## Fully supersymmetric chains

Advanced methods and approaches for solving Sudoku puzzles
denis_berthier wrote:As for the fact that the same chain is used twice to produce 2 deductions it could have produced at the same time, this is only an artifact of SudoRules and of the fact that the target is not attached to the chain: when two steps have the same complexity (here 2 xyzt6 chains), their order is arbitrary.

Thanks for responding Denis. Am I missing something obvious when I say that the next chain in your solution
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`nrczt6-chain {n9 n2}r6c1 - {n2 n6}r4c1 - {n6 n8}r5c1 - n9{r5c1 r4c3} - {n9 n7}r1c3 - {n7 n9}r2c1 ==> r8c1 <> n9 `

makes less sense if the 9 in r5c1 is not there?
re'born

Posts: 551
Joined: 31 May 2007

re'born wrote:Am I missing something obvious when I say that the next chain in your solution
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`nrczt6-chain {n9 n2}r6c1 - {n2 n6}r4c1 - {n6 n8}r5c1 - n9{r5c1 r4c3} - {n9 n7}r1c3 - {n7 n9}r2c1 ==> r8c1 <> n9 `

makes less sense if the 9 in r5c1 is not there?

No, you're not missing anything. If n9r5c1 is absent, this chain cannot exist.
I guess what's bothering you here: what if the next nrczt6 chain is applied (and the elimination done) before the present rule?
It is clear the next nrczt6 will no longer be applicable. But some other simpler rule will then apply.
denis_berthier
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Joined: 19 June 2007
Location: Paris

denis_berthier wrote:it is not true that, for my chain
nrczt6-chain {n9 n2}r6c1 - n2{r4c1 r4c7} - n2{r6c9 r2c9} - n4{r2c9 r2c7} - n9{r2c7 r3c7} - n9{r3c8 r5c8} ==> r5c3 <> n9, r5c2 <> n9
r5c123-9-r6c1-2-r6c9=2=r4c7=9=r5c78-9-r5c123, implies r5c123<>9
is much simpler: you are using ALS, when I am not. If you count the candidates involved, you get 12, as in my chain (not counting the targets).

Grouping candidates n9 in r5c78 is not ALS.
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` 4     29     79    | 3     257   579   | 6     1     8 789   289    5     | 2679  267   1     | 249   3     24 3     1      6     | 289   4     89    | 259   259   7--------------------+-------------------+------------------*269   7      349   | 459   135   359   |*12459 8     156 68-9  3468-9 348-9 | 4579  1357  2     |*1459 *4569  156*29    5      1     | 49    8     6     | 3     7     24--------------------+-------------------+------------------ 1567  346    2     | 567   9     357   | 8     456   1356 56789 34689  34789 | 1     2356  3578  | 2457  2456  356 15678 368    378   | 2568  2356  4     | 1257  256   9`

Using your notation, our alternative is {n9 n2}r6c1 - n2{r4c1 r4c7} - n9{r4c7 r5c78}. I count that as 6 candidates (in 3 bivalue/bilocation links), not 12. Well, maybe that's 7 candidates because of the grouping.

BTW your n9{r2c7 r3c7} bilocation link seems to ignore the existence of n9 in r4c7 and r5c7. How does that work
ronk
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Location: Southeastern USA

ronk wrote:Using your notation, our alternative is {n9 n2}r6c1 - n2{r4c1 r4c7} - n9{r4c7 r5c78}. I count that as 6 candidates (in 3 bivalue/bilocation links), not 12. Well, maybe that's 7 candidates because of the grouping.

I don't understand what you mean by {r4c7 r5c78}.
In my notation, there must be two distinguished (the left and right linking) candidates in each cell . Any addditional candidate must be justified by a t or z link.

ronk wrote:BTW your n9{r2c7 r3c7} bilocation link seems to ignore the existence of n9 in r4c7 and r5c7. How does that work

That's all the trick with xyzt chains: "bivalue" means "bivalue modulo something". I used to put a # in the cell, but it is redundant, once the type of the chain is declared before it.
Here, in n9{r2c7 r3c7, the bivalues modulo something are not those you may think, in block b3; they are in column c7.
n9r4c7 is allowed as an additional t candidate in column c7 because it is nrc-connected to n2r4c7, which is the right linking candidate of cell 2.
As for n9r5c7:
- it is allowed as an additional z candidate for the elimination of the n9r5c3 target because it nrc-connected to it,
- it is allowed as an additional z candidate for the elimination of the n9r5c2 target because it nrc-connected to it.[/quote]
denis_berthier
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Location: Paris

denis_berthier wrote:
ronk wrote:Using your notation, our alternative is {n9 n2}r6c1 - n2{r4c1 r4c7} - n9{r4c7 r5c78}. I count that as 6 candidates (in 3 bivalue/bilocation links), not 12. Well, maybe that's 7 candidates because of the grouping.

I don't understand what you mean by {r4c7 r5c78}.
In my notation, there must be two distinguished (the left and right linking) candidates in each cell . Any addditional candidate must be justified by a t or z link.

In your style, I think n9{r4c7 r5c78} would just be a grouped nrc-conjugate link. Just view the 9's in r5c78 as a right linking "quantum candidate" (to use a phrase from the early days of UR's). All of your work should pass through just fine and you can then define the general notion of a grouped nrczt-chain.
re'born

Posts: 551
Joined: 31 May 2007

re'born wrote:In your style, I think n9{r4c7 r5c78} would just be a grouped nrc-conjugate link. Just view the 9's in r5c78 as a right linking "quantum candidate" (to use a phrase from the early days of UR's). All of your work should pass through just fine and you can then define the general notion of a grouped nrczt-chain.

