||7||2.8118|
|5 B&O Railroad||3.0659||4||2.8930|
|6 Reading Railroad||2.9631||8||2.8010|
|8 Pennsylvania RR||2.9200||11||2.6354|
|9 Free Parking||2.8836||5||2.8253|
|11 Water Works||2.8074||10||2.6507|
|12 St. James Place||2.7924||9||2.6802|
Evidently, the jail strategy does affect the probabilities of visiting certain squares. Indeed, the strategy of immediately paying to get out of jail would decrease the chances of visiting squares that are 2, 4, 6 (St. James), 8 (Tennessee), 10 (Free Parking), or 12 spaces past the jail square, reached by rolling doubles.
The complete results and much additional information are available at http://www.teleport.com/~tcollins/monopoly.shtml.
Can these data be used to play a better game of Monopoly? "It should be possible to add up all of the expected income values for all of the properties in a partially completed game and get an estimate of who is likely to win in the end," Collins remarks. "Of course, this ignores lots of factors in a particular game, such as cash on hand, bargaining, and luck. The capricious nature of short-term probability insures that you never know what will happen in the next few rounds."
My article on Contra Dancing and Matrices introduced contra dancing as a highly structured descendant of English country dance, practiced with great devotion and abandon throughout the United States. The dancers typically line up in groups of four to produce a long column down the floor. Each block consisting of two couples can be thought of as a two-by-two matrix.
However, there's more to the matrix model of contra dancing than a mere resemblance between the couple configuration and a particular simple matrix. Ted Crane, a computer system analyst and contra dance caller, suggested that "matrix techniques are an effective way to analyze and model contra dances."
Simplifying matters somewhat, Crane proceeds in this fashion: There are 24 different ways that four people can be arranged in a square. One can then build a 24-by-24 matrix, in which the element in the ith row and jth column is a list of all the dance figures that go from configuration i to configuration j. Multiplying that matrix by itself provides a list of two consecutive figures that go from i to j.
Repeating the multiplication as many times as necessary fills out the entire dance. "You've now written every contra dance which can be written . . . using those figures and that limited set of formations," Crane says.
"At this point, it is desirable to filter the compound figures to obtain pleasing continuity between consecutive figures," he continues. "This is subjective, and every contra dancer could give you an opinion."
Anybody can write a dance, Crane says. It's really quite mechanical.
The success of a good author, however, depends on choosing continuity with widespread appeal. Of course, the appeal of a given dance also depends on the caller being able to teach the figures and the dancers being able to execute them!
Ted Crane calls contra dances in Ithaca, N.Y. (http://www.tedcrane.com/TCCD/main.htp).
So, happy dancing in the new year, and please keep those cards and letters (and e-mails) coming!
Copyright 1997 Ivars Peterson
Gardner, Martin. 1983. Nontransitive dice and other probability paradoxes. In Wheels, Life, and Other Mathematical Amusements. New York: W.H. Freeman.
Stewart, Ian. 1997. The lore and lure of dice. Scientific American (November):110-113.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.