Abstract
Puzzles and games have been used for centuries to nurture problem-solving skills. Although often presented as isolated brain-teasers, the desire to know how to win makes games ideal examples for teaching algorithmic problem solving. With this in mind, this paper explores one-person solitaire-like games.
The key to understanding solutions to solitaire-like games is the identification of invariant properties of polynomial arithmetic. We demonstrate this via three case studies: solitaire itself, tiling problems and a collection of novel one-person games. The known classification of states of the game of (peg) solitaire into 16 equivalence classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel algebraic formulation of the solution to a class of tiling problems. Finally, we introduce an infinite class of challenging one-person games inspired by earlier work by Chen and Backhouse on the relation between cyclotomic polynomials and generalisations of the seven-trees-in-one type isomorphism. We show how to derive algorithms to solve these games.
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References
Golomb, S.W.: Polyominoes. George Allen & Unwin Ltd. (1965)
Mackinnon, N.: An algebraic tiling proof. The Mathematical Gazette 73(465), 210–211 (1989)
Piponi, D.: Arboreal isomorphisms from nuclear pennies (September 2007), Blog post available at, http://blog.sigfpe.com/2007/09/arboreal-isomorphisms-from-nuclear.html
Chen, W., Backhouse, R.: From seven-trees-in-one to cyclotomics (2010) (submitted for publication), http://cs.nott.ac.uk/~wzc
de Bruijn, N.G.: A Solitaire game and its relation to a finite field. J. of Recreational Math. 5, 133–137 (1972)
Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways, vol. I, II. Academic Press, London (1982)
Reiss, M.: Beitrage zur Theorie der Solitär-Spiels. Crelle’s J. 54, 344–379 (1857)
Dijkstra, E.W.: The checkers problem told to me by M.O. Rabin (September 1992), http://www.cs.utexas.edu/users/EWD/ewd11xx/EWD1134.PDF
Piponi, D.: Using thermonuclear pennies to embed complex numbers as types (October 2007), Blog post available at, http://blog.sigfpe.com/2007/10/using-thermonuclear-pennies-to-embed.html
Blass, A.: Seven trees in one. Journal of Pure and Applied Algebra 103(1), 1–21 (1995)
Fiore, M.: Isomorphisms of generic recursive polynomial types. In: Proceedings of the 31st Annual ACM SIGPLAN-SIGACT Symposium on the Principles of Programming Languages, pp. 77–88. ACM Press, New York (2004)
Lawvere, F.W.: Some thoughts on the future of category theory. Lecture Notes in Mathematics, vol. 1488, pp. 1–13 (1991)
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Backhouse, R., Chen, W., Ferreira, J.F. (2010). The Algorithmics of Solitaire-Like Games. In: Bolduc, C., Desharnais, J., Ktari, B. (eds) Mathematics of Program Construction. MPC 2010. Lecture Notes in Computer Science, vol 6120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13321-3_1
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DOI: https://doi.org/10.1007/978-3-642-13321-3_1
Publisher Name: Springer, Berlin, Heidelberg
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