Magic square and arrangement of consecutive integers that avoids k-term arithmetic progressions
Document Type
Article
Publication Date
9-1-2021
Abstract
In 1977, Davis et al. proposed a method to generate an arrangement of n]={1,2, horizontal ellipsis ,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.
Keywords
Magic square, Arithmetic progression, Permutations
Divisions
MathematicalSciences
Funders
Fundamental Research Grant Scheme (FRGS) by Malaysia Ministry of Higher Education and Publication Support Scheme by Sunway University, Malaysia (FRGS/1/2020/STG06/SYUC/03/1)
Publication Title
Mathematics
Volume
9
Issue
18
Publisher
MDPI
Publisher Location
ST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND