A Sudoku puzzle is locked when it is still incomplete and there are no more fill steps to be made. The key to unlocking it is to use logical deduction and eliminate some possible locations for a digit or digits. The elimination step or steps would hopefully lead to a valid fill step.
There are two general types of elimination steps: a digit restricted by location (confinement: B, R or C ) and two or more digits filling up two or more locations (full house: N).
Elimination by confinement
The simplest case of confinement is when a digit is restricted within a block (B) to two or three cells that are also within a column or row. The digit can then be eliminated from the rest of the column or row. The SP Notation supports this by an observation step and one or more elimination steps. The observation step takes the form <locations>:<digit> <reason> and the elimination step takes the form <location>/<digit> <reason>.
The observation step is based on the fact that three cells make up the intersection of a block and column or of a block and row. A block restriction of a digit can thus lead to the elimination of the digit from the rest of the row or column. Similarly, a column or row restriction of a digit can lead to the elimination of the digit from the rest of the block.
A more complicated case of confinement involves 2 columns and rows. If a digit is restricted to 4 cells along 2 rows that are also in 2 columns (forming a rectangle), then the digit can be eliminated from the other cells in the 2 columns. The observation of 2 row restrictions of a digit can lead to the elimination of the digit from the other cells in the 2 columns. Similarly, the observation of 2 column restrictions of a digit can lead to the elimination of the digit from the other cells in the 2 rows.
More complicated cases of confinement encompass 3 rows and 3 columns or 4 rows and 4 columns.
Elimination due to full house
The simplest example of a full house is when 2 cells may only contain the same 2 digits and the 2 cells are within the same block, row or column. The digits can then be eliminated from the other cells in the block, row or column. The observation step takes the form <locations>:<digits> N and the elimination step takes the form <location>/<digit> <reason B, R or C>.
The full house can also apply to 3 digits in 3 cells or 4 digits in 4 cells.
A full house can also be the result of confinement, when 2 digits are restricted to the same 2 cells within a block, row or column. In this case, the other digits can be eliminated from the 2 cells. This can apply also to the confinement of 3 digits in 3 cells (or 4 digits in 4 cells) within the same block, column or row. The observation step takes the form <locations>:<digits> <reason B, C or R> and the elimination step takes the form <location>/<digit> N.
For more information on SP Notation, please see the 2 earlier blogs
The Sudoku puzzle from the Toronto Star on Saturday, May 7, 2011 is a very good example of a locked puzzle because it is locked twice. It is also a twice-jammed puzzle.
Sudoku puzzle from Toronto Star, Saturday, May 7, 2011
| 000006180 000000072 100090600 600870040 000402000 070053001 009030004 410000000 057600000 |
1 A1 = 7 BC
2 C6 = 4 BR
3 D6 = 9 N
4 E5 = 6 BC
5 G2 = 4 BC
6 H6 = 6 R
7 B7 = 6 R
8 C5 = 1 R
9 F4 = 1 BN
10 I1 = 9 B
11 I8 = 6 BC
12 C2 = 6 BCR
2 C6 = 4 BR
3 D6 = 9 N
4 E5 = 6 BC
5 G2 = 4 BC
6 H6 = 6 R
7 B7 = 6 R
8 C5 = 1 R
9 F4 = 1 BN
10 I1 = 9 B
11 I8 = 6 BC
12 C2 = 6 BCR
At this point, the puzzle is jammed … (Click here for continuation)
A | B | C | D | E | F | G | H | I | |
1 | 7 | 6 | 1 | 8 | 9 | ||||
2 | 6 | 4 | 7 | 2 | |||||
3 | 1 | 9 | 6 | ||||||
4 | 6 | 8 | 7 | 1 | 4 | ||||
5 | 1 | 4 | 6 | 2 | |||||
6 | 7 | 4 | 9 | 5 | 3 | 6 | 1 | ||
7 | 6 | 9 | 3 | 4 | |||||
8 | 4 | 1 | 6 | ||||||
9 | 5 | 7 | 6 |
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