Sudoku: Pincers & Chains

After the simple eliminations are completed in a Sudoku, we sometimes get to a point where the next step isn't obvious. The Pincer is a pattern that may help eliminate one more candidate and get past the impasse. A Chain is an 'extended' pincer and is a bit harder to detect. But once you get adept at finding Pincers and Chains, they become invaluable for solving Sudokus. The basic Pincer pattern is popularly called the 'XY Wing', and chains are called 'XY Chains'. This note shows a few methods to find Chains.

To get started, perform the simple eliminations and mark each cell with possible remaining candidates. Now the goal is to eliminate more candidates. For the basic Pincer, find a sequence of 3 cells, each with only 2 candidates, that have the pattern AB - BC - CA. Further, the cells containing AB and BC should be in the same zone, and BC and CA should be in a zone. 'Zone' means the same row, column or macro cell. If you find such a pattern, then other cells that are in the same zone as both AB and CA cannot have the digit A as a candidate. This is because one of either AB or CA will be the digit A. Think of AB and CA as the tips of the Pincer and they squeeze out any A that is in its grips. Applying this technique to cells with 3 candidates is possible, but is more tricky.

Here is a simple example.

Look at the chain of cells marked green: Either way, the candidate 4 is eliminated from the purple
cell Click on the puzzle to magnify it.

*b9* - *c8*
- *c6* (or 48 - 58 - 45). Note that the cell *b4* (containing
45) is controlled by both *c6* and *b9*. The cell *c8* can
be 5 or 8. If it is 5 then *c6* will be 4 and *b4* cannot be 4.
If it is 8 then *b9* will be 4 and *b4* cannot be 4.*b4* because it is common to *b9* (48) and *c6* (45).
It is 5.

In the above example, the green cells *b9* and *c6*
(48 and 45) are both 'controlled' by the green cell *c8* (58). That cell,
called the pivot, can either be a 5 or an 8. If it is 5, then *c6* will
be 4. If it is 8 then *b9* will be 4. Either way, one of the pincer tips
will be a 4. The purple cell marked 45 is controlled by both 45 and 48.
Therefore, the purple cell *b4* 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 are those with exactly 2 candidates. Start with one of them, say AB. Look for a BC (which has one shared candidate) that is in the same zone. Then look for a cell containing CA, which is what is left after joining BC & AB and dropping the common B. If one is found, check if the cells AB and CA control some other common cell, and if that cell has A as a candidate. If so, eliminate A.

Another simple example:

The red cells It becomes 29 and along with the other 29 in

*d4* -> *d9* -> *f7*
contain 24 -> 25 -> 45. The target cell *f4* (in purple)
cannot be a 4.*e4*,
it forces the red 24 in *d4* to be a 4.

Another example, with a rare double yield...

The cells circled green The chain is: 89 - 58 - 59.

*a2*, *c3* and *d3*
force the elimination of 9 from the cells circled in purple *d2* &
*e2*. So they become 5 and 2.

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

Cells circled in green force the cells circled in purple
to be NOT 5. So purple 35 in The chain is: 75 - 37 - 35

*e8* becomes 3.

Soon, we get to this situation. The green cells force the
purple cell in Alternatively, in this case, the pincer could start with

*g3* to be NOT 4. It becomes 6. The chain is: 46 - 63
- 34*c1* (36), then *i1* (34) and *g3* (46). Now the 6 in the green
circled *a3* (46) is eliminated. Either way...

Chains

Chains are like longer pincers. A chain is a sequence of connected pairs where the first and last cell have a common candidate and they control a common cell. Consider a chain like: AB - BC - CD - DA. If the two end cells AB and DA can both see a target cell, then the target cannot have an A as a candidate. This is because if the first cell is A, then the target cannot be A. If the first cell is B, then the last cell in the chain will be an A, and again the target cannot be A. The first cell can only be A or B. This is the same logic used while explaining pincers.

There are many types of chains. Like Pincers, they are made up of cells what have exactly 2 candidates. To find a chain like AB - BC - CD - DA, starting with AB, look for a cell with one common candidate, say BC, in the same zone. Next, from BC, look for a CA in the same zone as BC. If a CD is found instead, then look for a DA in the same zone as the CD. If a DE is found instead, then look for a EA in the same zone as the DE. It seems exhausting, but it becomes intuitive with a little practice.

Here is a simple chain which 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. View the chain as: 28 - 68 - 68 - 28.

