Links by Variant
"It seems like a weird use of time to catalog variations of
pig games, but we each have our priorities in life."
 Jim Koplow,
Koplow Games
 One Die, 1 is Bad (oldest variant in literature from "Scarne on
Dice" by John Scarne, 1945):
 Two Dice, 1 is Bad:
 NRICH
Online Maths Club Pig Game  ones  Pig with 2 dice. Throwing a 1
ends the turn with no score. Throwing two 1's sets the player score
to 0. (original
link)
 Durango Bill's
partial analysis of two dice Pig.  A hold value analysis is
performed assuming one is trying to reach 100 in N turns.
However, the analysis doesn't provide a policy for what N should be.
"In practice (in a game) you as a player have to determine how many
turns you should use to try to get to 100." This analysis is more
complex than Knizia's oddsbased analysis. Although there are regions
in our Pig state space where the optimal player is clearly trying to
reach the goal in one or two turns, the true optimal policy is more
complex than "win in N turns" over much of the state space. We
conjecture that the solution for this version of Pig would be
qualitatively similar to our own, and thus more complex than this
analysis indicates.
 SKUNK
is like NRICH's Pig variant but limited to 5 turns with the maximum
score winning. This page presents classroom teaching activities.
Note: We have also solved this variation of Pig. The number of dice
rolls is irrelevant to optimal play or score maximization.
 In R.C. Bell's Discovering Dice and Dominoes (ISBN
085263532X, 1980), a variant is described where double 1's simply
ends the turn with no score. That is double 1's are like rolling a
single 1.
 Two Dice, 1 is Bad, Doubles are Good:
 Two Dice, 6 is Bad:
 Two Dice, Doubles are Bad: Doubles lose the turn total and end the turn. All other rolls accumulate their sum to the turn total. This variant, "Piggy", appears in "Kids Travel: A Backseat Survival Kit" by the editors of Klutz Press, 1994, p. 11.
 Two PigDice: Pass the
Pigs® Milton Bradley / Winning Moves game played with "pig" dice.
Also known as PigMania®
(David Moffat Enterprises, 1977). Rolls are scored depending on
whether pigs land on their feet, sides, backs, etc. in various
combinations.
 Two Nonstandard Dice, Doubles are Bad: Ivars Peterson
describes a Pig variant called Piggy with nonstandard dice that
produce the same distribution of rolls but with a different
probability for rolling doubles. ("Weird Dice", Muse Magazine,
May/June 2000, p. 18). Related curricular resources are available at
the Teacher
Resource Exchange.
 Five Dice, 1 is Bad: This pig dice game
from Koplow Games, is not to
be confused with Parker Brother's Pig Dice®. The goal score is
200. One 1 reduces a player's turn total by 10. Two through four 1's loses
the turn total and ends the turn. Five 1's loses the turn total and
score, and ends the turn.
One can also order individual pig dice
from Koplow Games, which are
standard dice with the 1's replaced by pictures of pigs.
 n Dice, 1 is Bad: This variant is described by
Bohan, J. F., and Schultz, J. L. in "Revisiting and Extending the
Hog Game," The Mathematics Teacher, 89(9), pp. 728733, 1996 and by
Larry Feldman and Fred Morgan in "The
Pedagogy and Probability of the Dice Game HOG", Journal of Statistics
Education Volume 11, Number 2 (2003). In this variant, a player has only
one roll per turn, but can choose the number of dice to be rolled.
Bibliography
For a more complete
bibliography, see the our paper
listed below.
 Todd W. Neller and Clifton G.M. Presser. Optimal Play
of the Dice Game Pig, The UMAP Journal
25(1) (2004), pp. 2547. A paper detailing the Pig analysis
presented on this website.
 Todd W. Neller and Clifton G.M. Presser. Pigtail: A
Pig Addendum, The UMAP Journal 26(4) (2005), pp. 443458. An addendum with solutions to many Pig variants.
 Todd W. Neller and Clifton G.M. Presser. Practical Play of the Dice Game Pig, The UMAP Journal 31(1) (2010), pp. 519. Evaluation of humanplayable policies with respect to optimal play.
 Bell, Robert Charles. 1979. Board and Table Games from Many
Civilizations. Revised ed. New York: Dover Publications,
Inc. Excellent game history reference.
 Bellhouse, David R. 1999. Il campanile statistico: What I did on my summer
holidays. Chance 12 (1): 4850. An empirical study of ancient
astragali and modern pig dice.
 Bellman, Richard. 1957. Dynamic Programming. Princeton, NJ: Princeton University
Press. Bellman's optimality equation is the basis for the value iteration technique used to solve Pig.
 Bertsekas, D.P. 1987. Dynamic Programming: Deterministic and
Stochastic Models. Englewood Cliffs, NJ: PrenticeHall. A more
recent treatment of Bellman's optimality equation.

