1. 1 CS 410 Mastery in Programming Chapter 10 Hints for Simple Sudoku Herbert G. Mayer, PSU CS Status 7/29/2013

2. 2 Syllabus Requirements Requirements Level 0 Level 0 Level 0 Algorithm Level 0 Algorithm Data Structure Data Structure Check Row, Columns Check Row, Columns Check Subsquares Check Subsquares L0 Sudoku L0 Sudoku Sample Sample References References

3. 3 Requirements  The Sudoku game has a starting board S start  S start must be solvable, leading to a fully populated Sudoku board S end  The situation must be uniquely solvable; i.e. S start may not be open-ended so that multiple solutions S end will be possible; that would be interesting too, but is ambiguous  S end must be reachable without backtracking. I.e. it will not be necessary to discard an intermediate situation S inter since it fails to produce a solution, and then to continue where S inter has left off; backtracking is an option to pursue, but is not required for our assumed unique, deterministic situations  If a violating S start is encountered, the game cannot be solved with the programs proposed here; program aborts

4. 4 Level 0 Implement a simple Sudoku solution that uses solely the requirements for all rows, all columns, and all subsquares. Sometimes this leads to a solution, which we name level 0 Sudoku Other solvable situation are of higher level Initially we solely focus on level 0 start situations S start

5. 5 Level 0 Algorithm Precondition: Have an initial Sudoku board that is: Incomplete, i.e. not all fields are populated But has some fields filled in, but in a way to let the game be uniquely and deterministically solvable Without contradictions, i.e.: No row holds the same number more than once No column holds the same number more than once No sub-square holds the same number more than once Initial Steps: For each empty field, i.e. that has no defined number: Set all possible candidate numbers in some data structure The list of all candidate numbers for a 3*3 by 3*3 Sudoku is: 1, 2, 3, 4, 5, 6, 7, 8, 9

6. 6 Level 0 Algorithm Step1: Set “changes” to 0 Step2: For every undefined sudoku[row][col] field, i.e. that has candidates in subset of { 1, 2, 3, 4, 5, 6, 7, 8, 9 } For any number n found on same “row”, remove n from the candidate list This is reminiscent of “Sieve if Eratosthenes” For any number n found on same “column”, remove n from the candidate list, if not already gone For any number n found in the same “sub-square”, remove n from the candidate list, if not already gone if the candidate list of sudoku[row][col] is down to 1, then changes++ Set the field sudoku[row][col] to the sole remaining number that is left Step 3: go to step 1 until “changes” is 0 i.e. we go though all fields sudoku[row][col] repeatedly, as long as we see changes Step 4: If all 3*3 by 3*3 fields are set: print success and output the whole board; else the initial field is not a level 0 solvable situation

7. 7 Data Structure 1 #define BIG_N ( SMALL_N * SMALL_N ) #define EOL '\n' #define EMPTY 0 #define NO_CHAR ' ' #define FALSE 0 #define TRUE 1 #define P_BARprintf( "|" ) #define P_LINEprintf( "-" ) #define P_PLUSprintf( "+" ) #define P_EOL printf( "\n" ) #define P_TABprintf( "\t" ) #define P_BLANKprintf( " " )

8. 8 Data Structure 2 // each element of an n*n X n*n sudoku board has the following structure: // "field" has any of the values 0..n*n, with 0 meaning that it is empty. // "option_count" states: how many of n*n numbers could be options. // "can_be[]" is array of n*n+1 bools. Says whether element "i" is option // if can_be[i] is false, number "i" cannot be option any longer. // if option_count ever reaches 1, then we "know" the field value // General data structure layout // 0 1 2 3 8 9 // +-----+ +-----+ +-----+-----+-----+-----+-... -+-----+-----+ // | NAT | | NAT | | DC | | | |... | | | // +-----+ +-----+ +-----+-----+-----+-----+-... -+-----+-----+ // field option_count can_be[ BIG_N + 1 ] // int int bool

9. 9 Data Structure 3 // specific layout for empty field // 0 1 2 3 8 9 // +-----+ +-----+ +-----+-----+-----+-----+-... -+-----+-----+ // | 0 | | 9 | | DC | T | T | T |... | T | T | // +-----+ +-----+ +-----+-----+-----+-----+-... -+-----+-----+ // field option_count can_be[ BIG_N + 1 ] // // Specific layout for occupied field 3 // 0 1 2 3 8 9 // +-----+ +-----+ +-----+-----+-----+-----+-... -+-----+-----+ // | 3 | | 0 | | DC | F | F | T |... | F | F | // +-----+ +-----+ +-----+-----+-----+-----+-... -+-----+-----+ // field option_count can_be[ BIG_N + 1 ] // typedef struct board_tp { unsignedfield;// 0 if empty, else one of 1..n*n unsignedoption_count; // initially, all n*n are options unsignedcan_be[ BIG_N + 1 ];// if false, number "i" impossible } struct_board_tp;

10. 10 Data Structure 4 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// //////////// //////////// //////////// G l o b a l O j e c t s //////////// //////////// //////////// //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// // this is the complete sudoku board, n*n by n*n = BIG_N * BIG_N fields struct_board_tp sudoku[ BIG_N ][ BIG_N ];

11. 11 Check Rows, Columns // check all horizontal and vertical lines for empty spaces. // each empty space initially has BIG_N options // but for each existing value in that row or col, decrease the option. // if per chance the options end up just 1, then we can plug in a number. // return the number of fields changed from empty to a new value unsigned horiz_vert( row_col_anal_tp row_or_col ) { // horiz_vert unsigned changes = 0; unsigned options = 0; unsigned field = 0;// remember the next number to be excluded for ( int row = 0; row < BIG_N; row++ ) { for ( int col = 0; col < BIG_N; col++ ) { if ( SRC.field ) {// there is a number ASSERT( ( SRC.option_count == 0 ), "has # + option?" ); }else{ // field is EMPTY. Goal to count down options to 1 ASSERT( ( SRC.option_count ), "0 field must have opt" ); // go thru each field. For # found, exclude # from can_be[] for ( int i = 0; i < BIG_N; i++ ) { // continued on next page...

