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

Kai An Sim and Kok Bin Wong
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Kai An Sim: School of Mathematical Sciences, Sunway University, Selangor 47500, Malaysia
Kok Bin Wong: Institute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, Malaysia

Mathematics, 2021, vol. 9, issue 18, 1-14

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.

Keywords: magic square; arithmetic progression; permutations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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