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Solving Sudoku1 Website: www.nikoli.co.jp/en/ Labelling scheme.

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Presentation on theme: "Solving Sudoku1 Website: www.nikoli.co.jp/en/ Labelling scheme."— Presentation transcript:

1 Solving Sudoku1 Website: Labelling scheme

2 Solving Sudoku2 I refer to cells as column, then row, e.g. a5. The 3 cells where a row or column meets a block are referred to block first, e.g. A 8 or E h. Definition: two cells are in line if they are in the same row or column. Two cells are adjacent if they are in the same row or the same column or the same block.

3 Solving Sudoku3 Theorem 1: Elimination ( aka last symbol ). If an empty cell c is adjacent to cells that contain 8 of the digits, then the 9 th digit must go in cell c. In the written solution, Z denotes an elimination. E.g. / Z a5 = 7 /

4 Solving Sudoku4 1 1 Part solution: / i6 = 1 / f3 = 1 / Ex. 1: shooting and filling

5 Solving Sudoku5 2 2 Next part: / i2 = 2 / c7 = 2 / Continue:

6 Solving Sudoku6 Definition. Let x be any digit and c any cell. The cell c is not x if either c contains a digit different from x or c is adjacent to some other cell that contains x. Theorem 2. Shooting. Let x be any digit and S any row or column or block. If 8 cells of S are not x, then the 9 th cell must be x.

7 Solving Sudoku7 How to shoot. For x = 1 to 9 do < if you hit, repeat with x until you fail > end where is: Look for 3 blocks in line where 2 of them do contain x and the 3 rd does not. Where can x be in the 3 rd block ?

8 Solving Sudoku8 After 12 more shots, we reach this position. FIG 3: Filling Solution: / 6 / A /

9 Solving Sudoku9 Theorem 3. Let S be any row or column or block. If nearly all the cells of S are filled, then the digits not yet used in S must go into the remaining empty cells. In long form a filling is written as: name of S, colon, missing digits, results found In Fig 3: / A : 1,4 : b7 = 4, a7 = 1 /

10 Solving Sudoku10 Continuing from Fig 3: / C: 5,8,9: h7=9, g9=8, g7=5 / / D: 1,9 : a4=9, c4=1 / / F / 4 / I / 3 / c / 9 / 2 / 1 / d / rest /

11 Solving Sudoku11 Fig 4: crossfire. 1 2 / a1 = 1 = i2 / / c7 = 2 /

12 Solving Sudoku12 Theorem 4. Pairs. Let x be any digit and S any row or column or block. If 7 cells of S are not x, then either the 8 th cell or the 9 th must be x. In the solution: / P : a5 or a6 = 7 / Theorem 5. Twins. If there are two digits x and y that both want to go into the same pair of cells, then no other digit can go into either cell. Solution: / W : a5, a6 = 2, 7 /

13 Solving Sudoku13 Fig 5: Pairs and Twins / i5 = 7 / W e8,f8=9,7 / / a9=7 / W b2, b3 = 9, 7 / / W g5, h5 = 6, 2 / e4 = 2 = d3 /

14 Solving Sudoku14 Solution so far: / W e8,f8=9,7 / i5 = 7 / / a9=7 / W b2, b3 = 9, 7 / / W g5, h5 = 6, 2 / e4 = 2 = d3 / Continue: 16 more cells... / P h7 or h8 = 2 /

15 Solving Sudoku15 Solution so far: 1) / W e8,f8=9,7 / i5 = 7 / 2) / a9=7 / W b2, b3 = 9, 7 / 3) / W g5, h5 = 6, 2 / e4 = 2 = d3 /... 9) / P h7 or h8 = 2 / 10) / L3 L9: h5=6, g5=2 / g9=6/ 9 /

16 Solving Sudoku16 Last method: Cross pairs. Theorem 6. If p, q, r, s are 4 cells at the corners of a rectangle and there are two copies of digit x in these cells, then all other cells in both rows and both columns are not x. Write X in the solution.

17 Solving Sudoku17. Fig 6 Difficult. 1) h6/ P c7 or c9=2 / e7 or e9 = 2 / 2) / X h8 or i8 = 2 / h2 or i2 = 2 / 3) / X g5=2/ Z h4=1=g7 /

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