Pig Links ( home | game | solution | links )

• Links by Variant
• Jeopardy Dice Game Geeklist
• Optimal Play:
  • Todd W. Neller and Clifton G.M. Presser. Optimal Play of the Dice Game Pig, The UMAP Journal 25(1) (2004), pp. 25-47.
  • Todd W. Neller and Clifton G.M. Presser. Pigtail: A Pig Addendum, The UMAP Journal 26(4) (2005), pp. 443-458.
• Practical Play:
  • Todd W. Neller and Clifton G.M. Presser. Practical Play of the Dice Game Pig, The UMAP Journal 31(1) (2010), pp. 5-19.
• Computer Science Teaching Resources (CS1, AI)
• Annotated Bibliography
• Ivars Peterson's MathTrek Review Article
• Wikipedia Article (Simple English)
• Pig Variation Rules Handout

• One Die, 1 is Bad (oldest variant in literature from "Scarne on Dice" by John Scarne, 1945):
  • Optimal play for this variant is solved here. You can also play Pig online or see visualizations of optimal play. An iPhone Pig Dice app with optimal play is also available.
  • The Games Forum provides the same rules we use and also cites the source The Official World Encyclopedia of Sports and Games, Diagram Visual Information Ltd 1979.
  • The Interactive Mathematics Program uses The Game of Pig as the core example for its Year 1 probability curriculum. A downloadable Pig simulation tool to test simple strategies for long term scoring is available. Similar curricular materials are available on the NZMaths Statistics Web-site. In the NZMaths variant, one is limited to 10 turns and the highest score wins.
  • Roger Eastman implemented a Pig dice game with optimal play as part of his iDice 2.0 dice game compilation for the Macintosh.
  • Michael Plasmeier implemented a Pig dice game PLAYPIG for TI-83+/84+ graphing calculators.
  • Justin Boyan and Andrew Moore applied reinforcement learning to a variation of Pig where a roll of 1 scores one point. In this acyclic variation, you can't find yourself in the same situation twice. Such a game can be solved with dynamic programming (computing the best moves from the end of the game backwards to the beginning). This Pig variation description is available as HTML, PDF and PostScript (see reference below).
• 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 odds-based 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 0-85263-532-X, 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:
  • Skip Frey's rules (from How to Win at Dice Games, Wilshire Book Co., N. Hollywood, CA, 1975 (Reprinted 1997)) have doubles scoring double points (except for two 1's being 25 points). These rules are also given at the dice-play website.
  • Downloadable freeware of Frey's variant is available from NZP Puzzle and Games.
  • Kyle Bradshaw's Pigalator applet features complex four-player tournament rules with doubles doubled as in Frey's variant (but with re-roll required), double 1's eliminating all points and doubling game stakes, and all points eliminated if 100 is reached exactly.
• Two Dice, 6 is Bad:
  • NRICH Online Maths Club Pig Game - sixes - Pig with 2 dice. Throwing a 6 ends the turn with no score. Throwing two 6's sets the player score to 0. (original link). Another rule description of Piggy Six with variant suggestions is at Leonard's Bits of Stuff Homepage
  • Saskatchewan Education's Evergreen Curriculum has a Trying Not to Pig-Out mathematics rubric with a similar variant where double 6's only ends the turn with no score.
  This same variant was earlier described by S. Humphrey in the Mathematical Gazette, 63, December 1979, where a simple analysis was given.
• 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 Pig-Dice: 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.
  • About.com's Pass the Pigs Page
  • Freddie W.'s Pass the Pigs Page with probabilities for "snouters", "leaning jowlers", etc.
  • Font Face Java Game by Gary Kramer
  • Commercial source of pig dice from Koplow Games
  • Fabrice Derepas has an analysis of some simple Pass the Pigs strategies.
  • Todd Neller computed and Clif Presser visualized an approximation of the optimal strategy based on Freddie W.'s data.
• Two Non-standard Dice, Doubles are Bad: Ivars Peterson describes a Pig variant called Piggy with non-standard 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. 728-733, 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.
  • Exercise: For which number(s) of dice rolled in a Hog turn is one expected to maximize one's score gain? What is the expected maximal score gain per turn?

Jeopardy Dice Game Geeklist

• Todd W. Neller and Clifton G.M. Presser. Optimal Play of the Dice Game Pig, The UMAP Journal 25(1) (2004), pp. 25-47. 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. 443-458. 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. 5-19. Evaluation of human-playable 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): 48-50. 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: Prentice-Hall. 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): 28-33. 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 ruless
• Elliott, Nancy Lee. 1973. A Mathematical Approach to an Optimal Strategy for the Dice Game Pig, Masters Thesis, University of North Carolina at Greenboro. Early work focusing on optimal strategies among points-per-turn and rolls-per-turn constrained strategies.
• Falk, Ruma, and Maayan Tadmor-Troyanski. 1999. THINK: A game of choice and chance. Teaching Statistics 21 (1): 24-27. Partial analysis of Brutlag's SKUNK game. We have performed a more complete analysis, computing optimal scoring play, and demonstrating that single-decision odds-based 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 2-dice Pig variant
• Reiner Knizia. Dice Games Properly Explained. Elliot Right-Way 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 odds-based 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. McGraw-Hill, 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 2-dice 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): 30-34. Source for another 2-dice Pig variant
• Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: an introduction. MIT Press, Cambridge, Massachussetts, 1998. This is perhaps the most gentle undergraduate-level introduction I've found for value iteration. The value iteration algorithm used to solve Pig is described clearly and thoroughly.

Author: Todd Neller