Mind, Body, World- Foundations of Cognitive Science, 2013a

After two applications of the “last available label” constraint, the cage illustrated in
Figure 8-5A becomes the cage shown at the top of Figure 8-6. Note that this cage
has only two ambiguous cells, each with the possible labels 3 and 5. These two cells
define what Sudoku solvers call a naked pair, which can be used to define a third
rule called the “naked pair constraint.”
3
3
5
5
7 6
8 1
2 9 4
457
89
34
89
1
457
6
8
Figure 8-6. The “naked pair constraint.”
In the naked pair pointed out by the two arrows in Figure 8-6, it is impossible for
one cell to receive the label 3 and for the other cell not to receive the label 5. This
is because these two cells have only two remaining possible labels, and both sets of
labels are identical. However, this also implies that the labels 3 and 5 cannot exist
elsewhere in the part of the puzzle over which the two cells containing the naked
pair have control. Thus one can use this as a constraint to remove the possible labels
3 and 5 from the other cells in the same column as the naked pair, i.e., the cells
shaded in grey in the lower part of Figure 8-6.
The three constraints described above have been implemented as a working
model in an Excel spreadsheet. This model has confirmed that by applying only
these three constraints one can solve a variety of Sudoku problems of easy and
medium difficulty, and can make substantial progress on difficult problems. (These
three constraints are not sufficient to solve the difficult Figure 8-3 problem.) In
order to develop a more successful Sudoku solver in this framework, one would
have to identify additional constraints that can be used. A search of the Internet
for “Sudoku tips” reveals a number of advanced strategies that can be described as
constraints, and which could be added to a relaxation labelling model.
374 Chapter 8