Sudoku: Pincers & Chains

After the simple eliminations are completed in a Sudoku, we sometimes get to an point where the next step isn't obvious.  The Pincer is a pattern that may help eliminate another digit and get past this impasse. This pattern can be further extended to a Chain, though Chains are rare and harder to detect.  Others have called this pattern the 'XY Wing'.

To get started, mark each cell with possible options and perform the simple eliminations.  For the basic Pincer, find 3 pairs of the form AB - BC - CA, such that A & B are in the same zone, B & C share a different zone, and likewise C & A.  ('Zone' means the same row, column or macro cell.) If you find such a pattern, then no other cells that are in the same zone as AB and CA can be the digit A.  This is because, one of either AB or CA will be the digit A. 

Here is a simple example.

Cells circled in green force the purple cell to be a NOT 4.  So it is 5.

The chain is: 48 - 85 - 54, with 58 as the pivot.

 4 has to be eliminated from the purple cell common to 48 and 45.

That makes the 56 in the rightmost cell a 6 and the cell marked 456 must be a 4.

 

In this example, the green cells marked 48 and 45 are both 'controlled' by the green cell marked 58.  That cell can either be a 5 or an 8.  Either way, one of the controlled cells will be a 4.  The purple cell marked 45 is controlled by both 45 and 48.  Therefore, the purple cell cannot be a 4.

After this elimination, the rest of the puzzle can be rapidly solved.

The trick is to find these Pincers.  Firstly, all the cells that are part of a pincer (or chain) are those with exactly 2 options.  Start with one of them, say AB.  Look for a BC that is in the same zone.  Then look for a CA.  If one is found, check if AC and CA control a common cell.


Another simple example:

The cell in purple cannot be a 4.

Not that it helped that much !!

 

Another example, with a rare double yield...

Cells circled green force the cells circled in purple to be NOT 9.  So they become 5 and 2. 

The chain is: 98 - 85 - 59. 

 


Now a game where the Pincer is applied twice.  First:

Cells circled green force the cells circled in purple to be NOT 5.  So purple 35 becomes 3.

The chain is: 57 - 73 - 35

 

Then we get to this situation.  The green cells force the purple cell to be NOT 4.  It becomes 6.

The chain is: 46 - 63 - 34

Alternatively, in this case, the chain could start with the 36, then the 34 and 46.  Now the 6 in the green circled 46 is eliminated.

End result is the same...

 

 

Chains:

Chains are harder to find, but here is a simple chain.  It is easy to recognize because it has 2 reflecting pairs.  In the following, the chain is like: AB - BC - CB - BA.  The end points of the chain control a cell that contains A, which is then eliminated.  (AKA XY Chain)  View the chain as: 28 - 68 - 68 - 28

Here we have a 68 and 28.  We are looking for a 62, but we find another 68 & 28.  The cell in purple cannot be a 2.

That means the 28 in the bottom row must be a 2.

Note that there are 2 more 28 (Blue) in the middle block.

 

Here is a longer chain: AB - BC - CD - DA.  It eliminates A from cells common to AB and DA.

Start with the green circled 59 -> Red 58 -> Red 48 -> Red 49.  The chain is:

95 -> 58 -> 84 -> 49

So, the red 569 cannot be a 9.  It becomes a 56  (purple)

Then we notice the 3 purples: 56 -> 68 -> 58.

This eliminates the two 5s in the bottom row (circled yellow)

 

Shared link

There is a simple way to find 4 link chains while searching for Pincers.  In Pincers, when we find a potential AB and BC, we search for a suitable CA.  Often we don't find it and continue our search.  Sometimes we then find another pair AD and DC which also needs a CA.  When we note two separate pairs that need the same missing cell, we can try to join them together.  One cell of each pair needs to be share a zone.  Here is another example of a chain: AB - BC - CD - DA.  The chain is in green and the affected cell is in red.  The two pairs 69 & 46 and 48 & 89 both needed a 49.

The chain is: 96 - 64 - 48 - 89

The cell circled in red cannot be 9.

Think of this as 2 pairs that both need 94.

 

And another...

In the next piece above, we see Green 78 - Green 17 and we search for a 18.  It is just not there.  Then we see Blue 13 - Blue 28 and note that it too needs an 18.  We see that the 17 and 13 share a zone, and walla!, we have a chain that knocks out the 8 from Red 58.  The chain is 78 - 17 - 13 - 38, and the game is over. 

Attribution:

Most of the examples here are (c) Daily Sudoku.  I thank them for many relaxing hours.

Jul 2018