The Hidden Logic of Sudoku


Denis Berthier



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Online Supplements


Whips




Denis Berthier. All the material in this page and the pages it gives access to are the property of the author and may not be re-published or re-posted without his prior written permission.


1) DEFINITION OF WHIPS


Whips are mainly another view of chains and lassos, unifying them into a single concept.

Contrary to chains that must be closed on the target or to lassos that must be closed on a previous left- or right- linking candidate, whips remain open ended. Whence the name I chose for them.


Definition: an xyzt-whip of length n is a partial xy-chain of length n-1 plus a nth left-linking candidate (call it nrc) such that at least one of the following conditions is true:
- every candidate in final rc-cell rc is nrc-linked to a previous right-linking candidate [or, for whips containing the z-extension: to the pre-selected target]
- every candidate in final rn-cell rn is nrc-linked to a previous right-linking candidate [or, for whips containing the z-extension: to the pre-selected target]
- every candidate in final cn-cell cn is nrc-linked to a previous right-linking candidate [or, for whips containing the z-extension: to the pre-selected target]
- every candidate in final bn-cell bn is nrc-linked to a previous right-linking candidate [or, for whips containing the z-extension: to the pre-selected target], where b is the block of cell rc.


Theorem: given an xy-whip [resp. an xyt- xyz- xyzt- whip], any candidate z that is nrc-linked to the first left-linking candidate [resp. for whips containing the z-extension: z = the pre-selected target z] can be eliminated.

Proof:
first recall the partial xy-chain [resp. xyt- xyz- xyzt-] theorem: given a potential target, if it was true, then any left-linking candidate would be false and any right-linking candidate would be true.

In the given situation, we have a left-linking candidate nrc which leaves no possibility for having a candidate true in cell rc or no possibility for having a candidate true in rn-cell rn or no possibility for having a candidate true in cn-cell cn or no possibility for having a candidate true in bn-cell bn, where b is the block of rc-cell rc.
Conclusion: either z is false or the puzzle has no solution.
But if the puzzle has no solution, eliminating z from its candidates won't make it have fewer solutions.
We can therefore always conclude that z can be eliminated.


This theorem can be extended in an obvious way to the hidden counterparts of the above 2D chains [hxy, hxyt, hxyz, hxyzt] and to the fully supersymmetric chains [nrc, nrct, nrcz, nrczt], using the corresponding partial chain theorem. Let's do it for the nrczt 3D version.



But the most interesting whips are the 3D whips.

Definition: given a set S of candidates, it is convenient to say that a candidate C is compatible with S if C is nrc-linked to no element in S.

Definition: given a candidate z (the target of the whip), an nrczt-whip built on z is any sequence L1 R1 L2 R2 L3 R3 .... Ln (notice that there is no Rn) of candidates, each nrc-linked to the previous one, alternatively called left-linking and right-linking, such that:
- none of the Lk or Rk is equal to z;
- L1 is nrc-linked to the target z;
- for each 1 < k <= n: Lk is nrc-linked to Rk-1; this is the first part of the whip condition;
- for each 1 <= k < n, there is at least one of the four rc-, rn-, cn- or bn- cells containing Lk and Rk and such that, in this cell, Rk is the only candidate compatible with the target and the previous right-linking candidates; this is the second part of the whip condition;
- there is an rc-, rn-, cn- or bn- cell containing Ln such that there is no candidate in this cell compatible with the target and the previous right-linking candidates.


Theorem: given an nrczt-whip built on a target z, its target can be eliminated.


Remarks:

1) As was the case for the 2D or 3D chains and lassos, the additional t- or z- candidates are not considered part of the whips.
This is not an arbitrary convention; it has practical consequences: if I notice a whip but I don't use it immediately (e.g. because I've seen a shorter one), then, later, some z- or t- candidates may have disappeared; if the left-linking and right-linking ones haven't changed, this will still be the same whip. This is extensively used in SudoRules.

2) inner loops: as for nrczt-chains, inner loops are not excluded a priori by this definition. Inner loops can easily be proven to be useless (in the sense that they don't lead to more eliminations) for some types of whips (those that don't use the t- extension, i.e. in 2D: xy, hxy, xyz, hxyz; in 3D: nrc-, nrcz-).
It would therefore seem to be an arbitrary restriction to exclude inner loops a priori from whips with the t-extension. But, in practice, inner loops are useless. Moreover, nrczt-whips in which inner loops would be allowed are subsumed by nrczt-braids in which allowing loops is obviously a priori useless.
Let us therefore say that 3D (nrc, nrcz, nrct, nrczt) chains or whips that satisfy the no-loop condition are "standard". In all these web pages, only standard chains, lassos or whips will be considered.
To be explicit, the no-loop condition is expressed as: all the candidates in the chain are different.



2) NRCZT- CHAINS, LASSOS AND WHIPS



As many of Sudoku players, I've started with the idea that the two ends of a chain must be linked to any of its targets.
After introducing several new types of chains based on this idea, I've discovereed lr-lassos, whose tail is linked to a previous right-linking candidate instead of the target (and rl-lassos, which can be considered as a special case of lr-lassos).
Trying to see if other kinds of contradictions can be obtained from a partial chain, I found whips, which are open ended (a left-linking candidate in a 2D-cell with no possibility for a right-linking candidate in this cell, compatible with both the target and the previous right-linking candidates).

Let us show that the (non disjoint) union of chains and lassos is equivalent to whips.

Theorem: the set of nrczt-whips of length n contains the sets of nrczt chains and lassos of length n.

Abbreviations used: rlc (llc) rigth- (left-) linking candidate.
Proof:
It is obvious that chains and lassos can be considered as whips:
- for chains, the last rlc is nrc-linked to the target; it is therefore an impossible value in the last rc-, rn-, cn- or bn- cell of the partial chain;
- for lr-lassos, the last rlc is nrc-linked to a previous rlc; it is therefore an impossible value in the last rc-, rn-, cn- or bn- cell of the partial chain;
- for rl-lassos, the last rlc is equal to a previous llc (which is nrc-linked to an rlc: the next one); it is therefore an impossible value in the last rc-, rn-, cn- or bn- cell of the partial chain.
In each case, the last cell has no possible rlc non linked to a previous rlc or to the target.



The distinction between chains and lassos is based on whether there is a z-candidate in the last cell. But in the nrczt context, this may be considered irrelevant.
What becomes more interesting is whether the last cell (with no remaining possibility for a right-linking candidate) is an rc-, rn-, cn- or bn- cell. Depending on this 2D space, we get nrczt-whip-rc, nrczt-whip-rn, nrczt-whip-cn, nrczt-whip-bn.
 


3) THE SCOPE OF NRCZT-WHIPS (first concrete results)


sudogen0_1M is a collection of 1,000,000 random minimal puzzles (generated with the suexg program). See the ratings page for details.

All these puzzles can be solved using only nrczt-whips of maximal length 13.
(This is confirmed on the classification page: the random puzzles generated by random generators - and about 10 millions were thus generated - can be solved by whips).

I'm very surprised by this result - and I can't explain it:
- there is nothing in the suexg generator that could bias it in favour of puzzles solvable by nrczt-whips (the solver it uses to check the intermediate grids is based on a set cover algorithm, very different from the nrczt resolution rules).
- I've also checked that all the puzzles in the collection are different and uncorrelated (See the ratings page for details.).
All this tends to prove that the sudogen0_1M collection is indeed a random sample of the set of all minimal puzzles and that whips are a very powerful tool.


