Very Large Sudokus
In addition to the standard 9x9 puzzle, 16x16 and above puzzles have been created. An even larger puzzle can be constructed by starting with a normal 9x9 grid and subjecting it to transformations of different types including those that increase the size. This can be done provided the Sudoku properties of constraining cells in three groups are preserved under the chosen transformations. One method of expansion of a 9x9 puzzle to an indefinitely large puzzle consists of replacing each original number 1,2,...,9, by a square matrix and successive replication and swapping of rows and columns to expand the 9x9 puzzle to an nxn puzzle.
For example, if each number is replaced by a 2x2 matrix of numbers, chosen randomly from the integers 1,2,...,36, each integer being used only once, the 9x9 grid becomes an 18x18 grid. The integers 1,2,...,36 each appear once in each 6x6 square as required, and no integer appears twice in a row or column. The complete grid must be 36x36. This is accomplished by swapping each pair of sub-rows and replicating the result on the right, followed by swapping each pair of sub-columns of the resulting 18x36 grid and replicating it on the bottom, to form a complete 36x36 grid. All three constraints are now satisfied, for rows, columns and 6x6 squares. The resulting puzzle's starting grid is not satisfactory, however, since its numbers by construction will appear in clumps of four. This is fixed by permutation of columns and rows (permitted when the permutation is within a column or row of 6x6 squares). The final grid has 36 rows, 36 columns and 36 6x6 squares, and a total of 1,296 cells.
Of course the expansion method can be used for larger matrices of numbers. For example, we can replace each of the numbers 1,...,9 with a 3x3 matrix, permute (i.e., rotate) the 3 sub-rows and sub-columns as above, and expand by replicating twice to the right and twice to the bottom. Since the 81 integers are each used only once in the replacement process, the rows, columns and 9x9 squares resulting all satisfy the Sudoku constraints. The process could be continued to obtain a Sudoku puzzle with as large dimensions as desired:
|matrix size||grid size|
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