Still too vague for me to understand. What are the conditions:
- for n9{r4c7 r5c78} to be valid
- for n9r5c78 to be linked to the target
denis_berthier
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Joined: 19 June 2007
Location: Paris

denis_berthier wrote:
re'born wrote:In your style, I think n9{r4c7 r5c78} would just be a grouped nrc-conjugate link. Just view the 9's in r5c78 as a right linking "quantum candidate" (to use a phrase from the early days of UR's). All of your work should pass through just fine and you can then define the general notion of a grouped nrczt-chain.

Still too vague for me to understand. What are the conditions:
- for n9{r4c7 r5c78} to be valid
- for n9r5c78 to be linked to the target

Perhaps these won't be the correct definitions, but let me suggest at least an outline:

First one needs to define grouped candidates: Typically, this is a set of candidates all in a unit that are (nrc-)-linked in some uniform way. In our case, r5c78 lie in the same box (or row) and both contain 9. If we take your definition of nrc-linked and break it up as
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`L1: n1=n2 and the two rc-cells (r1, c1) and (r2, c2) are rc-linked (i.e. share a unit) in rc-space`

and
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`L2: n1 <> n2 and the rc-cells (r1, c1) and (r2, c2) are the same `

then I imagine we would want to say that a set of candidates form a group if there exists i in {1,2} such that the candidates are pairwise nrc-linked by Li (or is this too restrictive?) That way you would have L1-groups of candidates and L2-groups of candidates.

Next, for i in {1,2}, let A and B be disjoint Li-groups (if |A|=1, then A is a vacuously an L1 and L2-group). We say that A and B are nrc-linked if A union B is an Li-group.

Now we take the definition of nrc-conjugate and generalize it as follows:
For i in {1,2}, two disjoint Li-groups, A and B, are nrc-conjugate if they are nrc-linked and
if i = 1 (then there are numbers n_A and n_B describing the the L1-group structure of A and B):
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`C1: n_A=n_B,   there is a row, column or block containing A and B for which every instance of n_A occurs in either A or B`

if i=2 (then we get a set of numbers n_A and n_B describing the L2-group structure of A and B):
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`C2: n_A intersect n_B is empty and there is a cell that is the disjoint union of A and B`

Linking to the target would then be defined as before.

I think this should at least be a good start towards such a grouped nrczt-theory.
re'born

Posts: 551
Joined: 31 May 2007

re'born
This is not the way I work.
When I say I have a solution, and especially if I claim that it is shorter than another one, it means I have a solution using well defined rules that I have tested on thousands of cases, not imaginary rules.
Different ethics …
denis_berthier
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ronk wrote:Denis's "chain" looks like ...
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`                          +=9======================+r5c123-9-r6c1-2-r4c1=2=r4c7-2-r6c9=2=r2c9=4=r2c7=9=r3c7-9-r3c8=9=r5c8-9-r5c123     +-9-r5c7=9====================================+implies r5c123<>9`

Here is a much simpler alternative chain ...
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`r5c123-9-r6c1-2-r6c9=2=r4c7=9=r5c78-9-r5c123implies r5c123<>9`

denis_berthier wrote:re'born
This is not the way I work.
When I say I have a solution, and especially if I claim that it is shorter than another one, it means I have a solution using well defined rules that I have tested on thousands of cases, not imaginary rules.
Different ethics …

I can't imagine how the "way you work" and "ethics" relate to relative complexity.

You've merely chosen to use the multiple inferences caused by "t-links" and "z-links" which, for these exclusions, is clearly more complicated than using grouped candidates (6 bi-links instead of 3, 10 cells instead of 5, and your count of 12 candidates instead of my 6, and 2 multiple inferences instead of 0).
ronk
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Location: Southeastern USA

ronk wrote:Denis's "chain" looks like ...
Code: Select all
`                          +=9======================+r5c123-9-r6c1-2-r4c1=2=r4c7-2-r6c9=2=r2c9=4=r2c7=9=r3c7-9-r3c8=9=r5c8-9-r5c123     +-9-r5c7=9====================================+implies r5c123<>9`

Here is a much simpler alternative chain ...
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`r5c123-9-r6c1-2-r6c9=2=r4c7=9=r5c78-9-r5c123implies r5c123<>9`

denis_berthier wrote:re'born
This is not the way I work.
When I say I have a solution, and especially if I claim that it is shorter than another one, it means I have a solution using well defined rules that I have tested on thousands of cases, not imaginary rules.
Different ethics …

ronk wrote:I can't imagine how the "way you work" and "ethics" relate to relative complexity.

You've merely chosen to use the multiple inferences caused by "t-links" and "z-links" which, for these exclusions, is clearly more complicated than using grouped candidates (6 bi-links instead of 3, 10 cells instead of 5, and your count of 12 candidates instead of my 6, and 2 multiple inferences instead of 0).

Your post #1539 clearly shows that you hadn't understood the z and t extensions of nrc-chains:
ronk wrote:your n9{r2c7 r3c7} bilocation link seems to ignore the existence of n9 in r4c7 and r5c7. How does that work

But in the same post, you claim having a shorter solution with an extension of them that your re'born avatar later calls grouped nrc-chains, in a tentative definition.
That's where our ethics differ (but with yours you''ll probably not understand this).
denis_berthier
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### moderate level examples of nrc(z)(t) chains

Moderate level examples of nrc(z)(t) chains

Elementary and extreme puzzles are interesting but, if you want to get familair with these new chains, it is also worth having intermediate level examples.
Here are three pedagogic examples, in increasing order of complexity, taken from the Ocean collection, a vey nice collection, with all its puzzles starting with one or more swordfish.
http://forum.enjoysudoku.com/viewtopic.php?t=4144