These results also confirmed the strong correlation between the nrczt-rating and the SER : 0.895. (More on correlation between various aspects of a puzzle soon in the ratings page.) Although I've already mentioned this correlation result long ago, it remains very surprising, because these two ratings are based on very different sets of rules.

Here are now the results about the number of puzzles solved at each level of the NRCZT-hierarchy:

Level    Number     Total
1_0      417,624      417,624
1          120,618      538,242
2          138,371      676,613
3          168,355      844,968
4          123,153      968,121
5          24,187        992,308
6          5,511          997,819
7          1,514          999,333
8          473            999,806
9          130            999,936
10         38             999,974
11         15             999,989
12         9              999,998
13         2              1,000,000
14         0    

They confirm on a very large sample that:
- all the puzzles in th collzction can be solved with whips;
- more than 99% of the minimal puzzles can be solved with whips of length 5 or less
;
- more than 99.9% of the minimal puzzles can be solved with whips of length 7 or less;
- "almost all" the puzzles can be solved with nrczt-whips of length 13 or less (exceptions are less than one in a million);
- using more complex patterns, such as nrczt-braids, is not necessary for this collection.

For a better estimate of the NRCZT complexity distribution, see the classification page.




4) WHIP EXAMPLES


Remark:
The examples in this section use an improved version of the strict nrc notation.
It allows a very uniform presentation of all the 2D or 3D chains/whips/braids.
And it stresses more than ever the complementarity between the two possible views of chains: sequences of cells vs sequences of candidates.

As usual, each pattern is prefixed by its name and length.
Chains/whips/braids are still displayed as sequences A1 - A2 - A3 -
where each Ai is a pair of bivalue/bilocal candidates (modulo z and/or t)

Each Ai is written systematically in the form 2D-cell{Li Ri}, where:

- "2D-cell" is some explicitly named rc-, rn-, cn- or bn- cell; this cell is bivalue (or bivalue modulo the target and/or the previous right-linking candidates);

- {Li Ri} is the couple of values in the previous rc, rn, cn or bn cell; additional z or t candidates can be added in these cells, in an unchanged manner, with the # and * signs to justify them;

- if 2D-space = bn, then the block now appears explicitly, but Li and Ri are still written in the "rc style": r1c1, r2c2 ; putting instead a coordinate internal to the block (such as the relative position in the block) would have been simpler but it would make it harder to understand the linking conditions with the previous and next candidates;

- for whips or braids, the final inexistent a2 is still represented by a dot; as the final 2D-space, in which the contradiction occurs, appears at the end, it becomes useless do have it in the pattern name.

Notice that for 2D chains (xy, hxy,...), this slightly modified convention reverses the old order of each element, which was {Li Ri}2D-cell.



Notice also that the general shape of the resolution path gives a rough idea of how the difficulty varies with time.



4.1) First whip example

First easy example: Sudogen0 #23

016
070
009
005
409
800
000
000
004
040
000
020
000
540
060
029
000
108
001
000
000
000
000
080
030
605
000



***** SudoRules version 13.7wter *****
016005000
070409000
009800004
040000029
000540000
020060108
001000030
000000605
000080000

hidden-singles ==> r1c1 = 4, r4c1 = 6, r5c1 = 1, r5c2 = 9, r6c4 = 9, r5c6 = 2, r4c6 = 8, r5c3 = 8, r2c1 = 8, r6c8 = 4, r4c7 = 5, r7c5 = 5, r8c5 = 9, r3c6 = 6
interaction column c5 with block b2 for number 2 ==> r1c4 <> 2
interaction column c6 with block b8 for number 1 ==> r9c4 <> 1, r8c4 <> 1
hidden-single-in-a-column ==> r4c4 = 1
hidden-pairs-in-a-row r1{n8 n9}{c7 c8} ==> r1c8 <> 7, r1c7 <> 7, r1c7 <> 3, r1c7 <> 2

At this point, the PM is:


+----------------------------+-------------------------+--------------------------+
| 4          1       6             | 37      237    5          | 89         89      237    |
| 8          7       235         | 4       123     9          | 23        156     1236  |
| 235      35      9            | 8       1237    6         | 237      157     4        |
+----------------------------+-------------------------+--------------------------+
| 6           4       37          | 1        37      8          | 5          2         9        |
| 1           9       8            | 5        4        2          | 37        67       367    |
| 357       2       357        | 9        6        37        | 1          4         8        |
+----------------------------+-------------------------+--------------------------+
| 279       68      1           | 267     5       47        | 24789  3         27      |
| 237       38      2347     | 237     9       1347    | 6         178      5        |
| 23579   356    23457   | 2367   8       1347    | 2479   179     127     |
+----------------------------+-------------------------+--------------------------+


and we get our first whip:
nrczt-whip[2]  r4n3{c5 c3} - b1n3{r2c3 .} ==> r3c5 <> 3 (the dot in the last rn-cell indicates the impossibility of having any right-linking candidate)

naked-triplets-in-a-column c7{r2 r3 r5}{n3 n2 n7}  ==> r9c7 <> 7, r9c7 <> 2, r7c7 <> 7, r7c7 <> 2
interaction column c7 with block b3 for number 2 ==> r2c9 <> 2, r1c9 <> 2
hidden-single-in-row r1 ==> r1c5 = 2
naked-triplets-in-a-block b3{r1c9 r2c7 r3c7}{n7 n3 n2} ==> r3c8 <> 7, r2c9 <> 3
nrc-chain[3]  r7c9{n2 n7} - b3n7{r1c9 r3c7} - r3n2{c7 c1} ==> r7c1 <> 2
nrc-chain[3]  r8n1{c6 c8} - b9n8{r8c8 r7c7} - r7n4{c7 c6} ==> r8c6 <> 4 
hidden-single-in-row r8 ==> r8c3 = 4
nrc-chain[4]  b8n1{r8c6 r9c6} - c9n1{r9 r2} - r2c5{n1 n3} - b5n3{r4c5 r6c6} ==> r8c6 <> 3
nrc-chain[2]  c6n3{r9 r6} - r4n3{c5 c3} ==> r9c3 <> 3
nrc-chain[4]  b3n7{r1c9 r3c7} - r3n2{c7 c1} - c3n2{r2 r9} - b9n2{r9c9 r7c9} ==> r7c9 <> 7
naked-single ==> r7c9 = 2
nrct-chain[3]  r9n6{c2 c4} - b8n2{r9c4 r8c4} - b8n3{r8c4 r9c6} ==> r9c2 <> 3
xy-chain[4]  r1c4{n3 n7} - r7c4{n7 n6} - r7c2{n6 n8} - r8c2{n8 n3} ==> r8c4 <> 3
interaction row r8 with block b7 for number 3 ==> r9c1 <> 3
nrc-chain[4]  r8n8{c8 c2} - r7c2{n8 n6} - r7c4{n6 n7} - r8c6{n7 n1} ==> r8c8 <> 1
hidden-single-in-row r8 ==> r8c6 = 1
nrc-chain[4]  c3n2{r9 r2} - r3n2{c1 c7} - r3n7{c7 c5} - r4n7{c5 c3} ==> r9c3 <> 7
interaction column c3 with block b4 for number 7 ==> r6c1 <> 7
nrc-chain[3]  r6c1{n5 n3} - b7n3{r8c1 r8c2} - r3c2{n3 n5} ==> r3c1 <> 5
nrczt-whip[4]  r9c9{n7 n1} - r9c8{n1 n9} - c1n9{r9 r7} - r7n7{c1 .} ==> r9c4 <> 7
nrczt-whip[4]  r9c9{n7 n1} - r9c8{n1 n9} - c1n9{r9 r7} - r7n7{c1 .} ==> r9c6 <> 7
xy-chain[5]  r3c8{n1 n5} - r3c2{n5 n3} - r8c2{n3 n8} - r8c8{n8 n7} - r9c9{n7 n1} ==> r9c8 <> 1
hidden-single-in-block b9 ==> r9c9 = 1
naked-single ==> r2c9 = 6
hidden-single-in-block b6 ==> r5c8 = 6
nrc-chain[5]  r9n6{c2 c4} - c4n3{r9 r1} - r2c5{n3 n1} - r2c8{n1 n5} - r3n5{c8 c2} ==> r9c2 <> 5
naked-singles
GRID 23 SOLVED. LEVEL = L5, MOST COMPLEX RULE = NRC5
416325897
872419356
359876214
647138529
198542763
523967148
781654932
234791685
965283471

