Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles - Mehrab Najafian; T. Aaron Gulliver; Morteza Esmaeili (IEEE Access, 2022)
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 $m$ is a solid Latin cube of order $m$ with solid subcubes of order $x × y × z$ 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 $Z_{n}$ , and we make use of a vector $Z$ and a basic table $T$ to construct SSSCs. We obtain $m$ tables by multiplying all entries of $T$ by a number from the vector $Z$ . 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).