12. 12 Check Rows, Columns //... continued from previous page // go thru each field. For # found, exclude # from can_be[] for ( int i = 0; i < BIG_N; i++ ) { if ( row_or_col == row_analysis ) { field = sudoku[ row ][ i ].field; }else{ // column analysis field = sudoku[ i ][ col ].field; } //end if if ( field ) { // we found a field SRC.can_be[ field ] = FALSE; } //end if SRC.option_count = options = count_fields( row, col ); if ( options == 1 ) { // plug in only 1 of BIG_N numbers // and set option_count to 0 field = find_1_field( row, col ); FILL(); } //end if } //end for i } //end if } //end if } //end for col } //end for col } //end for row return changes; } //end horiz_vert

13. 13 Check Subsquare // check all horizontal and vertical lines for empty spaces. // each empty space initially has BIG_N options // But for each field value in subsquare, decrease options. // if per chance the options end up just 1, then we can plug in a number. // return the number of fields changed from empty to a new value unsigned subsquare( ) { // subsquare unsigned changes = 0; unsigned options = 0; unsigned field = 0;// remember the next number to be excluded for ( int row = 0; row < BIG_N; row++ ) { for ( int col = 0; col < BIG_N; col++ ) { if ( SRC.field ) {// there is a number ASSERT( ( SRC.option_count == 0 ), "has # + option?" ); }else{ // field is EMPTY. Goal to count down options to 1 ASSERT( ( SRC.option_count ), "subsquare must have opt" ); // analyze all fields in subsquare, exclude from can_be[] for ( int i = f( row ); i < ( f( row ) + SMALL_N ); i++ ) { ASSERT( ( i <= row+SMALL_N ), "wrong i_row in [][]" ); // continue on next page

14. 14 Check Subsquare // continued from previous page... for ( int i = f( row ); i < ( f( row ) + SMALL_N ); i++ ) { ASSERT( ( i <= row+SMALL_N ), "wrong i_row in [][]" ); for ( int j = f( col ); j < ( f( col ) + SMALL_N ); j++ ) { ASSERT( j <= col+SMALL_N, "wrong j_col in [][]" ); field = sudoku[ i ][ j ].field; if ( field ) { // we found a non-zero field SRC.can_be[ field ] = FALSE; } //end if SRC.option_count = options = count_fields( row, col ); if ( options == 1 ) { // plug in only 1 of BIG_N numbers // and set option_count to 0 field = find_1_field( row, col ); FILL(); } //end if } //end for j } //end for i } //end if } //end if } //end for col } //end for col } //end for row return changes; } //end subsquare

15. 15 L0 Sudoku // simplest sudoku strategy by eliminating options for a field // that would conflict with existing numbers in row, column, subsquare unsigned sudoku_level0() { //sudoku_level0 unsigned changes;// count fields filled in unsigned iterations = 0;// count times we go around unsigned errors = 0;// do final sanity check do { changes = 0; changes = horiz_vert( row_analysis ); changes += horiz_vert( col_analysis ); changes += subsquare(); ++iterations; } while ( changes ); try_single_option(); #ifdef DEBUG printf( "Iterated level0 %d times.\n", iterations ); errors = sanity_check(); #endif // DEBUG return changes; } //end sudoku_level0

16. 16 Sample Input After setting initial fields sudoku board. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8+------+------+------+ | 4 1| 5| 2 | 0 | 6 7| 3 | 8 5 | 1 | 5 3| 7 | 1 | 2 +------+------+------+ | 8 | 2 4 3| | 3 | | 9| 1| 4 | 4| 8 | 2| 5 +------+------+------+ | 3 | 6| 9 | 6 | | 2 | 5 8| 7 | 7 | 9 | 6| 8 +------+------+------+ Statistics initially: Total # of fields: 81 Fields filled: 31 empty fields: 50

17. 17 Sample Output Sudoku level 0 sudoku board. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8+------+------+------+ | 4 9 1| 6 8 5| 2 7 3| 0 | 6 2 7| 3 9 1| 8 5 4| 1 | 8 5 3| 4 7 2| 6 1 9| 2 +------+------+------+ | 1 8 9| 2 4 3| 7 6 5| 3 | 7 3 2| 5 6 9| 4 8 1| 4 | 5 6 4| 8 1 7| 3 9 2| 5 +------+------+------+ | 3 4 8| 1 5 6| 9 2 7| 6 | 9 1 6| 7 2 4| 5 3 8| 7 | 2 7 5| 9 3 8| 1 4 6| 8 +------+------+------+ Statistics Sudoku level 0 Total # of fields: 81 Fields filled: 81 empty fields: 0 Field was solvable with level 0

18. 18 References  Here you may play: http://www.websudoku.com/  The rules: http://en.wikipedia.org/wiki/Sudoku  History: http://www.sudokucentral.com/what-is- sudoku  C Solution: http://www.daniweb.com/software- development/cpp/threads/48788