Consider this only as a first illustrative example. This puzzle could be solved without whipping it, with the same most complex nrc-chain[5] rule.



4.2) Second whip example


I've chosen this example because it has whips in which the final contradiction occurs in the four types of cells: rc, rn, cn and bn.

This example is grid "282 hard" from the (late) Sudoku-factory forum.

100
000
040
000
000
506
200
304
000
070
008
090
008
040
200
040
100
060
000
507
003
107
000
000
080
000
009


***** SudoRules version 13.7wter *****
1.....2........3.4.4.5.6....7...8.4...8.4.1...9.2...6....1.7.8.5.7........3.....9
hidden-single-in-a-block ==> r1c9 = 6
interaction row r8 with block b8 for number 9 ==> r7c5 <> 9
interaction block b3 with row r3 for number 8 ==> r3c5 <> 8, r3c1 <> 8, r9c8 <> 5, r5c8 <> 5
nrc-chain[3]  c8n3{r8 r5} - c2n3{r5 r1} - r3n3{c1 c5} ==> r8c5 <> 3
nrc-chain[3]  r7n3{c9 c5} - r3n3{c5 c1} - c2n3{r1 r5} ==> r5c9 <> 3
nrc-chain[3]  r7n3{c5 c9} - c8n3{r8 r5} - c2n3{r5 r1} ==> r1c5 <> 3
nrc-chain[3]  c8n3{r5 r8} - r7n3{c9 c5} - r3n3{c5 c1} ==> r5c1 <> 3
nrct-chain[3]  c1n7{r2 r3} - r3n3{c1 c5} - r3n2{c5 c3} ==> r2c1 <> 2 

At this point, the PM is:

  +----------------------+--------------------------+----------------------+       
  | 1        358   59      | 34789  789     349    | 2     579      6       |       
  | 6789  2568 2569  | 789      12789 129    | 3     1579    4       |       
  | 2379  4       29      | 5          12379 6        | 789  179     178   |       
  +----------------------+--------------------------+----------------------+       
  | 236   7        1256  | 369      13569 8        | 59     4        235   |       
  | 26     2356  8        | 3679    4         359    | 1       2379  257   |       
  | 34     9        145    | 2         1357    135    | 578   6       3578  |       
  +----------------------+--------------------------+----------------------+       
  | 2469 26      2469  | 1         2356   7        | 456    8       235    |       
  | 5       1268  7        | 34689 2689   2349  | 46      123   123    |       
  | 2468 1268  3        | 468     2568   245    | 4567  127   9        |       
  +----------------------+--------------------------+----------------------+

and we get our first whip
nrczt-whip[3]  r3n3{c5 c1} - r6n3{c1 c9} - r7n3{c9 .} ==> r4c5 <> 3 
in full nrc-notation:
nrczt-whip[3]  r3n3{c5 c1} - r6n3{c1 c9 c5* c6*} - r7n3{c9 .} ==> r4c5 <> 3  
This is a very special whip, making no use of t-candidates. We could name it an nrcz-whip (nrcz-whips are not programmed independently in SudoRules).  

At this point, the PM is:

  +----------------------+--------------------------+----------------------+       
  | 1        358   59      | 34789  789     349    | 2     579      6       |       
  | 6789  2568 2569  | 789      12789 129    | 3     1579    4       |       
  | 2379  4       29      | 5          12379 6        | 789  179     178   |       
  +----------------------+--------------------------+----------------------+       
  | 236   7        1256  | 369      1569   8        | 59     4        235   |       
  | 26     2356  8        | 3679    4         359    | 1       2379  257   |       
  | 34     9        145    | 2         1357    135    | 578   6       3578  |       
  +----------------------+--------------------------+----------------------+       
  | 2469 26      2469  | 1         2356   7        | 456    8       235    |       
  | 5       1268  7        | 34689 2689   2349  | 46      123   123    |       
  | 2468 1268  3        | 468     2568   245    | 4567  127   9        |       
  +----------------------+--------------------------+----------------------+

and we get our second whip, still an nrcz-whip:
nrczt-whip[3]  c2n3{r5 r1} - c4n3{r1 r8} - c8n3{r8 .} ==> r5c6 <> 3
in full nrc-notation:
nrczt-whip[3]  c2n3{r5 r1} - c4n3{r1 r8 r4*} - c8n3{r8 . r5*} ==> r5c6 <> 3

nrct-chain[3]  r1c3{n9 n5} - b4n5{r6c3 r5c2} - r5c6{n5 n9} ==> r1c6 <> 9

At this point, the PM is:

  +----------------------+--------------------------+----------------------+       
  | 1        358   59      | 34789  789     34      | 2     579      6       |       
  | 6789  2568 2569  | 789      12789 129    | 3     1579    4       |       
  | 2379  4       29      | 5          12379 6        | 789  179     178   |       
  +----------------------+--------------------------+----------------------+       
  | 236   7        1256  | 369      1569   8        | 59     4        235   |       
  | 26     2356  8        | 3679    4         59      | 1       2379  257   |       
  | 34     9        145    | 2         1357    135    | 578   6       3578  |       
  +----------------------+--------------------------+----------------------+       
  | 2469 26      2469  | 1         2356   7        | 456    8       235    |       
  | 5       1268  7        | 34689 2689   2349  | 46      123   123    |       
  | 2468 1268  3        | 468     2568   245    | 4567  127   9        |       
  +----------------------+--------------------------+----------------------+

and we get our third whip, still an nrcz-whip:
nrczt-whip[3]  r3n3{c5 c1} - r4n3{c1 c9} - r7n3{c9 .} ==> r6c5 <> 3
in full nrc notation:
nrczt-whip[3]  r3n3{c5 c1} - r4n3{c1 c9 c4*} - r7n3{c9 . c5*} ==> r6c5 <> 3


nrc-chain[4]  r7n3{c9 c5} - r3n3{c5 c1} - c2n3{r1 r5} - c8n3{r5 r8} ==> r8c9 <> 3
nrct-chain[3]  r8c9{n1 n2} - b6n2{r4c9 r5c8} - c8n3{r5 r8} ==> r8c8 <> 1


At this point, the PM is:

