Interlocking Triples

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:

Comments

[Lachlan Duncan]

Hi Dan,
Thanks for your Suduko blog's - they have been really helpful and your note are very simple to follow.

Just a quick obervation on the "Interlocking Triples" puzzle where you hand drew the zone around the 2,4 and 7. The number 9 is not an option in column 8 row 2 cell as it occurs in column 7 row 3.

[David Decker]

Great blog, I forwarded it to two of my friends. Luv it !

David
http://www.cyberken.com/sudoku1.htm

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