Geometric Puzzles on a Circular Track: A Recreational Introduction to Quantization

Authors

  • Sayandip Pandit Indian Institute of Technology Delhi Author
  • Amit Priyadarshi Indian Institute of Technology Delhi Author
  • Mrinal Kanti Roychowdhury School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley (UTRGV), USA Author

DOI:

https://doi.org/10.67239/rmj.v01.n01.a002

Keywords:

Recreational mathematics, Geometry puzzles, Quantization, Symmetry, Voronoi regions

Abstract

Imagine a circular running track with several water stations placed around it. If only a few stations may remain open, or if a few new stations may be built, where should they be located so that runners travel the shortest possible distance to reach one? This simple question leads to a collection of recreational geometry puzzles involving distance, symmetry, optimization, Voronoi regions, and quantization. Quantization is the mathematical problem of replacing a large collection of points by a smaller set of representative points while minimizing the approximation error. Using only elementary mathematics, we investigate several configurations of points on a circle and discover elegant optimal arrangements. The article illustrates how sophisticated mathematical ideas can arise naturally from simple geometric games and puzzles, providing an accessible introduction to concepts that play important roles in geometry, optimization, data science, signal processing, and approximation theory.

References

M. Gardner, Mathematical Carnival, Mathematical Association of America, Washington, DC, 1989.

R. Honsberger, Mathematical Gems III, The Mathematical Association of America, 1976.

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics, Vol. 1730, Springer, Berlin, 2000.

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, Boston, 1992.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third Edition, Springer, New York, 1999.

A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Second Edition, Wiley, Chichester, 2000.

Downloads

Published

2026-08-10

Issue

Vol. 1 No. 1 (2026): Volume 1, Issue 1 (2026)

Section

Articles

License

Copyright (c) 2026 Sayandip Pandit, Amit Priyadarshi, Mrinal Kanti Roychowdhury (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors retain copyright in their work and grant the ISQGD Recreational Mathematics Journal (RMJ) the right of first publication.

Unless otherwise indicated, articles published in RMJ are licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) License. This license permits use, sharing, adaptation, distribution, and reproduction in any medium or format, provided appropriate credit is given to the author(s) and the original publication in RMJ.

Authors are responsible for obtaining permission to reproduce any copyrighted or third-party material included in their work when such permission is required.