Here we have a 28 in The cell Note the two blue cells containing 28 could not be used
for the chain.

*b6* and a 68 in *a5* (circled
in red). We are looking for a 62, but we find another 68 in *a9*. The
next search is for 28 from *a9*. This is 62 & 68 and remove the
shared 6 to get 28. This is found in *e9*. Yay!*b9* in purple is common to *b6* and *e9*,
the ends of the chain. It cannot be a 2. That means the cell *e9* must
be a 2.

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

Start with Next, we notice a Pincer the 3 purples starting at

*a2* (green 59) -> *i2* (red 58).
Looking for 89, but find *i8* (red 48). Now looking for 49, which is
found in *h7* (red 49). The chain is: 59 -> 58 -> 48 -> 49.
So, *a7* (red 569) cannot be a 9. It becomes a 56.*a7*
-> *d7* -> *e9* (56 -> 68 -> 58). This eliminates
the two 5s in the bottom row (*b9* & *c9*, circled yellow)

Sometimes we find a chain while searching for Pincers. In Pincers, when we find a potential AB and BC, we search for a suitable cell with candidates 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. The previous example had a common missing link, both 59 & 58, and 49 & 48 needed a 98. There is no cell with 98 as candidates, but the 2 pairs fit together.

Here is another example of such 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: 69 - 46 - 48 - 89 ( Think of this as 2 pairs that both need a 94.

*f3* - *f4* -
*f8* - *d7*). The cell *d3* circled in red cannot be 9.

And here is another...

*c1* (78) & *b3* (17) and *i3* (13)
& *i9* (38) need a 18. *b3* (17) and *i3* (13)
connect. The chain ends are *c1* (78) and *i9* (38). They wind
up eliminating candidate 8 from *i1 *in red.

In the 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 *i1* (red 58). The chain is 78
- 17 - 13 - 38, and it is game over.

The next example has a 'tunnel' connecting two pairs. The
Cells *e1* & *f2* (red) need a 23. So do *b4* & *b9*
(green). The two pairs are not directly connected, but there is a way.
Starting with *e1* & *f2* we are looking for a 23. We don't see
a 23, but there is a 25 in *a2*. Moving to *a2*, we are now looking
for a cell with 35. Moving to *a4*, we are looking for a 23. Next move
to *b4* and look for a 35, and there it is in *b9*. The chain is *e1*
-> *f2* -> *a2* -> *a4* -> *b4* -> *b9*.
The tips e1 and b9 each have a 3. So the common target cell *e9* cannot
be a 3.

Cells They wind up eliminating candidate 3 from

*e1* & *f2* (red) need a 23. So do *b4*
& *b9* (green). They are connect by a 'tunnel' *a2* -> *a4*
-> *b4*. *e9*.

Often, there are puzzles where many cells have the same candidate pairs. It helps to form chains out of them. These chains look like AB - BA - AB - BA. I like to mark them with slanted slashes. Again, this is only for like pairs of candidates. When a reflected chain is long, we can eliminate candidates if a cell sees both left & right slashes for the same candidates, it cannot contain those candidate numbers.

The following puzzle has several cells marked 26. Starting
at *c3*, we mark it with a upward slant. The matching cells *b2* and
*c8* get an downward slant. We find that *c3* and *i9* have
opposite slant. The common cell *i3* cannot contain a 2 or a 6 and must
be a 5. The rest is easy.

Mark chains with matching pairs with slashes,
alternating left and right slant, like / and \. Note two 26s circled in red. They have opposite slant markings.
Any cell common to them cannot have a 2 or a 6.

The next puzzle also has chains of 26, just a coincidence.

The force is strong in this puzzle, but you need to be
persistent. The cells *e3*, *e7* and *d8* circled in red are
clearly connected. If *e3* is 2 then *e7* must be 6 and *d8*
must be 2. Similarly if *e3* is 6, *d8* will be equal to *e3*
and *e7* will be opposite. Marked with right or left slashes (shown in
purple). Cells with the same slant will be equal.

Next, note that there is also a 26 in *c1*. In this
case it is not too hard to connect it with the chain. If *c1* is a 6,
then *c7* will be a 2 and so *e7* will be a 6. But if *c1* is a
2, *c7* is 7, *c4* is 6, *d4* is 7, *d8* is 8, *d1* is
4, *d5* is 2, *d8* is 6 and so *e7* will be 2. In both cases *c1*
will be the same as *e7* and so it should be marked like *e7*. We
can eliminate 6 as a candidate from *a3*, that does not do much, but here
comes the reward.