Puzzle #1uses only nrc chains (of length no more than 4).
Our first example, which is also #1 in the list uses only nrc-chains.
Remember that nrc-chains are merely another view of "basic" AICs, i.e. AICs with no subsets of any kind (groups, hinges, …).
The chains used in this example should therefore be familiar.
000010002
001000030
040005600
000006700
300000005
008400000
007800040
050000900
200030000
swordfish-in-columns n3{r7 r1 r6}{c2 c6 c7} ==> r7c9 <> 3, n3{r7 r1 r6}{c2 c6 c7} ==> r6c9 <> 3, r1c4 <> 3, ==> r1c3 <> 3
swordfish-in-columns n4{r4 r8 r2}{c1 c5 c9} ==> r8c6 <> 4, r8c3 <> 4, r4c3 <> 4, r2c7 <> 4, ==> r2c6 <> 4
swordfish-in-rows n5{r2 r6 r7}{c7 c1 c5} ==> r9c7 <> 5, r4c5 <> 5, r4c1 <> 5, r1c7 <> 5, r1c1 <> 5
nrc3-chain n5{r9c8 r1c8} - {n5 n8}r2c7 - {n8 n1}r9c7 ==> r9c8 <> 1
nrc3-chain {n1 n8}r9c7 - {n8 n4}r1c7 - n4{r1c6 r9c6} ==> r9c6 <> 1
nrc3-chain n2{r8c8 r7c7} - n3{r7c7 r7c2} - {n3 n6}r8c3 ==> r8c8 <> 6
nrc4-chain n3{r1c2 r1c6} - n4{r1c6 r2c5} - n4{r8c5 r8c1} - n8{r8c1 r9c2} ==> r1c2 <> 8
nrc3-chain n8{r9c2 r2c2} - {n8 n5}r2c7 - n5{r7c7 r9c8} ==> r9c8 <> 8
nrc4-chain n5{r1c3 r2c1} - n5{r2c7 r7c7} - n3{r7c7 r7c2} - {n3 n6}r8c3 ==> r1c3 <> 6
nrc4-chain {n9 n5}r1c3 - n5{r4c3 r4c4} - n3{r4c4 r3c4} - n3{r3c3 r1c2} ==> r1c2 <> 9
nrc4-chain n3{r1c6 r3c4} - n3{r4c4 r4c9} - n4{r4c9 r2c9} - n4{r2c5 r1c6} ==> r1c6 <> 9, r1c6 <> 8, r1c6 <> 7
nrc4-chain {n6 n3}r8c3 - n3{r3c3 r3c4} - {n3 n4}r1c6 - n4{r9c6 r8c5} ==> r8c5 <> 6
nrc3-chain n6{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r2c7 r2c1} ==> r2c1 <> 6
nrc3-chain n6{r2c5 r7c5} - n5{r7c5 r7c7} - {n5 n8}r2c7 ==> r2c5 <> 8
nrc4-chain n2{r8c8 r7c7} - n5{r7c7 r7c5} - n6{r7c5 r2c5} - n4{r2c5 r8c5} ==> r8c5 <> 2
nrc4-chain {n8 n5}r2c7 - n5{r7c7 r7c5} - n6{r7c5 r2c5} - n4{r2c5 r2c9} ==> r2c9 <> 8
nrc4-chain n6{r2c5 r7c5} - {n6 n1}r7c9 - {n1 n8}r9c7 - n8{r9c2 r2c2} ==> r2c2 <> 6
row r2 interaction-with-block b2 ==> r1c4 <> 6
nrc3-chain n6{r1c1 r1c2} - n3{r1c2 r7c2} - {n3 n6}r8c3 ==> r8c1 <> 6, r7c1 <> 6
nrc3-chain n6{r1c1 r6c1} - n5{r6c1 r2c1} - {n5 n9}r1c3 ==> r1c1 <> 9
nrc4-chain n6{r1c1 r6c1} - n5{r6c1 r4c3} - {n5 n9}r1c3 - {n9 n7}r1c4 ==> r1c1 <> 7
nrc4-chain {n7 n9}r1c4 - {n9 n5}r1c3 - n5{r4c3 r4c4} - n3{r4c4 r3c4} ==> r3c4 <> 7
nrc4-chain {n6 n3}r8c3 - n3{r3c3 r3c4} - {n3 n4}r1c6 - n4{r9c6 r9c3} ==> r9c3 <> 6
nrc3-chain n4{r4c1 r5c3} - {n4 n9}r9c3 - {n9 n1}r7c1 ==> r4c1 <> 1
nrc4-chain {n1 n8}r9c7 - {n8 n4}r1c7 - {n4 n3}r1c6 - n3{r6c6 r6c7} ==> r6c7 <> 1
nrc3-chain {n2 n3}r6c7 - n3{r4c9 r4c4} - n5{r4c4 r6c5} ==> r6c5 <> 2
nrc4-chain {n2 n3}r6c7 - n3{r7c7 r8c9} - n3{r8c3 r3c3} - n2{r3c3 r2c2} ==> r6c2 <> 2
nrc4-chain {n2 n3}r6c7 - n3{r6c6 r4c4} - n5{r4c4 r9c4} - n5{r9c8 r7c7} ==> r7c7 <> 2
…(NS+HS)…
635914872
981267534
742385619
419526783
326178495
578493261
197852346
853641927
264739158

Puzzle #3 uses nrct chains (still of length no more than 4)
The second example, #3 in the list, requires various kinds of chains, including nrct-chains of length upto 4.
Interestingly, it also uses 2D chains, i.e. chains in the rc-, rn- and cn- spaces. Remember that these chains are particular case of nrc(z)(t) chains.
I keep have in SudoRules as independent chains because they are easier to spot (in my view of things, more general does not mean better).
Remember that 2D chains are enough to solve 97% of the randomly generated puzzles - although nrc(z)(t) chains sometimes allow using shorter chains (but 3D instead of 2D).
This is a very good example for understanding what is allowed by the t extension.