  +----------------------+--------------------------+----------------------+       
  | 1        358   59      | 34789  789     34      | 2     579      6       |       
  | 6789  2568 2569  | 789      12789 129    | 3     1579    4       |       
  | 2379  4       29      | 5          12379 6        | 789  179     178   |       
  +----------------------+--------------------------+----------------------+       
  | 236   7        1256  | 369      1569   8        | 59     4        235   |       
  | 26     2356  8        | 3679    4         59      | 1       2379  257   |       
  | 34     9        145    | 2         157      135    | 578   6       3578  |       
  +----------------------+--------------------------+----------------------+       
  | 2469 26      2469  | 1         2356   7        | 456    8       235    |       
  | 5       1268  7        | 34689 2689   2349  | 46      23     12      |       
  | 2468 1268  3        | 468     2568   245    | 4567  127   9        |       
  +----------------------+--------------------------+----------------------+

and we get our fourth whip, a little longer than than the first three, but still an nrcz-whip:
nrczt-whip[4]  c1n7{r2 r3} - r3n3{c1 c5} - b2n2{r3c5 r2c6} - b2n1{r2c6 .} ==> r2c5 <> 7
in full nrc notation:
nrczt-whip[4]  c1n7{r2 r3} - r3n3{c1 c5} - b2n2{r3c5 r2c6 r2c5*} - b2n1{r2c6 . r2c5* r3c5*} ==> r2c5 <> 7


nrct-chain[5]  r3c3{n2 n9} - r1c3{n9 n5} - b4n5{r6c3 r5c2} - c2n3{r5 r1} - r3n3{c1 c5} ==> r3c5 <> 2
interaction row r3 with block b1 for number 2 ==> r2c3 <> 2, r2c2 <> 2

At this point, the PM is:

  +----------------------+--------------------------+----------------------+       
  | 1        358   59      | 34789  89       34      | 2     579      6       |       
  | 6789  568   569    | 789      12789 129    | 3     1579    4       |       
  | 2379  4       29      | 5          1379   6        | 789  179     178   |       
  +----------------------+--------------------------+----------------------+       
  | 236   7        1256  | 369      1569   8        | 59     4        235   |       
  | 26     2356  8        | 3679    4         59      | 1       2379  257   |       
  | 34     9        145    | 2         157      135    | 578   6       3578  |       
  +----------------------+--------------------------+----------------------+       
  | 2469 26      2469  | 1         2356   7        | 456    8       235    |       
  | 5       1268  7        | 34689 2689   2349  | 46      23     12      |       
  | 2468 1268  3        | 468     2568   245    | 4567  127   9        |       
  +----------------------+--------------------------+----------------------+

and we get our fifth whip, our first example with t candidates:
nrczt-whip[4]  c3n4{r7 r6} - c3n1{r6 r4} - b4n5{r4c3 r5c2} - c2n2{r5 .} ==> r7c3 <> 2
in full nrc notation:
nrczt-whip[4]  c3n4{r7 r6} - c3n1{r6 r4} - b4n5{r4c3 r5c2 r6c3#1} - c2n2{r5 . r7* r8* r9*} ==> r7c3 <> 2

The sequel is similar, although with longer whips.

nrczt-whip[4]  r7n9{c1 c3} - r1c3{n9 n5} - b4n5{r6c3 r5c2} - c2n2{r5 .} ==> r7c1 <> 2
nrct-chain[5]  r9n1{c2 c8} - r8c9{n1 n2} - r8c8{n2 n3} - b8n3{r8c6 r7c5} - r7n2{c5 c2} ==> r9c2 <> 2
nrczt-whip[5]  r7n9{c1 c3} - r1c3{n9 n5} - r2c3{n5 n6} - b4n6{r4c3 r5c2} - c2n5{r5 .} ==> r7c1 <> 6
nrczt-whip[5]  r5c1{n6 n2} - r4c1{n2 n3} - r4c4{n3 n9} - r5c6{n9 n5} - r5c2{n5 .} ==> r4c3 <> 6
nrct-chain[4]  c3n6{r2 r7} - r7n9{c3 c1} - b7n4{r7c1 r9c1} - c1n8{r9 r2} ==> r2c1 <> 6
nrczt-whip[5]  r6n7{c9 c5} - r1n7{c5 c4} - r1n4{c4 c6} - r1n3{c6 c2} - r5n3{c2 .} ==> r5c8 <> 7
nrct-chain[6]  c1n7{r2 r3} - b1n3{r3c1 r1c2} - r5n3{c2 c8} - r8c8{n3 n2} - r8c9{n2 n1} - r9c8{n1 n7} ==> r2c8 <> 7
nrct-chain[6]  c1n8{r9 r2} - b1n7{r2c1 r3c1} - b3n7{r3c9 r1c8} - c8n5{r1 r2} - r2c2{n5 n6} - r7c2{n6 n2} ==> r9c1 <> 2
interaction block b7 with column c2 for number 2 ==> r5c2 <> 2
nrc-chain[3]  c2n2{r7 r8} - r8c8{n2 n3} - r7n3{c9 c5} ==> r7c5 <> 2
nrct-chain[6]  c1n8{r9 r2} - b1n7{r2c1 r3c1} - b3n7{r3c9 r1c8} - c8n5{r1 r2} - r2c2{n5 n6} - c3n6{r2 r7} ==> r9c1 <> 6
interaction column c1 with block b4 for number 6 ==> r5c2 <> 6

;;; a whip in which the final contradiction occurs in a bn cell:
nrc-chain[6]  r9c1{n8 n4} - r6c1{n4 n3} - r5n3{c2 c8} - r8c8{n3 n2} - r8c9{n2 n1} - r9n1{c8 c2} ==> r9c2 <> 8
nrczt-whip[6]  b1n3{r3c1 r1c2} - r5n3{c2 c8} - r8c8{n3 n2} - r8c9{n2 n1} - r9c8{n1 n7} - b3n7{r3c8 .} ==> r3c1 <> 7
singles ==> r2c1 = 7, r9c1 = 8
interaction block b7 with row r7 for number 4 ==> r7c7 <> 4
hidden-pairs-in-a-row r7{n4 n9}{c1 c3} ==> r7c3 <> 6
hidden-single-in-column c3 ==> r2c3 = 6
xy-chain[3]  r1c3{n9 n5} - r2c2{n5 n8} - r2c4{n8 n9} ==> r1c5 <> 9, r1c4 <> 9
nrc-chain[3]  r2c4{n9 n8} - r1c5{n8 n7} - b5n7{r6c5 r5c4} ==> r5c4 <> 9
nrczt-whip[4]  r8n8{c4 c5} - r8n9{c5 c6} - r5n9{c6 c8} - c8n3{r5 .} ==> r8c4 <> 3
nrc-chain[2]  c2n3{r5 r1} - c4n3{r1 r4} ==> r4c1 <> 3
naked-pairs-in-a-block b4{r4c1 r5c1}{n2 n6} ==> r4c3 <> 2
hidden-single-in-column c3 ==> r3c3 = 2
nrczt-whip[5]  b5n3{r4c4 r6c6} - c6n1{r6 r2} - b2n2{r2c6 r2c5} - b2n9{r2c5 r3c5} - c7n9{r3 .} ==> r4c4 <> 9
nrczt-whip[4]  c5n6{r9 r4} - b5n9{r4c5 r5c6} - r8n9{c6 c5} - r8n8{c5 .} ==> r8c4 <> 6
nrczt-whip[5]  r9c4{n4 n6} - c5n6{r9 r4} - b5n9{r4c5 r5c6} - r8n9{c6 c5} - r8n8{c5 .} ==> r8c4 <> 4
naked-pairs-in-a-column c4{r2 r8}{n8 n9} ==> r1c4 <> 8
nrct-chain[6]  c9n1{r8 r3} - c9n8{r3 r6} - c9n7{r6 r5} - r5c4{n7 n6} - r4n6{c4 c1} - r4n2{c1 c9} ==> r8c9 <> 2
singles ==> r8c9 = 1, r9c2 = 1
nrct-chain[3]  b6n8{r6c7 r6c9} - r3c9{n8 n7} - b6n7{r6c9 r6c7} ==> r6c7 <> 5
nrct-chain[6]  b5n9{r5c6 r4c5} - r4c7{n9 n5} - r7c7{n5 n6} - c2n6{r7 r8} - c5n6{r8 r9} - r9n5{c5 c6} ==> r5c6 <> 5
singles ==> r5c6 = 9, r4c7 = 9
interaction column c7 with block b9 for number 5 ==> r7c9 <> 5
naked-pairs-in-a-block b9{r7c9 r8c8}{n2 n3} ==> r9c8 <> 2
naked-single ==> r9c8 = 7
interaction row r1 with block b2 for number 7 ==> r3c5 <> 7
interaction row r9 with block b8 for number 2 ==> r8c6 <> 2, r8c5 <> 2
naked-pairs-in-a-column c6{r1 r8}{n3 n4} ==> r9c6 <> 4, r6c6 <> 3
hidden-single-in-block b5 ==> r4c4 = 3
naked-pairs-in-a-row r1{c3 c8}{n5 n9} ==> r1c2 <> 5
hidden-pairs-in-a-block b8{r8c4 r8c5}{n8 n9} ==> r8c5 <> 6
xy-chain[3]  r4c9{n5 n2} - r5c8{n2 n3} - r5c2{n3 n5} ==> r5c9 <> 5
....singles...
GRID 0 SOLVED. LEVEL = NRCZT6, MOST COMPLEX RULE = NRCZT6
135784296
786912354
942536718
671358942
258649137
394271865
429167583
567893421
813425679




