• Ulil Albab Mu'min Departement of Mathematics, Faculty of Mathematics and Natural Science, IPB University, Indonesia
  • Bib Paruhum Silalahi Departement of Mathematics, Faculty of Mathematics and Natural Science, IPB University, Indonesia
  • Sugi Guritman Departement of Mathematics, Faculty of Mathematics and Natural Science, IPB University, Indonesia
Keywords: Arithmetic sequence, Magic cube, Magic square, Perfect magic cube


A magic square whose dimensions are expanded is called a magic cube. A magic cube whose properties are expanded is called a perfect magic cube. The perfect magic cube problem is how to arrange  numbers in an  cube (matrix) such that the sum of rows, columns, pillars, diagonals (planes and spaces) produces a magic constant of the cube. In this paper, it will be studied how to construct a perfect magic cube of order  for  whose entries contain an arithmetic sequence with the difference  which is set to find specific patterns, and the algorithm for constructing a perfect magic cube is then implemented into programming language to solve large orders.


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P. B. Dhandapani, V. Leiva, and C. Martin-Barreiro, “Construction of a repetitive magic square with Ramanujan’s number as its product,” Heliyon, vol. 9, no. 1, p. e12046, 2023, doi: 10.1016/j.heliyon.2022.e12046.

Z. Duan, J. Liu, J. Li, and C. Tian, “Improved even order magic square construction algorithms and their applications in multi-user shared electronic accounts,” Theor. Comput. Sci., vol. 607, pp. 391–410, 2015, doi: 10.1016/j.tcs.2015.07.053.

O. M. Baksalary, D. Trenkler, and G. Trenkler, “On most perfect magic square of order four,” Linear Multilinear Algebr., pp. 1–13, Jul. 2018.

B. Gupta, “Algorithm for doubly even magic square,” Int. Adv. Res. J. Sci. Eng. Technol., vol. 6, no. 3, pp. 112–117, 2019.

U. A. Mu’min and B. P. Silalahi, “Konstruksi persegi ajaib dengan entri bilangan bulat (magic square construction with integer entry),” J. Mat. dan Pendidik. Mat., vol. 6, no. 2, pp. 166–176, 2021.

S. Duangmoon, P. Kanokladarom, S. Pongprasert, T. Rungratgasame, and N. Srithaisong, “Affine magic cubes of order 4: concepts and applications,” Int. J. Math. Math. Sci., vol. 2022, 2022, doi: 10.1155/2022/2578562.

B. J. Freyberg, “Constructing a tournament with a magic cube,” Am. Math. Mon., vol. 128, no. 8, pp. 719–725, 2021.

C. Ashbacher and L. Widmer, “Magic cube classes,” J. Recreat. Math., pp. 120–121, 2010, doi: 10.5951/mt.43.6.0290.

S. H. Jamel, T. Herawan, and M. M. Deris, “Steps for constructing magic cube using two orthogonal Latin squares and a magic square,” Malaysian Tech. Univ. Conf. Eng. Technol., pp. 1–4, 2009.

L. U. Uko and T. L. Barron, “A generalization of Trenkler’s magic cubes formula,” Recreat. Math. Mag., vol. 4, no. 8, pp. 39–45, 2017, doi: 10.1515/rmm-2017-0019.

F. Liao and H. Xie, “On the construction of pandiagonal magic cubes,” Mathematics, vol. 11, no. 5, pp. 1–23, 2023, doi: 10.3390/math11051185.

O. A. Dawood, A. M. S. Rahma, and A. M. J. Abdul Hossen, “Generalized method for constructing magic cube by folded magic squares,” Int. J. Intell. Syst. Appl., vol. 8, no. 1, pp. 1–8, 2016, doi: 10.5815/ijisa.2016.01.01.

C. Boyer, “Perfect magic cubes,” Multimagie, 2020. (accessed Nov. 01, 2022).

G. Guardia and N. B. Do Nascimento, “Arithmetic sequence with multiple reasons,” Am. J. Eng. Res. (AJER, vol. 7, no. 5, pp. 370–376, 2018.

J. F. T. Rabago, “On sequences of numbers in generalized arithmetic and geometric progressions,” Palest. J. Math., vol. 4, no. 1, pp. 170–176, 2015.

J. Arroyo, “A proof of the " Magicness " of the siam construction of a magic square,” Rose-hulman Undergrad. Math. J., vol. 19, no. 1, 2018.

M. C. Tastan and H. Ilhan, “Magic Square Based SM for UAV Communication,” TechRxiv, pp. 1–11, 2023.

B. Alspach and K. Heinrich, “Perfect magic cubes of order 4m,” Fibonacci Q., vol. 19, no. 2, pp. 97–106, 1981.

S. H. Jamel, T. Herawan, and M. M. Deris, “A cryptographic algorithm based in hybrid cubes,” Int. Conf. Comput. Sci. its Appl., pp. 175–187, 2010.

M. Trenkler, “An algorithm for making magic cubes,” Pi ME J., vol. 12, no. 2, pp. 105–106, 2005.

H. . Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated. HDH, 2000.

E. W. Weisstein, “Perfect Magic Cube,” Wolfram Mathworld, 2003. (accessed May 01, 2022).

W. S. Andrews, Magic Squares and Cubes. USA: Open Court Publishing, 1917.

M. Trenkler, “Magic cubes,” Math. Gaz., vol. 82, no. 493, pp. 56–61, 1998, doi: 10.5840/monist190616312.

M. Trenkler, “A construction of magic cubes,” Math. Gaz., vol. 84, no. 499, pp. 36–41, 2000, doi: 10.2307/3621472.

How to Cite
U. Mu’min, B. Silalahi, and S. Guritman, “SOME CONSTRUCTION OF 8N-DIMENSIONAL PERFECT MAGIC CUBE WITH ARITHMETIC SEQUENCE”, BAREKENG: J. Math. & App., vol. 18, no. 1, pp. 0565-0578, Mar. 2024.