000010002
001000030
040005600
000007400
300000006
008900000
006300080
050000100
200040000
swordfish-in-columns n3{r9 r1 r6}{c2 c6 c7} ==> r9c9 <> 3, r9c3 <> 3, r6c9 <> 3, r6c5 <> 3, r1c3 <> 3
nrc3-chain n4{r6c6 r6c1} - n4{r5c3 r8c3} - n4{r8c8 r1c8} ==> r1c6 <> 4
nrc4-chain n3{r1c6 r3c5} - n3{r3c3 r8c3} - n4{r8c3 r5c3} - n4{r6c1 r6c6} ==> r6c6 <> 3
hidden singles ==> r4c5 = 3, r1c6 = 3, r3c3 = 3, r9c2 = 3, r8c9 = 3, r6c7 = 3, r8c1 = 8, r2c2 = 2, r1c2 = 8
column c2 interaction-with-block b4 ==> r6c1 <> 6, r4c1 <> 6
row r9 interaction-with-block b8 ==> r7c6 <> 1
hidden-pairs-in-a-block {n1 n8}{r9c4 r9c6} ==> r9c6 <> 9, r9c6 <> 6, r9c4 <> 7, r9c4 <> 6
hidden-single-in-a-row ==> r9c8 = 6
hidden-pairs-in-a-block {n1 n8}{r9c4 r9c6} ==> r9c4 <> 5
hidden-single-in-a-block ==> r7c5 = 5
block b8 interaction-with-row r8 ==> r8c8 <> 7, r8c3 <> 7
nrc2-chain n8{r2c7 r5c7} - n8{r4c9 r4c4} ==> r2c4 <> 8
nrczt2-chain n8{r2c7 r5c7} - n8{r5c5 r3c5} ==> r2c6 <> 8
nrc3-chain {n9 n2}r7c6 - n2{r7c7 r8c8} - n4{r8c8 r7c9} ==> r7c9 <> 9
nrct3-chain n2{r3c4 r3c5} - {n2 n8}r5c5 - n8{r2c5 r3c4} ==> r3c4 <> 7
nrczt3-chain n4{r6c1 r7c1} - {n4 n7}r7c9 - n7{r7c2 r5c2} ==> r6c1 <> 7
nrczt3-chain n5{r4c4 r5c4} - n5{r5c3 r1c3} - n5{r1c8 r6c8} ==> r4c9 <> 5
nrczt3-chain n5{r5c4 r4c4} - n5{r4c3 r1c3} - n5{r1c8 r6c8} ==> r5c7 <> 5
hxyt-cn4-chain {r1 r8}c8n4 - {r8 r5}c3n4 - {r5 r4}c3n2 - {r4 r1}c3n5 ==> r1c8 <> 5
column c8 interaction-with-block b6 ==> r6c9 <> 5
nrc3-chain {n1 n7}r6c9 - {n7 n4}r7c9 - n4{r7c1 r6c1} ==> r6c1 <> 1
nrc4-chain n5{r6c8 r6c1} - n4{r6c1 r7c1} - {n4 n7}r7c9 - {n7 n1}r6c9 ==> r6c8 <> 1
nrct4-chain {n2 n8}r5c5 - n8{r5c7 r4c9} - n8{r3c9 r3c4} - n2{r3c4 r3c5} ==> r8c5 <> 2, r6c5 <> 2
naked-single ==> r6c5 = 6
hidden-single-in-a-row ==> r4c2 = 6
naked-pairs-in-a-row {n1 n7}r6{c2 c9} ==> r6c8 <> 7, r6c6 <> 1
hidden-pairs-in-a-column {n1 n8}{r5 r9}c6 ==> r5c6 <> 4, r5c6 <> 2
swordfish-in-columns n4{r7 r6 r2}{c1 c6 c9} ==> r2c4 <> 4
nrc2-chain n2{r6c8 r6c6} - n2{r7c6 r7c7} ==> r8c8 <> 2
naked and hidden singles ==> r7c7 = 2, r7c6 = 9, r8c5 = 7, r5c2 = 9
column c1 interaction-with-block b1 ==> r1c3 <> 9
xy3-chain {n2 n4}r6c6 - {n4 n5}r6c1 - {n5 n2}r4c3 ==> r4c4 <> 2
xy3-chain {n1 n8}r5c6 - {n8 n7}r5c7 - {n7 n1}r6c9 ==> r5c8 <> 1
row r5 interaction-with-block b5 ==> r4c4 <> 1
xy3-chain {n2 n5}r4c3 - {n5 n8}r4c4 - {n8 n2}r5c5 ==> r5c3 <> 2
hidden-single-in-a-block ==> r4c3 = 2
nrc3-chain {n5 n4}r6c1 - n4{r5c3 r5c4} - n5{r5c4 r4c4} ==> r4c1 <> 5
naked and hidden singles ==> r4c1 = 1, r6c2 = 7, r7c2 = 1, r6c9 = 1, r3c8 = 1
x-wing-in-rows n7{r3 r7}{c1 c9} ==> r9c9 <> 7, r2c9 <> 7, r2c1 <> 7 r1c1 <> 7
nrc3-chain {n5 n9}r9c9 - {n9 n4}r8c8 - n4{r7c9 r2c9} ==> r2c9 <> 5
hidden-single-in-a-column ==> r9c9 = 5
nrc3-chain {n7 n5}r1c3 - n5{r1c7 r2c7} - n7{r2c7 r2c4} ==> r1c4 <> 7
hidden-single-in-a-block ==> r2c4 = 7
nrc3-chain n6{r2c1 r2c6} - n4{r2c6 r6c6} - {n4 n5}r6c1 ==> r2c1 <> 5
…(NS+HS)…
685413792
921786534
743295618
162837459
394521876
578964321
416359287
859672143
237148965

Puzzle #6 uses nrczt chains (still of length no more than 4)
Our third example, #6 in the list, uses nrczt4 chains. After you have understood with the previous example what is allowed by the t extension, it will allow you to understand what is allowed by the z extension.

