Annals of Operations Research

, Volume 130, Issue 1–4, pp 41–56 | Cite as

Constraint Programming and Hybrid Formulations for Three Life Designs

  • Robert Bosch
  • Michael Trick


Conway's game of Life provides an interesting testbed for exploring issues in formulation, symmetry, and optimization with constraint programming and hybrid constraint programming/integer programming methods. We consider three Life pattern-creation problems: finding maximum density still-Lifes, finding smallest immediate predecessor patterns, and finding period-2 oscillators. For the first two problems, integrating integer programming and constraint programming approaches provides a much better solution procedure than either individually. For the final problem, the constraint programming formulation provides the better approach.

integer programming constraint programming hybrid formulation cellular automata game of Life 


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  1. Berlekamp, E.R., J.H. Conway, and R.K. Guy. (1982). Winning Ways for Your Mathematical Plays,Vol.2: Games in Particular. London: Academic Press.Google Scholar
  2. Bosch, R.A. (1999). “Integer Programming and Conway's Game of Life.” SIAM Review 41(3), 594–604.Google Scholar
  3. Bosch, R.A. (2000). “Maximum Density Stable Patterns in Variants of Conway's Game of Life.” Operations Research Letters 27(1), 7–11.Google Scholar
  4. Buckingham, D.J. and P.B. Callahan. (1998). “Tight Bounds of Periodic Cell Configurations in Life.” Experimental Mathematics 7(3), 221–241.Google Scholar
  5. Callahan, P. (2001). Random Still Life Generator, stilledit.html.Google Scholar
  6. Cook, M. (2001). Still Life Theory, 2DOutTot/Life/StillLife/StillLifeTheory.html.Google Scholar
  7. Elkies, N.D. (1998). “The Still-Life Density Problem and Its Generalizations.” In P. Engel and H. Syta (eds.), Voronoi's Impact on Modern Science, Book 1. Kyiv: Institute of Mathematics.Google Scholar
  8. Gardner, M. (1970). “The Fantastic Combinations of John Conway's New Solitaire Game “Life”.” Scientific American 223, 120–123.Google Scholar
  9. Gardner, M. (1971). “On Cellular Automata, Self-Reproduction, the Garden of Eden and the Game “Life”.” Scientific American 224, 112–117.Google Scholar
  10. Gardner, M. (1983). Wheels, Life, and Other Mathematical Amusements. New York: W.H. Freeman.Google Scholar
  11. Niemiec, M.D. (2001). Mark D.Niemiec's Life Page, lifepage.htm.Google Scholar
  12. Smith, B.M. (2001). “Reducing Symmetry in a Combinatorial Design Problem.” In CP-AI-OR 2001,Wye College, Kent.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Robert Bosch
    • 1
  • Michael Trick
    • 2
  1. 1.Department of MathematicsOberlin CollegeOberlinUSA
  2. 2.Graduate School of Industrial AdministrationCarnegie Mellon UniversityPittsburghUSA

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