The New Sudoku Players' Forum

[Scott H]

Pairs can be incorporated into forcing chains to extend the scope of nice solutions. An example:

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..1|754|.8.
4..|836|51.
...|921|74.
-----------
123|5.8|.9.
9..|2.3|.58
...|6.9|32.
-----------
.39|16.|874
814|397|265
...|48.|.3.

{23}   {69}   {1}    {7}  {5}   {4}   {69}  {8}  {23}   
{4}    {79}   {27}   {8}  {3}   {6}   {5}   {1}  {29}   
{356}  {568}  {568}  {9}  {2}   {1}   {7}   {4}  {36}   
{1}    {2}    {3}    {5}  {47}  {8}   {46}  {9}  {67}   
{9}    {467}  {67}   {2}  {147} {3}   {14}  {5}  {8}   
{57}   {4578} {578}  {6}  {147} {9}   {3}   {2}  {17}   
{25}   {3}    {9}    {1}  {6}   {25}  {8}   {7}  {4}   
{8}    {1}    {4}    {3}  {9}   {7}   {2}   {6}  {5}   
{2567} {567}  {2567} {4}  {8}   {25}  {19}  {3}  {19}

rows 456, showing only the relevant cells:
*      *      *      *    *     *     *     *    *     
*      {467}  {67}   *    *     *     {14}  *    *     
{57}   {4578} {578}  *    *     *     *     *    {17} 

r5c23 is almost a naked pair (67). If candidate 4 in r5c2 were excluded it would be a naked pair.
Now look at r6c9, r5c7, and r5c23. This is like an xy-wing, except it has an almost-naked-pair on one end instead of a doubleton node. Either r6c9 = 7 or r5c23 = 67 (or both), which excludes 7 from Box4 & r6 (giving r6c1 = 5) and opens up an easy finish to this puzzle.

An almost-pair is a configuration which would be a pair except for one candidate in some cell. This example uses an almost-naked-pair. An almost-hidden-pair or almost-X-wing can also be used in a forcing chain. The extra candidate for an almost-naked-pair is in a pair cell, but is outside the pair or X-wing for an almost-hidden-pair or almost-X-wing. A forcing chain using the extra candidate in a link incorportes the almost-pair into the chain.

This example has the almost-pair at the end of the chain, but you can also have almost-pairs in the middle of a forcing chain.

It's possible to incorporate almost-triples, almost-quads, ... into forcing chains, but I'll stop with pairs for now

[Jeff]

Scott

This is exactly what we need. Simple forcing chain that can solve puzzle without filtering. Your technique is similar to the advanced xyz-wing application. This example, in particular, is a combination of xy-wing and xyz-wing. Thank you for extending our logical thinking.

[Scott H]

Thanks Jeff. I enclose a second (constructed) example of an Xwing-chain combination in 4s:

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***|**.|.**
***|**.|.**
*4*|**4|4**
-----------
***|**4|4**
***|**.|.**
***|**.|.**
-----------
.4.|**.|.**
..4|**.|4**
...|**.|.**

where any of the *s can be 4. There is an almost-Xwing in r34c67 and r8c7, and a two way relation

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r8c7<>4  iff   r34c67 is an Xwing in 4s

This relation can be used as a strong link in a double implication chain:

r8c3=4 => r8c7<>4 => r34c67 is Xwing => r3c2<>4
r8c3<>4 => r7c2=4 => r3c2<>4

I placed this Xwing in adjacent rows/columns for easier visibility, but this is (obviously) not necessary in general.

[Jeff]

Another advanced application of x-wing. Well done.

I notice that a property of the almost x-wing is to turn a weak link into a strong link. The 4s in column 7 is normally a weak link, but turn into a strong link because of the almost x-wing. This is an important property that should be shared under a new thread.

Once this property is understood, it can be used in other boolean type techniques such as Angus' colouring, Colour chains and Turbot chain. I can see a turbot fish in your example.