Let's look for pincers. Looking at column 3, *c1* and *c7*
(26 and 27) form a pincer that eliminates 2 in any other cell common to them.
Well, there are no target cells containing a 2. However, we just showed that *c1*
is equivalent to *e7*. So *c7* and *e7* cannot allow any 2s.
This eliminates 2 in *a7*, *b7* and *h7*, the cells marked with
a red underline. The puzzle falls apart easily after this.

One more cool example of reflection, this time from
SudokuEssentials.com. After basic filling, we get the solution below on the
left. We find the pincer marked in red which forces *a1* to 2, *h1*
to 5, *h3* to 24 and *a4* to 45. Then the pincer marked in green further
reduces *i6* to 1, *e5* to 1 and *b6* to 2. Now it looks like
the figure on the right. There are many cells with the same two choices. We
can show that cells containing 56 *d2* and *e6* will be the same and
opposite to cells *e3* and *d5*. Also that cells containing 45 *a4*
and *g5* will be the same and *b3* will be opposite. But the jackpot
is in the cells containing 24. Clearly *h3* and *h8* are opposites.
How about cell *i4* ? If *i4* is 2 then cell *h8* must also be
2. If cell *h8* is 4, then *g7* must be 7, *i8* must be 2 and *i4*
must be 4. Therefore, cells *i4* and *h8* will be the same, and
opposite to *h3*. Mark these 3 cells with green slashes. Now we see that
cells *h3* and *i4* will always be opposite, so the common cell *i3*
cannot contain 2 or 4. Cell *i3* must be 7. The puzzle falls apart
easily after this, filling the solver with satisfaction.

It may seem that we did something shady here while showing
that cells *h8* and *i4* will be equal. We showed that if *i4*
was 2 then *h8* would be 2. Ideally, we should show that if *i4* was
4 then *h8* would also be 4. But we went backwards, showing that if *h8*
was 4, then *i4* would also be 4. This is ok because it can be proved
that if a relation (equal or opposite) is true in one direction for one value,
then it must be true for the reverse direction, *BUT ONLY IF* we can also show
that there is a similar relation for the other value as well.

Occasionally, while trying to establish a relation between cells, we may find that a target cell has the same value regardless of the value of the origin cell. This means that we were sloppy somewhere and missed eliminating a value from the target cell. Regardless, we got lucky and can lock that value into the target cell.

The lesson here is that it often helps to find related cells that contain like pairs and mark them with slashes. They are easy to find and can help sometimes.

Finding chains is not easy. Here is one way to do it. But
first, to set the stage, here is a long chain: 18 -> 28 -> 28 -> 38
-> 23 -> 21 (*a5* -> *e5* -> *e9* -> *d8*
-> *f7* -> *b7*). The 2 ends 18 & 21 eliminate candidate 1
from *b5* and *b6*, which forces *i6* to be 1 and then
everything else falls into place. Later, I found a shorter chain: 18 -> 28
-> 23 -> 23 -> 12, with the same effect.

BTW, not all pincers result in useful eliminations. There
is a pincer 69 -> 16 -> 19 that eliminates the 9 in *e1* below, but
nothing exciting came out of that.

This is going to be a brute force search with a way to avoid
dead ends. Say we start with 18 in *a5*. Find another cell in the same
zone with two candidates, one of which is common to the current cell. Cell *e5*
has a 28, where 8 is common to 18 and 28. Drop the 8 and look for 12 starting
from *e5*. If there is no 12, then look for a cell with a common
candidate. The 23 in *f5* has a 2 that is common to 12 and 23. Drop the
2 and look for the remaining candidate pair: 13. There is no 13, but there is
a 23 in *f7*. Drop the 3 that is common and look for the remaining candidate
pair: 12. Wow, there is one at *b7* which completes the chain. Next,
verify that the start and end cells in the chain have a common candidate.
Start cell 18 and 12 share the candidate 1. Then check if there are any cells
common to the start and end cells. Yup, cells *b5* & *b6* are
common and they both have 1 as a candidate. Eliminate the 1 from those cells
and move on.

I add new examples when I see interesting ones. Most of the examples here are from (c) Daily Sudoku. I thank them for many relaxing hours.

*Nov 2021*

All in fun...