# Elementary Approaches to the Teaching of the Combinatorial Problem of Rectangular Islands

• Attila Máder
• Róbert Vajda
Computer Math Snapshots - Column Editor: Uri Wilensky*

## Introduction

### The Topic of Islands

In this snapshot we consider the combinatorial problem of rectangular islands by elementary means. Although some of the corresponding results are quite new, the topic of islands and the methods for its investigation is suitable also for high school students. We offer a bunch of exercises, representations and methods usable by a math teacher who wants to consider some of the problems from the field with students. The questions which arise need no advanced mathematical knowledge. Because most of the problems are of finitary type, experimental mathematics with computer support proves to be useful for the formulations of general conjectures related to the bounds of the number of islands in a particular configuration.

Intuitively the combinatorial problem can be described as follows: Our world is a rectangular grid with cells. Each cell has a height and an island consisting of cells rises up from its neighbourhood, i.e., all of its cells are higher than the...

## Keywords

Computer Algebra System Rectangular Grid Height Function Island System General Conjecture
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Barát, J., Hajnal, P., & Horváth, E. K. (2010). Elementary proof techniques for the maximum number of islands. European Journal of Combinatorics. (submitted). Available at: http://www.math.u-szeged.hu/~horvath .
2. Beucher, S., & Lantuejoul, C. (1979). Use of watersheds in contour detection. International workshop on image processing: Real-time edge and Motion Detection/Estimation, Rennes, France.Google Scholar
3. Borwein, J. M. (2005). The experimental mathematician: The pleasure of discovery and the role of proof. International Journal of Computers for Mathematical Learning, 10(2), 75–108. New York: Springer.Google Scholar
4. Buchberger, B. (1990). Should students learn integration rules? ACM SIGSAM Bulletin, 24(1), 10–17.
5. Carducci, O. M. (2009). The Wolfram demonstrations project. MAA Focus, 28(1), 8–9.Google Scholar
6. Czédli, G. (2009). The number of rectangular islands by means of distributive lattices. European Journal of Combinatorics, 30, 208–215.
7. Földes, S., & Singhi, N. M. (2006). On instantaneous codes. Journal of Combinatorics, Information and System Science, 31, 317–326.Google Scholar
8. Horváth, E. K., Máder, A., & Tepavčević, A. (2011). Introducing Czédli-type islands. The College Mathematical Journal. (to appear).Google Scholar
9. Horváth, E. K., Németh, Z., & Pluhár, G. (2009). The number of triangular islands on a tiangular grid. Periodica Mathematica Hungarica, 58, 25–34.
10. Windsteiger, W., Buchberger, B., & Rosenkranz, M. (2006). Theorema. In F. Wiedijk (Eds.), The seventeen provers of the world, LNAI 3600, 96-107. New York: Springer.Google Scholar
11. Wolfram, S. (1999). The Mathematica book. Champaign, IL, USA: Wolfram Media Inc. Cambridge: Cambridge University Press. See http://www.wolfram.com/.