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Indecomposability: polyominoes and polyomino tilings

Published online by Cambridge University Press:  01 August 2016

Simone Rinaldi  and
D. G. Rogers
Simone Rinaldi
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche “R.Magari”, Università Siena, Plan dei Mantellini 44, 53100 Siena, Italy, e-mail: rinaldi@unisi.it
D. G. Rogers
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche “R.Magari”, Università Siena, Plan dei Mantellini 44, 53100 Siena, Italy, e-mail: rinaldi@unisi.it
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Extract

When we were preparing our earlier article [1], we thought to look back to see what else had appeared in the Gazette on the subject of polyominoes. A polyomino is a finite collection of cells in the square grid with connected interior - so it is insufficient that cells be connected only corner to corner. Some authors require polyominoes to have a simply connected interior, that is, to be without holes, as was appropriate, for example, in [1] for stack polyominoes. The classic text on polyominoes is Solomon Golomb's engaging book [2], first published in 1965, now supplemented at an even more accessible level by [3].

Type
Articles
Information
The Mathematical Gazette , Volume 92 , Issue 524 , July 2008 , pp. 193 - 204
Copyright
Copyright © The Mathematical Association 2008

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References

1. Rinaldi, S. and Rogers, D. G., How the odd terms in the Fibonacci sequence stack up, Math. Gaz. 90 (November 2006), pp. 431–442.CrossRefGoogle Scholar
2. Golomb, S. W., Polyominoes: Puzzles, patterns, problems, and packings (2nd edn.) Scribner’s, New York (1965), Princeton University Press, Princeton (1994).Google Scholar
3. Martin, G. E., Polyominoes: A guide to puzzles and problems in tiling Math. Assoc. Amer., Washington (1991).Google Scholar
4. Allenby, R. B. J. T., Almost pavings of patios, Math. Gaz. 83 (November 1999) pp. 490–492.Google Scholar
5. Silvester, J. R., Painting by numbers or polyominoes revisited, Math. Gaz. 86 (March 2002) pp. 68–72.CrossRefGoogle Scholar
6. Rajesh, V. and Leversha, G., Some properties of the odd terms of the Fibonacci sequence, Math. Gaz. 88 (March 2004) pp. 85–86.Google Scholar
7. Silvester, J. R., The r-subsequences of the Fibonacci sequence, Math. Gaz. 90 (July 2006) pp. 263–266.Google Scholar
8. Silvester, J. R., personal communication, (July 2004).Google Scholar
9. Rymer, N. W., Projects, problems and patience, Math. Gaz. 63 (March 1979) pp. 1–7.Google Scholar
10. MacKinnon, N. I. P., Some thoughts on polyomino tiles, Math. Gaz. 74 (March 1990) pp. 31–33.Google Scholar
11. Stanley, R. P., Enumerative combinatorics Vol. 2, Cambridge University Press (1999); supplementary material available at: http://www-math.mit.edu/~rstan/ec CrossRefGoogle Scholar
12. Sloane, N. J. A., Online encyclopedia of integer sequences, available at: http://www.research.att.com/~njas/sequences Google Scholar
13. Lobb, A., Deriving the n-th Catalan number, Math. Gaz. 83 (March 1999) pp. 109–110.Google Scholar
14. Swylan, E., Proving numbers Catalan, Math. Gaz. 86 (November 2002) pp. 460–463.Google Scholar
15. Hirst, A., Roll-a-penny probabilities, Math. Gaz. 73 (March 1989) pp. 101–106.Google Scholar
16. Hill, R., Counting and the reflection principle, Math. Gaz. 75 (November 1991) pp. 308–312.Google Scholar
17. Cobb, R. H., (a) Problems solved and unsolved, Math. Gaz. 60 (March 1976) pp. 1–10; (b) Problems: comments, corrections and further solutions, Math. Gaz. 60 (December 1976) pp. 284–291.Google Scholar
18. Griffiths, M., ‘Catch-up’ numbers, Math. Gaz. 91 (November 2007) pp. 500–509.Google Scholar
19. Zhou, L., (a) Problem 11187: Tiling 4-rowed rectangles with dominoes, Amer. Math. Monthly, 112 (2005) p. 929; (b) Solutions separately by (i) Northwestern University Math Problem Solving Group; (ii) C. C. Heckman; and, (iii) J. Minkus; with editorial comments, Amer. Math. Monthly, 112 (2007) pp. 554–556.Google Scholar
20. Rogers, D. G., An application of renewal sequences to the dimer problem, in Horadam, A. F. and Wallis, W. A., eds, Combinatorial Mathematics VI: Proc. Sixth Austral. Confi, August, 1978. Lecture Notes in Math., 748 Springer, Berlin (1979), pp. 143–153.Google Scholar
21. Cross, T., Student Problem 2003.3, Math. Gaz., 85 (July 2003) p. 336; (November 2003) pp. 527–528.CrossRefGoogle Scholar
22. MacKinnon, N. I. P., Personal communication, (October 2006).Google Scholar
23. Lyness, R. C., Al Capone and the death ray, Math. Gaz. 25 (December 1941) pp. 283–287.CrossRefGoogle Scholar
24. Lawden, D. F., On the solution of linear difference equations, Math. Gaz. 36 (September 1952) pp. 193–196.Google Scholar
25. Abbott, S., A difference method for Σmkpm , Math. Gaz. 79 (July 1995), pp. 355–359.Google Scholar
26. Feller, W., An introduction to the theory of probability (3rd. edn.) (J. Wiley and Sons (1968).Google Scholar
27. Renault, M. S., Lost (and found) in translation: Andre’s actual method and its application to the generalized ballot problem, Amer. Math. Monthly, 115 (2008) pp. 358–363; also available at http:// webspace.ship.edu/msrenault/ballotproblem/ Google Scholar
28. Renault, M. S., Four proofs of the ballot theorem, Math. Mag. 80 (2007) pp. 345–352.Google Scholar
29. Atkinson, M. D. and Lunnon, W. F., Regular fault-free rectangles, Math. Gaz. 64 (June 1980) pp. 99–106.Google Scholar