000010002001000030040005600000007800100000005003400000006300090050000400200080000
swordfish-in-columns n3{r9 r1 r5}{c2 c6 c7} ==> r9c9 <> 3, r5c5 <> 3, r1c1 <> 3
swordfish-in-columns n4{r4 r7 r2}{c1 c5 c9} ==> r7c6 <> 4, r4c8 <> 4, r4c3 <> 4, r2c6 <> 4
swordfish-in-rows n5{r2 r6 r7}{c7 c1 c5} ==> r9c7 <> 5, r4c5 <> 5, r4c1 <> 5, r1c7 <> 5, r1c1 <> 5
nrczt3-chain n1{r4c4 r6c6} - n1{r6c7 r7c7} - n1{r7c2 r9c2} ==> r9c4 <> 1
nrc4-chain n5{r7c7 r7c5} - n5{r6c5 r4c4} - n1{r4c4 r8c4} - {n1 n2}r7c6 ==> r7c7 <> 2
hidden-single-in-a-block ==> r8c8 = 2
column c8 interaction-with-block b3 ==> r3c9 <> 8, r2c9 <> 8
nrc4-chain n3{r1c6 r3c5} - n3{r4c5 r4c9} - n4{r4c9 r2c9} - n4{r2c5 r1c6} ==> r1c6 <> 9, r1c6 <> 8, r1c6 <> 6
nrc4-chain n4{r1c8 r2c9} - n4{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r9c8 r1c8} ==> r1c8 <> 8
hidden singles ==> r3c8 = 8, r3c9 = 1
row r8 interaction-with-block b8 ==> r9c6 <> 1, r7c6 <> 1
naked-single ==> r7c6 = 2
nrct3-chain n3{r5c7 r4c9} - n4{r4c9 r2c9} - n9{r2c9 r6c9} ==> r5c7 <> 9
nrc4-chain n4{r1c8 r2c9} - n4{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r9c8 r1c8} ==> r1c8 <> 7
nrc4-chain n3{r4c9 r5c7} - n3{r5c6 r1c6} - n4{r1c6 r1c8} - n4{r5c8 r4c9} ==> r4c9 <> 9
block b6 interaction-with-row r6 ==> r6c6 <> 9, r6c5 <> 9, r6c2 <> 9, r6c1 <> 9
nrc4-chain n9{r6c9 r2c9} - n4{r2c9 r4c9} - n3{r4c9 r5c7} - n2{r5c7 r6c7} ==> r6c7 <> 9
hidden-single-in-a-block ==> r6c9 = 9
nrc3-chain {n7 n4}r2c9 - {n4 n5}r1c8 - n5{r2c7 r2c1} ==> r2c1 <> 7
nrc4-chain n5{r9c4 r9c8} - {n5 n4}r1c8 - {n4 n7}r2c9 - {n7 n6}r9c9 ==> r9c4 <> 6
nrc4-chain n3{r1c2 r1c6} - n4{r1c6 r1c8} - {n4 n7}r2c9 - {n7 n9}r1c7 ==> r1c2 <> 9
nrc4-chain n3{r4c9 r5c7} - n3{r5c6 r1c6} - n4{r1c6 r1c8} - n4{r5c8 r4c9} ==> r4c9 <> 6
column c9 interaction-with-block b9 ==> r9c8 <> 6
nrc4-chain n4{r7c5 r9c6} - n4{r1c6 r1c8} - n5{r1c8 r9c8} - n5{r9c4 r7c5} ==> r7c5 <> 7
nrct4-chain {n8 n7}r7c9 - {n7 n4}r2c9 - n4{r4c9 r4c1} - {n4 n8}r7c1 ==> r7c2 <> 8
nrczt4-chain {n7 n9}r1c7 - {n9 n5}r2c7 - n5{r1c8 r9c8} - n7{r9c8 r5c8} ==> r6c7 <> 7
nrc3-chain {n2 n1}r6c7 - n1{r4c8 r4c4} - n5{r4c4 r6c5} ==> r6c5 <> 2
nrc4-chain {n6 n5}r6c5 - {n5 n4}r7c5 - n4{r2c5 r1c6} - n3{r1c6 r5c6} ==> r5c6 <> 6
nrczt4-chain {n7 n9}r1c7 - {n9 n5}r2c7 - n5{r1c8 r9c8} - n7{r9c8 r6c8} ==> r5c7 <> 7
block b6 interaction-with-column c8 ==> r9c8 <> 7
nrc3-chain n1{r4c4 r4c8} - {n1 n5}r9c8 - n5{r9c4 r4c4} ==> r4c4 <> 9, r4c4 <> 6
nrc3-chain n1{r4c4 r4c8} - {n1 n5}r9c8 - n5{r9c4 r4c4} ==> r4c4 <> 2
xy3-chain {n6 n5}r6c5 - {n5 n1}r4c4 - {n1 n6}r4c8 ==> r6c8 <> 6, r4c5 <> 6
nrc3-chain {n1 n5}r9c8 - n5{r9c4 r4c4} - n1{r4c4 r4c8} ==> r6c8 <> 1
naked-single ==> r6c8 = 7
nrc3-chain {n2 n3}r5c7 - {n3 n4}r4c9 - n4{r5c8 r5c3} ==> r5c3 <> 2
nrczt3-chain n2{r6c2 r6c7} - n1{r6c7 r4c8} - n6{r4c8 r4c1} ==> r6c2 <> 6
nrczt3-chain {n2 n3}r5c7 - n3{r5c6 r4c5} - n2{r4c5 r4c3} ==> r5c2 <> 2
hxy-rn4-chain {c2 c6}r1n3 - {c6 c8}r1n4 - {c8 c3}r5n4 - {c3 c2}r5n7 ==> r1c2 <> 7
