| Summary: | 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).
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