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

Autor: Kai An Sim, Kok Bin Wong
Jazyk: angličtina
Rok vydání: 2021
Předmět:
Zdroj: Mathematics, Vol 9, Iss 18, p 2259 (2021)
Druh dokumentu: article
ISSN: 2227-7390
24440787
DOI: 10.3390/math9182259
Popis: 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 2k−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.
Databáze: Directory of Open Access Journals
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