4.3) A hard whip example

This is the hardest (in the NRCZT sense) puzzle in the controlled-bias suexg-cb collection of ~ 250,000 minimal puzzles.


500
004
080
000
020
105
072
000
600
000
003
200
009
000
030
400
098
700
010
026
000
300
910
056
900
050
000



*****  SudoRules version 13.7wter  *****
5......72..4.2.....8.1.56.......94....3....982...3.7...1.3..9...2691..5.....56...
singles ==> r3c3 = 2, r7c3 = 5
interaction row r1 with block b2 for number 4 ==> r3c5 <> 4
nrc-chain[2]  r4n2{c4 c8} - r7n2{c8 c6} ==> r9c4 <> 2
hidden-single-in-block b8 ==> r7c6 = 2
nrczt-whip[3]  r4n3{c8 c9} - r8n3{c9 c1} - r3n3{c1 .} ==> r9c8 <> 3
nrczt-whip[3]  c7n5{r2 r5} - b6n2{r5c7 r4c8} - c8n3{r4 .} ==> r2c7 <> 3
nrczt-whip[3]  c7n3{r8 r1} - r3n3{c9 c1} - b7n3{r9c1 .} ==> r9c9 <> 3
nrczt-whip[5]  r8c7{n8 n3} - r1c7{n3 n1} - r1c3{n1 n9} - r2n9{c2 c9} - r2n5{c9 .} ==> r2c7 <> 8
nrczt-whip[6]  r3n7{c1 c5} - c6n7{r2 r5} - c6n1{r5 r6} - r6c8{n1 n6} - r7n6{c8 c9} - r7n7{c9 .} ==> r8c1 <> 7
nrczt-whip[7]  r3n7{c1 c5} - r7n7{c5 c9} - r7n6{c9 c8} - r6c8{n6 n1} - r4n1{c9 c3} - r1c3{n1 n9} - c5n9{r1 .} ==> r4c1 <> 7
nrczt-whip[8]  r8c7{n8 n3} - r1c7{n3 n1} - r2n1{c9 c1} - r1c3{n1 n9} - c1n9{r3 r9} - c1n3{r9 r3} - r3n7{c1 c5} - c5n9{r3 .} ==> r9c7 <> 8
nrczt-whip[9]  c4n2{r4 r5} - b6n2{r5c7 r4c8} - r4n3{c8 c9} - r4n5{c9 c2} - c4n5{r4 r6} - c9n5{r6 r2} - c9n9{r2 r3} - c5n9{r3 r1} - c5n6{r1 .} ==> r4c4 <> 6
nrczt-whip[10]  b1n7{r3c1 r2c2} - c6n7{r2 r8} - c4n7{r9 r4} - r4n2{c4 c8} - r4n3{c8 c9} - r8c9{n3 n4} - r3c9{n4 n9} - r2n9{c9 c1} - c1n6{r2 r4} - c1n1{r4 .} ==> r5c1 <> 7
nrczt-whip[11]  c4n2{r4 r5} - b6n2{r5c7 r4c8} - r4n3{c8 c9} - r4n5{c9 c2} - r5n5{c2 c7} - c9n5{r6 r2} - c9n9{r2 r3} - r3c5{n9 n7} - b5n7{r5c5 r5c6} - c6n1{r5 r6} - b6n1{r6c9 .} ==> r4c4 <> 8
nrczt-whip[11]  b1n7{r3c1 r2c2} - b2n7{r2c6 r3c5} - b8n7{r7c5 r8c6} - r5n7{c6 c4} - r5n2{c4 c7} - r5n5{c7 c2} - r4c2{n5 n6} - c1n6{r5 r2} - r2c4{n6 n8} - b8n8{r9c4 r7c5} - r4c5{n8 .} ==> r9c1 <> 7
nrczt-whip[12]  c3n7{r9 r4} - c2n7{r5 r2} - r3n7{c1 c5} - c4n7{r2 r5} - r5n2{c4 c7} - r5n5{c7 c2} - r4c2{n5 n6} - r4c5{n6 n8} - r7c5{n8 n4} - r9c4{n4 n8} - r2c4{n8 n6} - b1n6{r2c2 .} ==> r9c9 <> 7
nrczt-whip[9]  r6c8{n6 n1} - c6n1{r6 r5} - r5c1{n1 n4} - b7n4{r9c1 r9c2} - r9c9{n4 n1} - r6c9{n1 n5} - c7n5{r5 r2} - r2n1{c7 c1} - c1n6{r2 .} ==> r6c2 <> 6
nrczt-whip[9]  c6n1{r5 r6} - r6c8{n1 n6} - r7n6{c8 c9} - c9n7{r7 r8} - r8c6{n7 n8} - c7n8{r8 r1} - c5n8{r1 r4} - r6c4{n8 n5} - r6c9{n5 .} ==> r5c6 <> 4
nrczt-whip[11]  r3c8{n3 n4} - r3c9{n4 n9} - r3c5{n9 n7} - r2c6{n7 n8} - r2c8{n8 n1} - r6c8{n1 n6} - r7n6{c8 c9} - r7n7{c9 c1} - r7n4{c1 c5} - r8c6{n4 n7} - b9n7{r8c9 .} ==> r2c9 <> 3
nrczt-whip[11]  r6c8{n6 n1} - r6c9{n1 n5} - r6n6{c9 c4} - c5n6{r5 r1} - c5n9{r1 r3} - c9n9{r3 r2} - r2n5{c9 c7} - r2n1{c7 c1} - r1c3{n1 n9} - r6c3{n9 n8} - r4c1{n8 .} ==> r4c9 <> 6
nrczt-whip[11]  r6c8{n6 n1} - r6c9{n1 n5} - b6n6{r6c9 r4c8} - c5n6{r4 r1} - c5n9{r1 r3} - c9n9{r3 r2} - r2n5{c9 c7} - r2n1{c7 c1} - r1c3{n1 n9} - r6c3{n9 n8} - r4c1{n8 .} ==> r6c4 <> 6
interaction row r6 with block b6 for number 6 ==> r4c8 <> 6
nrczt-whip[13]  c9n9{r2 r3} - r3c5{n9 n7} - b1n7{r3c1 r2c2} - b1n6{r2c2 r1c2} - r4c2{n6 n5} - r5c2{n5 n4} - r5c5{n4 n6} - r5c1{n6 n1} - c6n1{r5 r6} - r6n4{c6 c4} - r1c4{n4 n8} - c6n8{r2 r8} - c7n8{r8 .} ==> r2c1 <> 9
nrct-chain[12]  r2n9{c9 c2} - r6n9{c2 c3} - r1n9{c3 c5} - r3c5{n9 n7} - r2n7{c4 c1} - r3c1{n7 n3} - c2n3{r1 r9} - b7n7{r9c2 r9c3} - c3n8{r9 r4} - c5n8{r4 r7} - r9c4{n8 n4} - r9c9{n4 n1} ==> r2c9 <> 1