nrc4-chain {n8 n2}r6c2 - n2{r6c7 r5c7} - n3{r5c7 r5c6} - n3{r1c6 r1c2} ==> r1c2 <> 8
nrc4-chain {n6 n3}r1c2 - {n3 n4}r1c6 - n4{r1c8 r5c8} - n6{r5c8 r4c8} ==> r4c2 <> 6
nrc3-chain n6{r4c1 r4c8} - {n6 n4}r5c8 - n4{r5c3 r4c1} ==> r4c1 <> 9
nrc4-chain n6{r4c1 r4c8} - {n6 n4}r5c8 - {n4 n5}r1c8 - n5{r2c7 r2c1} ==> r2c1 <> 6
nrc4-chain {n9 n2}r4c2 - n2{r6c2 r6c7} - {n2 n3}r5c7 - n3{r9c7 r9c2} ==> r9c2 <> 9
nrc4-chain {n9 n2}r4c2 - n2{r6c2 r6c7} - {n2 n3}r5c7 - n3{r5c6 r4c5} ==> r4c5 <> 9
row r4 interaction-with-block b4 ==> r5c3 <> 9, r5c2 <> 9
nrc3-chain {n2 n3}r4c5 - n3{r3c5 r1c6} - n4{r1c6 r2c5} ==> r2c5 <> 2, r2c5 <> 2
nrc4-chain n1{r7c2 r7c7} - {n1 n2}r6c7 - {n2 n3}r5c7 - n3{r9c7 r9c2} ==> r9c2 <> 1
hidden-single-in-a-block ==> r7c2 = 1
naked-triplets-in-a-column {n7 n9 n5}{r1 r2 r7}c7 ==> r9c7 <> 7
nrc3-chain n3{r8c9 r9c7} - {n3 n7}r9c2 - {n7 n6}r9c9 ==> r8c9 <> 6
hidden-single-in-a-block ==> r9c9 = 6
nrczt3-chain n3{r8c9 r9c7} - {n3 n7}r9c2 - n7{r7c1 r7c7} ==> r8c9 <> 7
block b9 interaction-with-row r7 ==> r7c1 <> 7
nrc3-chain {n8 n4}r7c1 - {n4 n5}r7c5 - n5{r6c5 r6c1} ==> r6c1 <> 8
naked-pairs-in-a-row {n5 n6}r6{c1 c5} ==> r6c6 <> 6
hxyzt4-rn-chain-type-1 {c3 c2}r5n7 - {c2 c4}r9n7 - {c4 c8}r9n5 - {c8 c3}r1n5 ==> r1c3 <> 7
nrc4-chain n7{r2c9 r7c9} - n8{r7c9 r8c9} - n3{r8c9 r8c1} - {n3 n7}r9c2 ==> r2c2 <> 7
nrc4-chain {n6 n5}r6c1 - n5{r6c5 r4c4} - n1{r4c4 r4c8} - n6{r4c8 r4c1} ==> r5c2 <> 6
column c2 interaction-with-block b1 ==> r1c1 <> 6
nrc3-chain n6{r1c4 r1c2} - n3{r1c2 r1c6} - n4{r1c6 r2c5} ==> r2c5 <> 6
xyzt4-chain {n7 n4}r2c9 - {n4 n5}r1c8 - {n5 n9}r2c7 - {n9 n7}r2c5 ==> r2c4 <> 7
nrc4-chain n6{r1c4 r1c2} - n3{r1c2 r1c6} - n4{r1c6 r1c8} - {n4 n6}r5c8 ==> r5c4 <> 6
block b5 interaction-with-column c5 ==> r8c5 <> 6
hidden-pairs-in-a-block {n1 n6}{r8c4 r8c6} ==> r8c6 <> 9, r8c4 <> 9, r8c4 <> 7
nrc3-chain n7{r8c5 r9c4} - n5{r9c4 r7c5} - n4{r7c5 r2c5} ==> r2c5 <> 7
row r2 interaction-with-block b3 ==> r1c7 <> 7
naked-single ==> r1c7 = 9
nrc2-chain n7{r8c5 r3c5} - n7{r1c4 r1c1} ==> r8c1 <> 7
column c1 interaction-with-block b1 ==> r3c3 <> 7
nrc3-chain {n8 n5}r1c3 - {n5 n4}r1c8 - n4{r5c8 r5c3} ==> r5c3 <> 8
block b4 interaction-with-column c2 ==> r2c2 <> 8
nrc3-chain n7{r1c1 r1c4} - n6{r1c4 r1c2} - n3{r1c2 r3c1} ==> r3c1 <> 7
hidden-single-in-a-block ==> r1c1 = 7
hidden-pairs-in-a-block {n5 n8}{r1c3 r2c1} ==> r2c1 <> 9
hidden-pairs-in-a-column {n3 n9}{r3 r8}c1 ==> r8c1 <> 8
xy3-chain {n7 n9}r8c5 - {n9 n3}r8c1 - {n3 n7}r9c2 ==> r9c4 <> 7
hidden singles ==> r8c5 = 7, r3c4 = 7
row r8 interaction-with-block b7 ==> r9c3 <> 9
xy3-chain {n9 n4}r2c5 - {n4 n5}r7c5 - {n5 n9}r9c4 ==> r2c4 <> 9
nrc3-chain {n8 n6}r1c4 - n6{r1c2 r2c2} - n2{r2c2 r2c4} ==> r2c4 <> 8
xy4-chain {n8 n7}r5c2 - {n7 n3}r9c2 - {n3 n6}r1c2 - {n6 n8}r1c4 ==> r5c4 <> 8
…(NS+HS)…
765813942
821649537
349725681
692537814
174298365
583461279
416352798
958176423
237984156
denis_berthier
2010 Supporter

