Here's another Sudoku rule that can be useful in solving puzzles. If
you followed the discussion about the rule of pairs, you should be
ready to grok the rule of *triples*. Simply put, if three
related cells (cells in the same row, column, or box) can only contain
a set of three possible values (that is, we've eliminated all other
possible values using other rules), then no other cell in the same
group can contain any of those values. For now, we'll only consider the
case where all three cells can contain exactly the same three values;
next time I'll discuss what I call *interlocking triples*.

Triples don't come up terribly often in practice, at least not just by chance. Of course you can design a puzzle to contain triples. My puzzle generator typically needs to search for several minutes in order to find a puzzle that requires the use of the rule of triples to be solved. Here's such a puzzle:

Using elimination, uniques, and pairs, we can get to this point:

Notice that the triple (1,3,6) appears in three cells in row 8. These values can be removed from cells (3,8), (5,8), and (7,8), leaving the grid:

Now cell (7,8) has only the value 2 left, so we can fill it in and use it to eliminate a bunch more values. Eventually, we arrive at the solution to the puzzle:

Here are a couple of puzzles that can be solved only using the rules I've discussed so far, including the rule of triples.

Triples are the hardest thing for me to figure out. I've been looking for answers and will now have to put them into practice.

Thanks, Deena

Posted by: Deena | February 05, 2006 at 10:04 PM

Very good! Well explained.

Posted by: Brian | December 18, 2008 at 03:50 PM