Introduction

The program loads, solves, generates and saves Sudoku puzzles of sizes up to 30x30 (this restriction is due to the use of 32 bit integers - the java default size - for flags). After solving, the original puzzle may be restored.

A 9x9 Sudoku puzzle is a 9x9 array of cells, which is further divided as a 3x3 array of blocks of 3x3 cells. In a 9x9 Sudoku puzzle the digits 1 to 9 are the symbols and each symbol must occur exactly once in each row, each column and each sub-block. Some of the cells are already given, and the problem is to find the unique solution satisfying the constraints.

Sudoku puzzles can be of any size - another common size is 16x16 divided into 4x4 sub-blocks. However unless the size is a perfect square, the sub-block rule cannot apply and the puzzle is less attractive.

Normally a 9x9 Sudoku puzzle uses sub-blocks, and in addition the blanks are arranged so the puzzle has some symmetry. This can be half-turn symmetry or both vertical and horizontal symmetry.

File Format

The file format is as follows.

Line 1 --- contains the puzzle size. A negative size indicates that sub-blocks are not to be used (in the case of perfect square sizes) A size of 1 or greater than 30 is not permitted. So the only permitted sizes are -25, -16, -9, -4 and 2 to 30. Line 2 --- the list of symbols used in the puzzle prefixed by the symbol used in the file to denote a blank Remaining lines contain the puzzle. For example the file

4
_1234
1_3_
23_1
3__2
_1_3

would denote the 4x4 puzzle

1?3?
23?1
3??2
?1?3

where I have replaced blanks by questionmarks. Note that the final line must be terminated by an end-of-line.

Loading and Saving files

When loading a file, the program verifies that no number occurs more than once in any row or column (or sub-block, if applicable), and set the flags indicated under the "Solution Technique" section. See the README file for information on Fedora 4 compatablity.

Preferences

The user-defined preferences (used for generating puzzles) are:-

size
sub-blocks: If sub-blocks are to be used (only valid when size is a perfect square)
Symbol set
Symmetry: Symmetry can be none, half-turn, quarter-turn or vert/horiz (vertical and horizontal).
Symmetry affects the difficulty level of puzzles generated to some degree. For 9x9 puzzles mostly trivial, easy puzzles come from quarter-turn symmetry, but for 25x25 puzzles quarter-turn symmetry often generates hard, diabolical puzzles.

Note:- The difficulty part of the Preferences Dialog has been commented out, as the time taken to generate is highly variable, and there is no code in the generation section to use the difficulty preference setting.

Note also that when the preferences are being edited, if the "Enter" key is pressed when the size field is being edited, the symbols will automatically be set to the required number of symbols from the string "1234567890ABCDEFGHiJKLMNoP".

Difficulty

The difficulty of a puzzle is determined by the level of rule (described below) in solving the puzzle.

Trivial means only phase 0 used.
Easy means only phases 0 and 1 used.
Moderate means phases 0,1 and 2 used.
Hard means phases 0 through 3 are used.
Diabolical means that in at least one case a choice had to be made (and possibly retracted and another choice made)

The reason for these choices are as follows.

Trivial - look at numbers fitting only one cell.
Easy - look at cells in a (row) where a number fits.
Moderate - need to look at two cells.
Hard - the rules here don't say which numbers go in cells, but only eliminate numbers from cells.
Diabolical - having to make a random choice may cause backtracking.

Solution Explanation

Note that after the difficulty button has been pressed, its label changes to Explain. Pressing this brings up a frame showing how the puzzle was solved.

Solving

When solving, the program assumes a single solution exists, and displays the first solution found. If no solution exists, it will display a partial solution (as much of a solution as it could find).

Solution Technique

The solution method is recursive, using an array of flags which indicate the symbols valid in each cell. Each cell has a flag (a 0 bit) indicating which symbols are possible. A 1 bit indicates that the corresponding symbol already occurs in the same row or column (or sub-block), or is not allowable for any of the additional rules below.