Posts: 1259
Joined: 19 June 2007
Location: Paris

denis_berthier wrote:re'born
This is not the way I work.
When I say I have a solution, and especially if I claim that it is shorter than another one, it means I have a solution using well defined rules that I have tested on thousands of cases, not imaginary rules.
Different ethics...

denis_berthier wrote:Your post #1539 clearly shows that you hadn't understood the z and t extensions of nrc-chains:
ronk wrote:your n9{r2c7 r3c7} bilocation link seems to ignore the existence of n9 in r4c7 and r5c7. How does that work

But in the same post, you claim having a shorter solution with an extension of them that your re'born avatar later calls grouped nrc-chains, in a tentative definition.
That's where our ethics differ (but with yours you''ll probably not understand this).

Wow Denis, are you sure this is the manner and tone you like to communicate with your fellow members here, who has spent much time and energy reading through your extra-long posts and replying? Or are you more comfortable having nobody else responding and just you talking to yourself in your own thread?

If you disagree with the ideas from others, that is perfectly fine. Just write out your arguments with logical reasoning. But I don't see the necessity in commenting about the "ethics" of others (gee that's a pretty strong word) or accusing others of using an alternative avatar/identity to gang up on you without any convincing evidence at all.

(As a matter of fact, you can feel free to imagine me, udosuk, as another "avatar" of ronk and re'born to gang up on you, if that makes you feel better, and if this is the way of your thinking you want others to see in this forum.)

As I always said, this forum is established for it's logical and intelligent discussions, but personal remarks and accusations against fellow members are both unnecessary and inappropriate here. If you thrive for personal flame wars, I'm sure the Eureka! forum is the place for you. But please don't try to bring their culture and atmosphere over here, which is always a calm and peaceful place.
udosuk

Posts: 2698
Joined: 17 July 2005

### Re: moderate level examples of nrc(z)(t) chains

denis_berthier wrote:Puzzle #1uses only nrc chains (of length no more than 4).
Our first example, which is also #1 in the list uses only nrc-chains.
Remember that nrc-chains are merely another view of "basic" AICs, i.e. AICs with no subsets of any kind (groups, hinges, …).
The chains used in this example should therefore be familiar.
000010002
001000030
040005600
000006700
300000005
008400000
007800040
050000900
200030000
swordfish-in-columns n3{r7 r1 r6}{c2 c6 c7} ==> r7c9 <> 3, n3{r7 r1 r6}{c2 c6 c7} ==> r6c9 <> 3, r1c4 <> 3, ==> r1c3 <> 3
swordfish-in-columns n4{r4 r8 r2}{c1 c5 c9} ==> r8c6 <> 4, r8c3 <> 4, r4c3 <> 4, r2c7 <> 4, ==> r2c6 <> 4
swordfish-in-rows n5{r2 r6 r7}{c7 c1 c5} ==> r9c7 <> 5, r4c5 <> 5, r4c1 <> 5, r1c7 <> 5, r1c1 <> 5
nrc3-chain n5{r9c8 r1c8} - {n5 n8}r2c7 - {n8 n1}r9c7 ==> r9c8 <> 1
nrc3-chain {n1 n8}r9c7 - {n8 n4}r1c7 - n4{r1c6 r9c6} ==> r9c6 <> 1
nrc3-chain n2{r8c8 r7c7} - n3{r7c7 r7c2} - {n3 n6}r8c3 ==> r8c8 <> 6
nrc4-chain n3{r1c2 r1c6} - n4{r1c6 r2c5} - n4{r8c5 r8c1} - n8{r8c1 r9c2} ==> r1c2 <> 8
nrc3-chain n8{r9c2 r2c2} - {n8 n5}r2c7 - n5{r7c7 r9c8} ==> r9c8 <> 8
nrc4-chain n5{r1c3 r2c1} - n5{r2c7 r7c7} - n3{r7c7 r7c2} - {n3 n6}r8c3 ==> r1c3 <> 6
nrc4-chain {n9 n5}r1c3 - n5{r4c3 r4c4} - n3{r4c4 r3c4} - n3{r3c3 r1c2} ==> r1c2 <> 9
nrc4-chain n3{r1c6 r3c4} - n3{r4c4 r4c9} - n4{r4c9 r2c9} - n4{r2c5 r1c6} ==> r1c6 <> 9, r1c6 <> 8, r1c6 <> 7
nrc4-chain {n6 n3}r8c3 - n3{r3c3 r3c4} - {n3 n4}r1c6 - n4{r9c6 r8c5} ==> r8c5 <> 6
nrc3-chain n6{r2c5 r7c5} - n5{r7c5 r7c7} - n5{r2c7 r2c1} ==> r2c1 <> 6
nrc3-chain n6{r2c5 r7c5} - n5{r7c5 r7c7} - {n5 n8}r2c7 ==> r2c5 <> 8
nrc4-chain n2{r8c8 r7c7} - n5{r7c7 r7c5} - n6{r7c5 r2c5} - n4{r2c5 r8c5} ==> r8c5 <> 2
nrc4-chain {n8 n5}r2c7 - n5{r7c7 r7c5} - n6{r7c5 r2c5} - n4{r2c5 r2c9} ==> r2c9 <> 8
nrc4-chain n6{r2c5 r7c5} - {n6 n1}r7c9 - {n1 n8}r9c7 - n8{r9c2 r2c2} ==> r2c2 <> 6
row r2 interaction-with-block b2 ==> r1c4 <> 6
nrc3-chain n6{r1c1 r1c2} - n3{r1c2 r7c2} - {n3 n6}r8c3 ==> r8c1 <> 6, r7c1 <> 6
nrc3-chain n6{r1c1 r6c1} - n5{r6c1 r2c1} - {n5 n9}r1c3 ==> r1c1 <> 9
nrc4-chain n6{r1c1 r6c1} - n5{r6c1 r4c3} - {n5 n9}r1c3 - {n9 n7}r1c4 ==> r1c1 <> 7
nrc4-chain {n7 n9}r1c4 - {n9 n5}r1c3 - n5{r4c3 r4c4} - n3{r4c4 r3c4} ==> r3c4 <> 7
nrc4-chain {n6 n3}r8c3 - n3{r3c3 r3c4} - {n3 n4}r1c6 - n4{r9c6 r9c3} ==> r9c3 <> 6
nrc3-chain n4{r4c1 r5c3} - {n4 n9}r9c3 - {n9 n1}r7c1 ==> r4c1 <> 1
nrc4-chain {n1 n8}r9c7 - {n8 n4}r1c7 - {n4 n3}r1c6 - n3{r6c6 r6c7} ==> r6c7 <> 1
nrc3-chain {n2 n3}r6c7 - n3{r4c9 r4c4} - n5{r4c4 r6c5} ==> r6c5 <> 2
nrc4-chain {n2 n3}r6c7 - n3{r7c7 r8c9} - n3{r8c3 r3c3} - n2{r3c3 r2c2} ==> r6c2 <> 2
nrc4-chain {n2 n3}r6c7 - n3{r6c6 r4c4} - n5{r4c4 r9c4} - n5{r9c8 r7c7} ==> r7c7 <> 2
…(NS+HS)…
635914872
981267534
742385619
419526783
326178495
578493261
197852346
853641927
264739158

