Sudoku Toolkit
1. Functional Requirements
- The user should be able to solve a Sudoku board at a specified level.
- The user should be able to generate a Sudoku board of a specified difficulty.
- The user should be able to convert a Sudoku board to a printable format.
2. Implementation Details
2.1 Sudoku Board
First, we begin with a simple representation of a Sudoku grid. This is a two-dimensional array of numbers, of a given size N x N, with numbers from 1 to N filling the grid. This stores the values in the solved portion of the grid. There is also a three-dimensional array of boolean flags, of size N x N x N, pertaining to decisions about various locations on the board, called the markup board. A ‘true’ in cell (I, J, K) in the markup board means that a value of K can potentially appear in cell (I, J) of the grid. This is the foundation for the entire Sudoku program. As values in the markup board need to be concurrently maintained with the values in the grid, methods have been created to place a value in the grid, that takes markup board entries into consideration.
2.2 Sudoku Solver
The Sudoku Solver extends the Sudoku Board class, adding solving methods. There are roughly twelve to fifteen uniquely different methods for a human to solve a Sudoku grid. Of those, the ones with the most uniqueness were chosen to be implemented by the program. The solve method takes a difficulty level, and it begins at the bottom. It attempts the most basic of strategies, and continues using those strategies until they are no longer effective. Then, the program attempts more difficult strategies. Whenever a strategy is effective, we start back at the bottom of the difficulty chain and try the most basic strategies again. When we finally reach the difficulty level that was specified to the solve routine, or, if we have completed the puzzle, we stop.
2.3 Sudoku Generator
The Sudoku Generator simply generates a filled grid (more on this in a moment), then uses a dynamic programming algorithm to remove candidates until a threshold is reached. This is the concept of generating a “unique” grid, wherein there is only one legal solution. Since the Solver routines can only solve a board if it has a unique solution (no guesswork is ever employed), the ability to solve a grid proves the uniqueness of the board. If a cell is cleared, and no solution can be reached, we replace it and try another cell. The threshold specified is the number of “givens” the generated board should have. It is conjectured that the minimum number of givens required for a Sudoku board to be unique is 17, but no formal proof exists to this effect. Nonetheless, any value less than roughly 25 for this threshold may take considerable time to generate.
A filled grid is generated by using a legal filled board (which will the refered to as the canonical solved grid), then performing elementary row, column, and element operations. The three elementary operations are:
- Any row can be switched with any other row in the same tower.
- Any column can be switched with any other column in the same tower.
- Any value can be switched with any other value, as long as all values are exchanged.
Where a tower is defined to be a group of three (in the case of 9×9 Sudoku) rows or columns in the same row- or column-group, respectively. If only these elementary operations are performed, the legality of the solved grid is not compromised. Then, removing the elements in order to create an unsolved grid is described above.
3. Source Code
The source is available on coderprofile.com: SudokuToolkit v1.0
