﻿ Triplets

Triplets (Method 5b)

Triplets occur frequently in puzzles and are very useful in eliminating possibilities.  Triplets follow essentially the same rules as twins, except they involve three cells and three possible numbers.  The definition of a triplet is three cells in an element with a total of three possibilities between them.

Triplets can be more complicated than twins, though, as there are more variations.  We show this on Figure Triplets 1.  Can you pick out the triplet in column three, highlighted in blue?  That’s right, cells 32, 37 and 38 all have the possibilities 134.  Because there are a total of three possibilities over three cells in an element, they are triplets.  One cell must be a 1, the other a 3, and the other a 4.  You can eliminate the other 1, 3 and 4s in that column.

But now it gets a little trickier.  Column 6, highlighted in green, also contains triplets.  Only this time, they are a little harder to find.  For convenience sake, we’ve placed them in the same row as the previous example, but notice the pattern is different.  This time, it’s 134, 134 and 34.  But still, following the rules of triplets, this is a triplet.  We know that one cell must be a 1, the other a 3, and the other a 4.  The only difference is, cell 68 will be a 3 or a 4, but can’t be a 1.  So you can eliminate the 1, 3 and 4 from the other cells in the column.

And now, the real tricky triplet is found in column nine.  It’s so tricky that we’ve coined the name “two-faced triplets” in describing it.  Why two-faced?  For two reasons.  First, each cell only has two possibilities.  Second, it’s two-faced in that it masquerades as a potential pair, not as a triplet.  But in reality, a triplet it is.

So back to column nine, again, in the same rows (2, 7, 8) as the previous examples, notice that the possibilities are 13, 14, and 34.  How are these triplets?  Remembering the rule, there are three cells, and three possibilities between them; in this case, 1, 3 and 4.  One cell has to be a 1, the other a 3, and the other a 4.  Therefore, this is a triplet, and you can eliminate those possibilities for the other cells in the column.

Two-faced triplets occur somewhat infrequently; perhaps in about 10-15 % of puzzles.  Because of this, and because they parade as potential twins by containing two digits, they are very easy to overlook.  I’ve been stuck on many a puzzle for 10, 20 or 30 minutes, until I finally detected these masters of disguise.  So you may want to be on the lookout for them, especially if you get stuck, because finding one can get you out of a jam.

And remember, triplets can occur in several forms, even more than the three examples we’ve shown.  134, 134, 134 is an obvious triplet.  And we’ve already seen two-faced triplets 13, 14, 34.  But 134, 134, 13 is a triplet, and 134, 13, 14 is a triplet.  Just remember the rule:  three cells, total of three possibilities . . . that’s a triplet.