Alternatively, after the 3 swordfish there are two nice loops:
[r8c5]-6-[r8c3]-3-[r8c9]=3=[r4c9]=4=[r2c9]-4-[r2c5]=4=[r8c5], =>r8c5<>6
and
[r7c7]=3=[r6c7]-3-[r6c6]=3=[r4c4]=5=[r6c5]-5-[r7c5]=5=[r7c7], =>r7c7<>1,2 (as well as r4c4<>1,2,9).
From here it is a naked quad and singles to solve the puzzle.
re'born

Posts: 551
Joined: 31 May 2007

udosuk wrote:
denis_berthier wrote:re'born
This is not the way I work.
When I say I have a solution, and especially if I claim that it is shorter than another one, it means I have a solution using well defined rules that I have tested on thousands of cases, not imaginary rules.
Different ethics...

denis_berthier wrote:Your post #1539 clearly shows that you hadn't understood the z and t extensions of nrc-chains:
ronk wrote:your n9{r2c7 r3c7} bilocation link seems to ignore the existence of n9 in r4c7 and r5c7. How does that work

But in the same post, you claim having a shorter solution with an extension of them that your re'born avatar later calls grouped nrc-chains, in a tentative definition.
That's where our ethics differ (but with yours you''ll probably not understand this).

Wow Denis, are you sure this is the manner and tone you like to communicate with your fellow members here, who has spent much time and energy reading through your extra-long posts and replying? Or are you more comfortable having nobody else responding and just you talking to yourself in your own thread?

If you disagree with the ideas from others, that is perfectly fine. Just write out your arguments with logical reasoning. But I don't see the necessity in commenting about the "ethics" of others (gee that's a pretty strong word) or accusing others of using an alternative avatar/identity to gang up on you without any convincing evidence at all.

(As a matter of fact, you can feel free to imagine me, udosuk, as another "avatar" of ronk and re'born to gang up on you, if that makes you feel better, and if this is the way of your thinking you want others to see in this forum.)

As I always said, this forum is established for it's logical and intelligent discussions, but personal remarks and accusations against fellow members are both unnecessary and inappropriate here. If you thrive for personal flame wars, I'm sure the Eureka! forum is the place for you. But please don't try to bring their culture and atmosphere over here, which is always a calm and peaceful place.

Well, udosuk, if my posts are "extra-long", it might be the case that there is some substance in them.

If you think there is no problem is stating that we have a solution when the rule we claim to use in it isn't even defined, that's a strange conception of "logical reasoning".

As "flame wars" are not my cup of tea, I won't comment on your sayings about the Eureka forum - I know it too well.

I leave to participants in this forum the freedom to judge by themselves whether my rules and examples are interesting or not.
denis_berthier
2010 Supporter

Posts: 1259
Joined: 19 June 2007
Location: Paris

Hello Denis,

You invited participants of that forum to express views.

I'll do it, although as newcomer I could stay as viewver.

Several reasons to that :

- Coming from the same country (which can explain my poor wording) we should be in a better position to open the dicussion,
- I am following thru a French forum (in which I am pure viewer) a group of guys experiencing your method,
- I am not supposed to be part of any gang.
- I have a method widely disclosed in French forums, not only by me,
- I have specific expectations coming here which I expressed in my first post.

Your way to enter grid solving is for sure very interesting. My feeling is that it's a complementary way to speed up solving.

I compared in details our two paths to solve your first example. Nearly from the beginnng, we follow different tracks, which means we are breaking different logic weaknesses of the grid.

This is not enough to add your logic in my computer. To do so, I need either a strong feeling that the new method is easier to handle or a true breakthrough with examples of grids solved on your side and not by my program.

I proposed earlier in this thread four grids not too difficult in gsf's ranking , I didn't get any answer.

Anyway, I was amazed when I saw discussion switching to insults. To stay purely in the logic, this a couterproductive way to go ahead.

To end positively, I got to see yesterday a grid which seems to be the "worst case" for your solving procedure.

Here is the grid (not an extreme one).

1= 000080904
2= 001007000
3= 000000006
4= 003200050
5= 040060000
6= 000000020
7= 090000008
8= 000105000
9= 007803000

It receives a quick and nice solution in my program (Alternate chains exclusively) using ALS and multi-chains.

As it has been proposed on the forum where a group is working on xyzt chains, I'll see if they have a solution.

kind regards

Champagne
champagne
2017 Supporter

Posts: 6547
Joined: 02 August 2007
Location: France Brittany

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