Boyan,
J. A. and A. W. Moore. Learning Evaluation Functions for Large
Acyclic Domains. In L. Saitta (ed.), Machine Learning:
Proceedings of the Thirteenth International Conference (ICML). Morgan
Kaufmann, 1996. Reinforcement learning is applied here to a
variation of Pig where a roll of 1 scores one point and thus game
states do not repeat.
 Brutlag, Dan. 1994. Choice and chance in life: The game of "skunk". Mathematics
Teaching in the Middle School 1 (1): 2833. Describes a pig variant
as a tool for teaching probabilistic concepts at the middle school
level. Unfortunately incorporates the "gambler's fallacy" in
teaching.
 Diagram Visual Information Ltd. 1979. The Official World Encyclopedia of Sports
and Games. London: Paddington Press. Additional source for our Pig
rules
 Falk, Ruma, and Maayan TadmorTroyanski. 1999. THINK: A game of choice
and chance. Teaching Statistics 21 (1): 2427. Partial analysis of
Brutlag's SKUNK game. We have performed a more complete analysis,
computing optimal scoring play, and demonstrating that singledecision
oddsbased analysis does not yield optimal scoring play.
 Fendel, Dan, Diane Resek, Lynne Alper, and Sherry Fraser. 1997. The
Game of Pig. Teacher's Guide. Interactive Mathematics Program, Year
1. Berkeley, CA: Key Curriculum Press. An excellent teacher's
resource which uses the game of Pig to teach probabalistic reasoning
at the secondary education level.
 Frey, Skip. 1975. How to Win at Dice Games. North Hollywood, CA: Wilshire
Book Co. 1997. Reprint. Source for a 2dice Pig variant

Reiner
Knizia. Dice Games Properly Explained. Elliot RightWay Books,
Brighton Road, Lower Kingswood, Tadworth, Surrey, KT20 6TD U.K., 1999.
The rules for Pig and many other dice games can be found here.
Knizia makes an oddsbased argument for Pig tactics which I call the
"Hold at 20" policy. He then goes on to point out that there are
situations where one should deviate from this policy. Knizia made me
curious about what optimal Pig play would look like.

Tom
M. Mitchell. Machine Learning. McGrawHill, New York,
1997. A good advanced undergraduate or graduate text on the machine
learning in the field of artificial intelligence. Chapter 13 gives an
overview of Reinforcement Learning and covers convergence properties
relevant to value iteration.
 Parlett, David. 1991. A History of Card Games. New York: Oxford University
Press. Excellent reference for card game history.
 Parlett, David. 1999. The Oxford History of Board Games. New York: Oxford University
Press. Excellent reference for board game history.
 Peterson, Ivars. 2000. Weird dice. Muse Magazine. (May/June):
18. Source for 2dice Pig variant

Stuart Russell
and Peter Norvig. Artificial Intelligence: a modern
approach, 2nd ed. Prentice Hall, Upper Saddle River, NJ, USA,
2003. A very good and comprehensive undergraduate textbook on
artificial intelligence.
 Scarne, John. 1945. Scarne on Dice. Harrisburg, PA: Military
Service Publishing Co. 1980. 2nd ed. New York: Crown Publishers,
Inc. The earliest published rules for our game of Pig
 Shi, Yixun. 2000. The game PIG: Making decisions based on
mathematical thinking. Teaching Mathematics and Its Applications 19
(1): 3034. Source for another 2dice Pig variant

Richard S. Sutton and Andrew G. Barto. Reinforcement Learning:
an introduction. MIT Press, Cambridge, Massachussetts,
1998. This is perhaps the most gentle undergraduatelevel
introduction I've found for value iteration. The value iteration
algorithm used to solve Pig is described clearly and thoroughly.
Author: Todd Neller
Translation to Belarusian by Martha Ruszkowski