Wednesday, October 28, 2009

Sudoku: ‘Locked’ Candidate Reduction Method Clarified

To solve a sudoku puzzle you need to fill each empty square with one of the numbers 1 to 9. In some squares there may more than one candidate number, and it may be difficult to decide which of these candidate numbers would be the correct one. The number of candidates needs to be reduced. You need to perform a 'candidate reduction’ in the sudoku puzzle. So, you should try to locate candidates, which may safely be removed from the squares. There are various methods for doing just that. A relatively easy method is to locate a ‘locked’ candidate in the sudoku puzzle. It works like this:

Sudoku Locked Candidate: Row - Block

If in a row a candidate number is confined to a single block, it is 'locked' inside the block. Since the block can only have this number once, that candidate number can be removed from the other rows within that block.
Here you can see that within row 3 candidate number 7 only occurs inside block K (blue numbers in squares E3 and F3). So, within block K number 7 is ‘locked’ in row 3. Therefore candidate number 7 can be deleted in other rows within block K, i.e. in squares D1, E1 and F1 (red numbers).
In the small picture you can see the final solution for this part of the board. Within block K number 7 should be in square F3.

Sudoku Locked Candidate: Column - Block

Similarly, if in a column a candidate number is confined to a single block, it is ‘locked’ inside the block. Since the block can only have this number once, that candidate number can be removed from the other columns within that block.

Here you can see that within column C candidate number 1 only occurs inside block P (blue number in squares C8 and C9). So, within block P number 1 is ‘locked’ in column C. Therefore, candidate number 1 can be deleted in other columns within block P, i.e. in squares A9, B7 and B8 (red numbers). In the small picture you can see the final solution for this part of the board. Within block P number 1 should be in square C9.

Sudoku Locked Candidate: Block - Row

Conversely, if within a block a candidate number is confined to one row, it is 'locked' in that row. Since the number can occur only once in the row, that candidate number can be removed from that row outside the block.

Here you can see that within block K candidate number 1 only occurs in row 1 (blue numbers in squares E1 and F1) - it is 'locked' in row 1. Therefore, outside block K candidate number 1 can be deleted in row 1, i.e. in squares G1, H1 and I1 (red numbers).
In the small picture you can see the final solution for this part of the board. Within block K number 1 should be in square E1.

Sudoku Locked Candidate: Block - Column

Similarly, if a candidate number in a block is confined to one column, it is 'locked' in that column. Since the number can occur only once in that column, that candidate number can be removed from the column outside the block.

Here you can see that within block P candidate number 7 only occurs in column A (blue number in squares A7 and A8) - it is 'locked' in column A. Therefore, in column A candidate number 7 can be deleted outside block P, i.e. in squares A4 and A5 (red numbers).
In the small picture you can see the final solution for this part of the board. Within block P number 7 should be in square A8.

By repeating finding ‘locked’ candidates in the sudoku puzzle, the number of candidates can be reduced considerably. Good luck!