nrczt-whip[5]  b3n8{r1c7 r2c8} - c8n3{r2 r4} - b6n2{r4c8 r5c7} - c7n5{r5 r2} - b3n1{r2c7 .} ==> r1c7 <> 3
interaction column c7 with block b9 for number 3 ==> r8c9 <> 3
nrczt-whip[8]  c1n9{r9 r3} - r1c3{n9 n1} - r1c7{n1 n8} - r8c7{n8 n3} - c1n3{r8 r2} - b1n7{r2c1 r2c2} - r2c6{n7 n8} - r8n8{c6 .} ==> r9c1 <> 8
nrct-chain[9]  r9c9{n1 n4} - r8c9{n4 n7} - r7c9{n7 n6} - r7c8{n6 n8} - b3n8{r2c8 r1c7} - c5n8{r1 r4} - c1n8{r4 r8} - r8n4{c1 c6} - r6c6{n4 n1} ==> r6c9 <> 1
nrc-chain[3]  c9n1{r9 r4} - r4n3{c9 c8} - c8n2{r4 r9} ==> r9c8 <> 1
nrczt-whip[10]  c7n8{r1 r8} - b8n8{r8c6 r7c5} - b7n8{r7c1 r9c3} - c3n7{r9 r4} - r4c5{n7 n6} - r1n6{c5 c2} - r1n3{c2 c6} - r2c6{n3 n7} - b8n7{r8c6 r9c4} - c2n7{r9 .} ==> r1c4 <> 8
nrczt-whip[10]  c9n9{r2 r3} - r3n4{c9 c8} - r3n3{c8 c1} - c1n9{r3 r9} - b7n3{r9c1 r9c2} - r1c2{n3 n6} - r1c4{n6 n4} - r9n4{c4 c9} - c9n1{r9 r4} - c9n3{r4 .} ==> r2c2 <> 9
singles ==> r2c9 = 9, r2c7 = 5
naked-pairs-in-a-block b3{r3c8 r3c9}{n3 n4} ==> r2c8 <> 3
interaction block b3 with row r3 for number 3 ==> r3c1 <> 3
nrc-chain[3]  r5c6{n7 n1} - r5c7{n1 n2} - c4n2{r5 r4} ==> r4c4 <> 7
nrc-chain[5]  r6c9{n5 n6} - r6c8{n6 n1} - r2n1{c8 c1} - r1c3{n1 n9} - b4n9{r6c3 r6c2} ==> r6c2 <> 5
nrczt-whip[5]  r3n7{c1 c5} - r7n7{c5 c9} - r7n6{c9 c8} - r6c8{n6 n1} - r2n1{c8 .} ==> r2c1 <> 7
nrczt-whip[6]  c3n7{r4 r9} - r7n7{c1 c9} - c9n6{r7 r6} - c9n5{r6 r4} - c2n5{r4 r5} - r5n7{c2 .} ==> r4c5 <> 7
interaction row r4 with block b4 for number 7 ==> r5c2 <> 7
nrczt-whip[6]  b3n8{r1c7 r2c8} - r7n8{c8 c1} - r4n8{c1 c3} - r4c5{n8 n6} - r4c1{n6 n1} - r2n1{c1 .} ==> r1c5 <> 8
nrct-chain[4]  r1n8{c6 c7} - r2c8{n8 n1} - r1n1{c7 c3} - r6n1{c3 c6} ==> r6c6 <> 8
nrczt-whip[5]  c4n2{r5 r4} - c4n5{r4 r6} - r6n4{c4 c2} - r6n9{c2 c3} - r6n8{c3 .} ==> r5c4 <> 4
nrct-chain[6]  c6n3{r2 r1} - r1n8{c6 c7} - r1n1{c7 c3} - r2n1{c1 c8} - r6n1{c8 c6} - r5c6{n1 n7} ==> r2c6 <> 7
swordfish-in-rows n7{r2 r4 r9}{c4 c2 c3} ==> r5c4 <> 7
nrc-chain[5]  c9n6{r7 r6} - r6c8{n6 n1} - c6n1{r6 r5} - c6n7{r5 r8} - r8c9{n7 n4} ==> r7c9 <> 4
nrc-chain[5]  r6c8{n1 n6} - c9n6{r6 r7} - b9n7{r7c9 r8c9} - c6n7{r8 r5} - b5n1{r5c6 r6c6} ==> r6c3 <> 1
nrc-chain[2]  c3n1{r4 r1} - r2n1{c1 c8} ==> r4c8 <> 1
nrczt-whip[5]  c5n8{r4 r7} - r8n8{c6 c7} - r1c7{n8 n1} - r1c3{n1 n9} - r6c3{n9 .} ==> r4c1 <> 8
interaction column c1 with block b7 for number 8 ==> r9c3 <> 8
nrc-chain[3]  r9c3{n7 n9} - c1n9{r9 r3} - c1n7{r3 r7} ==> r9c2 <> 7
nrc-chain[3]  r9c3{n9 n7} - c1n7{r7 r3} - c1n9{r3 r9} ==> r9c2 <> 9
hxyt-cn-chain[4]  c4n7{r2 r9} - c3n7{r9 r4} - c3n8{r4 r6} - c4n8{r6 r2} ==> r2c4 <> 6
interaction row r2 with block b1 for number 6 ==> r1c2 <> 6
naked-triplets-in-a-column c2{r1 r6 r9}{n3 n9 n4} ==> r5c2 <> 4,  r2c2 <> 3
nrc-chain[3]  b4n4{r5c1 r6c2} - r6n9{c2 c3} - r9n9{c3 c1} ==> r9c1 <> 4
nrc-chain[4]  c2n4{r9 r6} - b4n9{r6c2 r6c3} - r6n8{c3 c4} - r9n8{c4 c8} ==> r9c8 <> 4
nrct-chain[4]  r2c4{n7 n8} - c6n8{r2 r8} - b7n8{r8c1 r7c1} - b7n7{r7c1 r9c3} ==> r9c4 <> 7
singles ==> r2c4 = 7, r3c5 = 9,  r3c1 = 7, r2c2 = 6,  r5c2 = 5, r4c2 = 7,  r9c3 = 7, r9c1 = 9
interaction block b2 with column c6 for number 8 ==> r8c6 <> 8
naked-pairs-in-a-row r8{c6 c9}{n4 n7} ==> r8c1 <> 4
naked-pairs-in-a-block b2{r1c4 r1c5}{n4 n6} ==> r1c6 <> 4
nrc-chain[2]  c6n4{r8 r6} - c2n4{r6 r9} ==> r9c4 <> 4
singles
GRID 0 SOLVED. LEVEL = NRCZT13, MOST COMPLEX RULE = NRCZT13
539468172
164723589
782195643
671289435
453671298
298534716
815342967
326917854
947856321




4.4) Another hard whip example

This is the hardest (in the NRCZT sense) puzzle in the gsf-cb collection of 5,926,343 controlled-bias minimal puzzles.

