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Aligned Pairs

Warning! The aligned pairs technique is an advanced technique, meaning that it is only used when you ask Puzzle Tiger for a "Very Hard" puzzle. As such you will almost certainly never come across a published puzzle in a newspaper or magazine which requires this or any of the more advanced techniques. If, however, you're interested in really stretching yourself, read on...

Aligned pairs is a candidate elimination technique, meaning that it eliminates possible values from cells. The aligned pairs technique states that no pair of cells can take on two values that replicate the possible candidates in a simple pair that they can see (a simple pair is a cell with only two possible candidates). A cell can "see" another if they share a group (a row, column or box). This technique requires a little further explanation, but an example is worth a thousand words:

In this example, the cells highlighted in dark blue and red can both see the two pairs highlighted in orange. The technique allows the digit "3" to be eliminated from the cell highlighted in red. To see how this works, let's write out a list of all the permutations of values that the red and blue cell could have:

So there are six possible ways we could fill the dark blue and red cell, when taken together. Now, let's consider the simple pairs that both cells can see (the important ones are highlighted in orange in the picture above). Suppose we were to choose option #2 for the values of the cells. This would mean that the red cell would be a "3" and the blue cell an "8". The cell at the bottom of that column, highlighted in orange, could therefore neither be a "3" or an "8". So there would be no values left to go there. So option #2 can't possibly be correct. Going through the table in this way, we can eliminate some of the possible options:

Option #1 isn't possible, because both cells share a box with a {3, 4} simple pair, and option #2 isn't possible because they both share a column with {3, 8}. This leaves us with only the last four options in the table. Which are {4, 3}, {4, 8}, {9, 3} and {9, 4}. So the red cell can be a 4 or a 9, but none of these options allow it to be "3". Below is another example:

In this example, the cell highlighted in red cannot be a "3". If we consider the possible options for the red and dark blue cells, we get {2, 8}, {2, 9}, {3, 8}, {3, 9} and {8, 9}. But both of these cells can see the two simple pairs highlighted in orange, {3, 8} and {3, 9}. This means that we can eliminate these permutations, leaving us with {2, 8}, {2, 9} and {8, 9}. So the cell highlighted in red cannot be a "3". This elimination can also be made using the Y-wing pattern, but the first example cannot.

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