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Magic Square and Arrangement of Consecutive Integers That Avoids k -Term Arithmetic Progressions

Author

Listed:
  • Kai An Sim

    (School of Mathematical Sciences, Sunway University, Selangor 47500, Malaysia
    These authors contributed equally to this work.)

  • Kok Bin Wong

    (Institute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, Malaysia
    These authors contributed equally to this work.)

Abstract

In 1977, Davis et al. proposed a method to generate an arrangement of [ n ] = { 1 , 2 , … , n } that avoids three-term monotone arithmetic progressions. Consequently, this arrangement avoids k -term monotone arithmetic progressions in [ n ] for k ≥ 3 . Hence, we are interested in finding an arrangement of [ n ] that avoids k -term monotone arithmetic progression, but allows k − 1 -term monotone arithmetic progression. In this paper, we propose a method to rearrange the rows of a magic square of order 2 k − 3 and show that this arrangement does not contain a k -term monotone arithmetic progression. Consequently, we show that there exists an arrangement of n consecutive integers such that it does not contain a k -term monotone arithmetic progression, but it contains a k − 1 -term monotone arithmetic progression.

Suggested Citation

  • Kai An Sim & Kok Bin Wong, 2021. "Magic Square and Arrangement of Consecutive Integers That Avoids k -Term Arithmetic Progressions," Mathematics, MDPI, vol. 9(18), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:18:p:2259-:d:635399
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    Citations

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    Cited by:

    1. Kai An Sim & Kok Bin Wong, 2022. "Minimum Number of Colours to Avoid k -Term Monochromatic Arithmetic Progressions," Mathematics, MDPI, vol. 10(2), pages 1-10, January.
    2. Pedro García-del-Valle-y-Durán & Eduardo Gamaliel Hernandez-Martinez & Guillermo Fernández-Anaya, 2022. "The Greatest Common Decision Maker: A Novel Conflict and Consensus Analysis Compared with Other Voting Procedures," Mathematics, MDPI, vol. 10(20), pages 1-39, October.

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