Generation of all magic squares of order 5 and interesting patterns finding

Abstract

This paper presents an enumeration algorithm to generate all magic squares of order 5 based on the ideas of basic form (Schroeppel [7]) and generating vector which is extension of Frénicle Quads (Ollerenshaw and Bondi [6]). The results lead us to extend Frénicle-Amela patterns from the case of order 4 to the case of order 5, which we refer to Frénicle-Amela-Like patterns. We show that these interesting Frénicle-Amela-Like patterns appear simultaneously. The number of these patterns is also calculated.

Some results on magic squares based on generating magic vectors and R-C similar transformations

“…The complementary pairs are: (1,16), (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), (8,9). For the order ve, the complementary sum is 26 and the complementary pairs are: (1,25), (2,24), (3,23),· · · , (11,15), (12,14).…”

“…In this paper, we focus on the classi cation of magic squares of order 5. Lin, Liu, Fang and Deng [5] discovered some properties of M that can be considered as an extension of Frénicle-Amela patterns in order 5. In this paper, we employ two important tools: GMV and the R-C similar transformation (that will be de ned in Section3) for classi cation of M .…”

“…So the left pair of two elements is also a CP and it lies in the same GMV. Thus the GMV class set [4,3,5] cannot exist.…”

“…In the above 6 matrices, there are same row-GMVs: (1,6,8), (3,5,7), and (2,4,9); 3 column-GMVs: (3,4,8), (1,5,9), and (2,6,7); and the same main diagonal GMV (2,5,8). There are 3 di erent vice diagonal GMVs: (4,5,6), (1,2,3), and (7,8,9) (1,4,7), (2,5,8) and (3,6,9).…”

“…There are 3 di erent vice diagonal GMVs: (4,5,6), (1,2,3), and (7,8,9) (1,4,7), (2,5,8) and (3,6,9). If we consider the transpose of M , , i.e., M , and implement the 6 R-C similar transformations to M , , we obtain 2 magic squares (SMSs), M , and M , and 4 semi-magic squares with vice diagonal GMVs: (4,5,6), (1,2,3), and (7,8,9).…”