The Sudoku rule of triples I discussed in the previous entry is
actually stricter than it needs to be. Suppose you have a row that
looks like this:

There are 3 cells that together contain only 3 values: {2,4}, {4,8},
and {2,8}. Sippose the first cell actually contains a 2. Then the third
cell must contain an 8 and the middle cell must contain a 4.
Alternatively, if the first cell contains a 4, the second cell must
contain an 8 and the third cell must contain a 2. Either way, these
three cells are forced to contain the values 2, 4, and 8 in some order.
Therefore, none of the other cells in the same group can have any of
those values, and they can be eliminated.

Here‘s a puzzle that can take advantage of this rule.

After some elimination and filling in unique values, we arrive at this point:

Notice the values {4,8}, {1,8}, and {1,4} in column (the ones circled
in red). We can remove 1‘s, 4‘s, and 8‘s from the rest of the column,
and we find out that cell (7,3) must contain a 9.

Later in the same puzzle, we arrive at the grid below. Another interlocking triple is present, (appallingly badly) highlighted.

We can remove the numbers 2, 4, and 7 from the rest of the box:

The rest of the solution proceeds using elimination and uniques:

There's a subtle variation in the rule that can also come in handy. As
long as the three cells contain exactly three possible values between
them, the rule is valid:

For example, the values {2,4}, {4,8}, and {2,4,8} form an
interlocking triple as well. If the first cell contains a 4, the second
must contain an 8 and the third must contain a 2. If the first cell
contains a 2, the other two cells form a pair on {4,8}. Either way, the
cells in the rest of the group can't contain any of the values 2, 4, or
8. Cells {2,4}, {2,4,8}, and {2,4,8} would also form an interlocking
triple: if the first cell contains a 2, the other two cells form the
pair {4,8}; if it contains a 4, the other two cells form the pair {2,8}.

The
rule of pairs and triples actually generalizes to N-tuples: if any N
related cells contain exactly N possible values between them, those
values can be removed from the rest of the group. The rule of
elimination is just the case where N=1. The rule of pairs is the case
where N=2. For N=3, as we have seen, there are several ways in which 3
cells can hold 3 values. Ignoring ordering, the ones we‘ve seen so far
look like this:

{A} {B} {C} -> nothing to do

{ABC} {ABC} {ABC} -> rule of triples

{AB} {AC} {BC} -> interlocking triples

{AB} {AC} {ABC} -> interlocking triples (variation)

The rule can be extended to quadruples,
quintuples, etc. But I expect that these cases are rare in practice,
and quite difficult to spot.

Here are a few puzzles
that can benefit from the interlocking triples pattern. At least that's
the case in my solver program. Your experience may differ since the
order in which rules are applied affects which rules may come into
play. An interesting future project would be to examine all possible
solution orders, and find puzzles that absolutely require a particular
rule for their solution (given a particular set of allowable rules).

Puzzle 1:

Puzzle 2:

Puzzle 3:

Puzzle 4:

## Recent Comments