Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles

• Published in: IEEE access Vol. 10; pp. 30180 - 30188
• Main Authors: Najafian, Mehrab, Gulliver, T. Aaron, Esmaeili, Morteza
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Columnar structure; Cubes; Games; Latin cube; solid Sudoku cube; Solids; Stacking; Sudoku puzzle; Sudoku table; Three-dimensional displays; Two dimensional displays
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2022.3159027

In this paper, standard solid Sudoku cubes (SSSCs), a three-dimensional (3D) extension of Sudoku tables, are introduced, and a method to construct these cubes is presented. This is the first class of standard solid Sudoku cubes. An SSSC of order <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> is a solid Latin cube of order <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> with solid subcubes of order <inline-formula> <tex-math notation="LaTeX">x \times y \times z </tex-math></inline-formula> in which each element occurs exactly once in each row, column, depth, and subcube. The structure of these cubes is based on cyclotomic cosets of <inline-formula> <tex-math notation="LaTeX">\mathbf {Z}_{n} </tex-math></inline-formula>, and we make use of a vector <inline-formula> <tex-math notation="LaTeX">Z </tex-math></inline-formula> and a basic table <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> to construct SSSCs. We obtain <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> tables by multiplying all entries of <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> by a number from the vector <inline-formula> <tex-math notation="LaTeX">Z </tex-math></inline-formula>. Then, these tables are converted to an SSSC by stacking them in order. Based on this method of construction, a perfect set of strongly mutually distinct standard solid Sudoku cubes is designed. We also provide a two-dimensional (2D) representation of these SSSCs in a table with numbers placed on the main diagonal of its subtables. Finally, a new class of 3D Sudoku puzzles based on SSSCs is presented as standard solid Sudoku puzzles (SSSPs).