Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles
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="LaT...
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| Format: | Article |
| Language: | English |
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IEEE
2022-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9733326/ |
| _version_ | 1811341891340337152 |
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| author | Mehrab Najafian T. Aaron Gulliver Morteza Esmaeili |
| author_facet | Mehrab Najafian T. Aaron Gulliver Morteza Esmaeili |
| author_sort | Mehrab Najafian |
| collection | DOAJ |
| description | 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). |
| first_indexed | 2024-04-13T19:01:59Z |
| format | Article |
| id | doaj.art-fcc4cfc62d8849fd8e5a97bde6524ba4 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-13T19:01:59Z |
| publishDate | 2022-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-fcc4cfc62d8849fd8e5a97bde6524ba42022-12-22T02:34:05ZengIEEEIEEE Access2169-35362022-01-0110301803018810.1109/ACCESS.2022.31590279733326Construction of Standard Solid Sudoku Cubes and 3D Sudoku PuzzlesMehrab Najafian0https://orcid.org/0000-0002-3503-9920T. Aaron Gulliver1https://orcid.org/0000-0001-9919-0323Morteza Esmaeili2https://orcid.org/0000-0002-8894-7017Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, CanadaDepartment of Electrical and Computer Engineering, University of Victoria, Victoria, BC, CanadaDepartment of Electrical and Computer Engineering, University of Victoria, Victoria, BC, CanadaIn 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).https://ieeexplore.ieee.org/document/9733326/Latin cubesolid Sudoku cubeSudoku puzzleSudoku table |
| spellingShingle | Mehrab Najafian T. Aaron Gulliver Morteza Esmaeili Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles IEEE Access Latin cube solid Sudoku cube Sudoku puzzle Sudoku table |
| title | Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles |
| title_full | Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles |
| title_fullStr | Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles |
| title_full_unstemmed | Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles |
| title_short | Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles |
| title_sort | construction of standard solid sudoku cubes and 3d sudoku puzzles |
| topic | Latin cube solid Sudoku cube Sudoku puzzle Sudoku table |
| url | https://ieeexplore.ieee.org/document/9733326/ |
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