100
000
870
050
700
000
009
030
004
008
530
000
000
090
302
000
600
400
060
005
001
004
900
000
800
310
000



*****  SudoRules version 13.7wter  *****
1...5...9...7...3.87......4..8......53..9.6.....3.24...6...48....59..31...1......
singles ==> r4c9 = 3, r2c2 = 5, r9c8 = 4
interaction column c1 with block b7 for number 3 ==> r7c3 <> 3
interaction row r6 with block b6 for number 5 ==> r4c8 <> 5, r4c7 <> 5
interaction column c8 with block b3 for number 6 ==> r2c9 <> 6
nrc-chain[2]  b2n4{r2c5 r1c4} - r5n4{c4 c3} ==> r2c3 <> 4
nrczt-whip[2]  c2n9{r6 r9} - c7n9{r9 .} ==> r4c1 <> 9
nrczt-whip[4]  r7n1{c5 c4} - r3n1{c4 c7} - c7n5{r3 r9} - b8n5{r9c6 .} ==> r2c5 <> 1
nrczt-whip[12]  r3n9{c6 c3} - c3n3{r3 r1} - r1c6{n3 n8} - r8c6{n8 n7} - r5c6{n7 n1} - r2c6{n1 n9} - r2n8{c6 c9} - c9n1{r2 r6} - r6n5{c9 c8} - c8n8{r6 r5} - r5c4{n8 n4} - c3n4{r5 .} ==> r3c6 <> 6
nrczt-whip[14]  r2n4{c5 c1} - c3n4{r1 r5} - c2n4{r4 r8} - r1c2{n4 n2} - b4n2{r4c2 r4c1} - r8n2{c1 c9} - c7n2{r9 r3} - c7n5{r3 r9} - c9n5{r9 r6} - r7n5{c9 c4} - r7n1{c4 c5} - r6n1{c5 c2} - r4c2{n1 n9} - c7n9{r4 .} ==> r2c5 <> 2
nrczt-whip[10]  r5n2{c9 c3} - c3n4{r5 r1} - r1c2{n4 n2} - r2n2{c1 c9} - b3n8{r2c9 r1c8} - c8n2{r1 r7} - b9n9{r7c8 r9c7} - r9c2{n9 n8} - c4n8{r9 r5} - r5n4{c4 .} ==> r4c7 <> 2
nrczt-whip[8]  b9n9{r7c8 r9c7} - c7n5{r9 r3} - c8n5{r3 r6} - c8n9{r6 r4} - c2n9{r4 r6} - c2n1{r6 r4} - r4c7{n1 n7} - c9n7{r6 .} ==> r7c8 <> 7
nrczt-whip[7]  b9n7{r7c9 r9c7} - b9n9{r9c7 r7c8} - r4c8{n9 n2} - r5n2{c9 c3} - c3n4{r5 r1} - r1c2{n4 n2} - r1c7{n2 .} ==> r6c9 <> 7
nrczt-whip[7]  b9n7{r7c9 r9c7} - b9n9{r9c7 r7c8} - r4c8{n9 n2} - r5n2{c9 c3} - c3n4{r5 r1} - r1c2{n4 n2} - r1c7{n2 .} ==> r5c9 <> 7
interaction column c9 with block b9 ==> r9c7 <> 7
nrczt-whip[10]  b2n2{r3c4 r1c4} - c7n2{r1 r9} - b9n9{r9c7 r7c8} - r4c8{n9 n7} - r5c8{n7 n8} - c4n8{r5 r9} - r9c2{n8 n9} - r6c2{n9 n1} - r6c9{n1 n5} - b9n5{r9c9 .} ==> r3c8 <> 2
nrczt-whip[10]  r3c8{n6 n5} - r6n5{c8 c9} - b9n5{r9c9 r9c7} - c6n5{r9 r4} - b5n6{r4c6 r4c4} - c4n5{r4 r7} - r7n1{c4 c5} - r6n1{c5 c2} - r4n1{c2 c7} - c7n9{r4 .} ==> r3c5 <> 6
nrczt-whip[14]  r1c2{n2 n4} - c3n4{r1 r5} - r5n2{c3 c9} - b9n2{r9c9 r9c7} - c7n9{r9 r4} - r4c8{n9 n7} - r5c8{n7 n8} - r5c4{n8 n1} - r7n1{c4 c5} - r4n1{c5 c2} - c2n2{r4 r8} - c5n2{r8 r3} - r3c4{n2 n6} - c8n6{r3 .} ==> r1c8 <> 2
nrczt-whip[14]  r1c2{n2 n4} - c3n4{r1 r5} - b4n2{r5c3 r4c2} - c2n1{r4 r6} - c2n9{r6 r9} - c7n9{r9 r4} - c7n7{r4 r1} - b3n2{r1c7 r3c7} - r9c7{n2 n5} - c6n5{r9 r4} - c4n5{r4 r7} - r7n1{c4 c5} - r4n1{c5 c4} - c4n4{r4 .} ==> r2c1 <> 2
nrczt-whip[13]  c7n5{r3 r9} - c7n9{r9 r4} - c7n1{r4 r2} - r2n2{c7 c3} - r1c2{n2 n4} - c3n4{r1 r5} - c4n4{r5 r4} - r4n5{c4 c6} - b8n5{r9c6 r7c4} - c9n5{r7 r6} - c9n1{r6 r5} - c6n1{r5 r3} - c4n1{r3 .} ==> r3c7 <> 2
nrczt-whip[11]  r3c8{n6 n5} - r6n5{c8 c9} - r7n5{c9 c4} - r9n5{c6 c7} - r3c7{n5 n1} - b6n1{r4c7 r5c9} - c4n1{r5 r4} - c6n1{r5 r2} - r2n6{c6 c5} - b5n6{r6c5 r4c6} - r4n5{c6 .} ==> r3c3 <> 6
nrczt-whip[13]  b2n4{r2c5 r1c4} - r5n4{c4 c3} - c2n4{r4 r8} - r1c2{n4 n2} - c3n2{r3 r7} - c3n7{r7 r6} - c3n6{r6 r1} - r1n3{c3 c6} - b2n8{r1c6 r2c6} - r8n8{c6 c5} - r6c5{n8 n1} - c6n1{r5 r3} - c6n9{r3 .} ==> r2c5 <> 6
nrczt-whip[16]  r1c2{n2 n4} - c3n4{r1 r5} - c4n4{r5 r4} - c1n4{r4 r8} - r8c2{n4 n8} - r9c2{n8 n9} - b9n9{r9c7 r7c8} - r4c8{n9 n7} - r4c1{n7 n6} - r4c5{n6 n1} - r7n1{c5 c4} - c4n5{r7 r9} - c7n5{r9 r3} - r3n1{c7 c6} - r3n9{c6 c3} - r2c1{n9 .} ==> r4c2 <> 2
nrczt-whip[14]  r8n4{c1 c2} - c2n8{r8 r9} - c2n2{r9 r1} - b1n4{r1c2 r1c3} - r5n4{c3 c4} - c4n8{r5 r1} - r2n8{c6 c9} - r2n2{c9 c7} - b3n1{r2c7 r3c7} - c7n5{r3 r9} - c7n9{r9 r4} - r4c2{n9 n1} - c4n1{r4 r7} - r7n5{c4 .} ==> r4c1 <> 4 
nrc-chain[3]  b7n4{r8c1 r8c2} - b4n4{r4c2 r5c3} - b4n2{r5c3 r4c1} ==> r8c1 <> 2
nrczt-whip[10]  b4n2{r4c1 r5c3} - b4n4{r5c3 r4c2} - r8c2{n4 n8} - c2n2{r8 r1} - r9c2{n2 n9} - r9c7{n9 n5} - b8n5{r9c6 r7c4} - c4n2{r7 r3} - r3n6{c4 c8} - r3n5{c8 .} ==> r9c1 <> 2
nrczt-whip[10]  r6n5{c8 c9} - r6n8{c9 c5} - r6n1{c5 c2} - b4n9{r6c2 r4c2} - b4n4{r4c2 r5c3} - r5c4{n4 n1} - c9n1{r5 r2} - c6n1{r2 r3} - c6n9{r3 r2} - r2n8{c6 .} ==> r6c8 <> 9
interaction row r6 with block b4 for number 9 ==> r4c2 <> 9
nrc-chain[3]  b4n1{r4c2 r6c2} - c2n9{r6 r9} - c7n9{r9 r4} ==> r4c7 <> 1
interaction column c7 with block b3 for number 1 ==> r2c9 <> 1
nrczt-whip[6]  c9n1{r5 r6} - c2n1{r6 r4} - b4n4{r4c2 r5c3} - r5c4{n4 n8} - r6n8{c5 c8} - r6n5{c8 .} ==> r5c6 <> 1
nrczt-whip[5]  r2n6{c1 c6} - c6n9{r2 r3} - c6n1{r3 r4} - r4c2{n1 n4} - c3n4{r5 .} ==> r1c3 <> 6
interaction block b1 with row r2 for number 6 ==> r2c6 <> 6
nrczt-whip[5]  r5n1{c9 c4} - b5n8{r5c4 r6c5} - r2n8{c5 c6} - c6n9{r2 r3} - c6n1{r3 .} ==> r5c9 <> 8
nrczt-whip[5]  c6n9{r3 r2} - c6n1{r2 r4} - r4c2{n1 n4} - c3n4{r5 r1} - r1n3{c3 .} ==> r3c6 <> 3
nrczt-whip-cn[5]  n9{r2c6 r3c6} - n1{r3c6 r4c6} - {n1 n4}r4c2 - n4{r5c3 r5c4} - {n8r5c4 .} ==> r2c6 <> 8
naked-pairs-in-a-block b2{r2c6 r3c6}{n1 n9} ==> r3c5 <> 1, r3c4 <> 1
interaction block b2 with column c6 for number 1 ==> r4c6 <> 1
nrct-chain[4]  r3c4{n2 n6} - r3c8{n6 n5} - c7n5{r3 r9} - b8n5{r9c6 r7c4} ==> r7c4 <> 2
nrct-chain[4]  r3c4{n2 n6} - r3c8{n6 n5} - c7n5{r3 r9} - r9c4{n5 n2} ==> r1c4 <> 2
interaction block b2 with row r3 for number 2 ==> r3c3 <> 2
nrc-chain[5]  c6n3{r9 r1} - r3n3{c5 c3} - r3n9{c3 c6} - r3n1{c6 c7} - c7n5{r3 r9} ==> r9c6 <> 5
hidden-single-in-column c6 ==> r4c6 = 5
nrc-chain[4]  r3c5{n3 n2} - c4n2{r3 r9} - c4n5{r9 r7} - b8n1{r7c4 r7c5} ==> r7c5 <> 3
singles ==> r7c1 = 3, r4c1 = 2, r5c8 = 2, r7c8 = 9, r4c8 = 7, r4c7 = 9, r1c7 = 7
interaction row r1 with block b1 for number 2 ==> r2c3 <> 2
interaction row r4 with block b5 for number 6 ==> r6c5 <> 6
xyz-chain[3]  r1c8{n8 n6} - r1c4{n6 n4} - r2c5{n4 n8} ==> r1c6 <> 8
hidden-pairs-in-a-block b2{r1c4 r2c5}{n4 n8} ==> r1c4 <> 6
naked-pairs-in-a-column c4{r1 r5}{n4 n8} ==> r4c4 <> 4
x-wing-in-columns n4{c3 c4}{r1 r5} ==> r1c2 <> 4
singles ==> r1c2 = 2, r7c3 = 2
interaction column c3 with block b4 for number 7 ==> r6c1 <> 7
naked-pairs-in-a-column c5{r6 r7}{n1 n7} ==> r9c5 <> 7, r8c5 <> 7, r4c5 <> 1
nrc-chain[3]  r8c2{n8 n4} - c1n4{r8 r2} - r2c5{n4 n8} ==> r8c5 <> 8
xy-chain[3]  r1c6{n6 n3} - r3c5{n3 n2} - r8c5{n2 n6} ==> r9c6 <> 6, r8c6 <> 6
singles
GRID 0 SOLVED. LEVEL = NRCZT16, MOST COMPLEX RULE = NRCZT16
123456789
456789132
879231564
218645973
534897621
697312458
362174895
745928316
981563247