Phase 0 of solution. Check each cell and determine if there is a cell which permits only one symbol. This can occur as illustrated in the following diagram of a 3x3 puzzle. A _ indicates a blank.
```
__3
2__
___
```
Of the three symbols 1,2,3, simply concentrating on the top left cell indicates that neither 2 can occur (it is already in the first column), nor a 3 can occur (it is already in the first row). The flags for the top left cell indicate that only the symbol 1 can be placed there. If such a cell is found, progress to the recursive step (below).
Phase 1 of the solution. In this phase we consider symbols. We examine rows, then columns, then blocks for the following type of situation.
```
___
_1_
__1
```
Looking at the first row, we can see that the symbol 1 cannot occur in either of the second or third cells (as it already occurs in columns 2 and 3). Hence it must occur in cell 1. If this situation is found, we proceed to the recursive stage.
Phase 2 of the solution. In this phase we now need to look at pairs of cells. For simplicity we look at a small example which is best shown in a puzzle where sub-blocks apply. We look for two cells each of which can contain only the came two symbols.
```
____
___2
__21
__1_
```
In the first row, the flags for the third or fourth cells will indicate that symbols 3 and 4 (only) can occur. In this case this is because the symbols 1,2 already occur in the third and fourth columns. So cells 3 and 4 both can only contain the two symbols 3,4. This means we can adjust the flags for cells 1 and 2 to exclude symbols 3 and 4. If we adjust any flag we can go back and re-start phase 0
Phase 3 of the solution. This parallels phase 1, in that we look at the numbers that can occur in entire rows (ditto columns, then blocks. If in a row we discover that some two symbols can ooccur in only 2 cells, (even if on first sight the flags may indicate other symbols can occur also in those cells), the symbols must occur in those cells. So we adjust the flags for the row to permit only those two symbols in the two places and bar the symbols from the other places. If we adjust any flag we can go back and re-start phase 0.
Phase3b of the solution In this phase we use particularly the relation between blocks and rows/columns. Consider the puzzle
```
____
__3_
__12
____
```
In the bottom right corner subblock it is clear that 3 can only go in the bottom right cell. This means that 3 cannot go in the bottom row of the lower left cell. (This illustrates the interplay between rows and columns. Although in this small example phase 0 would apply, in larger puzzles a number may only fit in a particular row of a subblock, and so be eliminated from that row in other subblocks.)
Note.Phases 2 and 3 can be extended to considering 3 cells, 4 cells and so on but with probably diminishing return.
Recursive step. Choose the first cell found with the least number of possible symbols. Many times (indeed every time, except for "Diabolical" puzzles) there will be a cell with only one symbol. For the chosen cell, for every possible symbol in that cell place the symbol in the cell copy the array of flags adjust the flags to ban the symbol in that cell's row, column, block. Call solve recursively with the current puzzle state and the new flags. (Initially solve is called with the original puzzle and the original flags set when the puzzle is validated.)

Validating

When a puzzle is loaded, the puzzle is automatically validated to set the flags for each cell - this also helps to check that no symbol has been placed twice in any row, column or block.

Generating Puzzles

A default layout with all symbols is generated - for example in a 9x9 puzzle the lines 123456789, 234567891, 345678912, etc would be laid out in the suitable order to satisfy the block condition.
The symbol set is then permuted randomly, then rows and columns are permuted randomly so the generated solved puzzle is random.
Then symbols are removed (keeping vertical and horizontal symmetry) until the puzzle has more than one solution when the last set of removed symbols is restored.
If only half-turn or no symmetry is required, more symbols are removed until no further removal is possible.

Notes

Although to determine the difficulty of a puzzle the puzzle must actually be solved, the solution will not be displayed until the solve button is pushed. Generating puzzles can take a long time, especially for hard or diabolical puzzles with symmetry. See also the README file