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

this location. The affected locations are shaded dark grey in Figure 8-4. One can
propagate the “there can be only one” constraint through these locations, removing
the digit 5 as a possible label for any of them.
This constraint can be propagated iteratively through the puzzle. During one
iteration, any cell with a unique label can be used to eliminate that label from all
of the other cells that it controls (e.g., as in Figure 8-4). When this constraint is
applied in this way, the result may be that some new cells have unique labels. In this
case the constraint can be applied again, from these newly unique cells, to further
disambiguate the Sudoku puzzle.
The “there can be only one” constraint is important, but it is not powerful enough
on its own to solve any but the easiest Sudoku problems. This means that other constraints
must be employed as well. Another constraint is called “last available label,”
and is illustrated in Figure 8-5.
Figure 8-5A illustrates one of the cages of the Figure 8-3 Sudoku problem partway
through being solved (i.e., after some iterations of “there can be only one”).
The cells containing a single number have been uniquely labelled. The other cells
still have more than one possible label, shown as multiple digits within the cell.
Note the one cell at the bottom shaded in grey. It has the possible labels 1, 3, 5, and
9. However, this cell has the “last available label” of 9—the label 9 is not available
in any other cell in the cage. Because a 9 is required to be in this cage, this means
that this label must be assigned here and the cell’s other three possible labels can
be removed. Note that when this is done, the “last available label” constraint applies
to a second cell (shown in grey in Figure 8-5B), meaning that it can be uniquely
assigned the label 1 by applying this constraint a second time.
3 5
3 5
1
5
7 6
8
3
9
1 5
2 4
3 5
A
7 6
8
3 5 1 5
2 9 4
B
Figure 8-5. The “last available label” constraint.
Seeing and Visualizing 373