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

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 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.