5) AN EXCEPTIONALLY HARD PUZZLE WITH SER = 9.4 AND NRCZT = 17



Although the 10,000,000 puzzles in randomly generated collections can be solved with whips no longer than 13, there are extremely rare puzzles that need much longer whips.


The first example is #10 in gsf's list of 14 puzzles with backdoor size 3. It has SER = 9.4 and NRCZT = 17.

001
030
500
000
000
030
200
040
006
000
040
000
107
000
902
000
080
000
300
060
002
000
050
000
008
030
700



*****  SudoRules version 13.7wter  *****
001000200030000040500030006000107000040000080000902000300000008060050030002000700
singles ==> r5c5 = 6, r1c9 = 3
interaction block b5 with column c5 for number 8 ==> r9c5 <> 8, r2c5 <> 8, r1c5 <> 8
interaction block b5 with column c5 for number 4 ==> r9c5 <> 4, r7c5 <> 4, r1c5 <> 4
interaction block b5 with row r5 for number 5 ==> r5c9 <> 5, r5c7 <> 5, r5c3 <> 5, r5c7 <> 3, r5c3 <> 3
nrc-chain[3]  r8n2{c4 c9} - r5n2{c9 c1} - c2n2{r4 r3} ==> r3c4 <> 2
hidden-single-in-row r3 ==> r3c2 = 2
nrczt-whip[9]  r5c7{n1 n9} - r5c3{n9 n7} - c9n7{r5 r2} - r3n7{c8 c4} - r1c5{n7 n9} - r9c5{n9 n1} - c2n1{r9 r7} - c8n1{r7 r3} - b3n9{r3c8 .} ==> r6c9 <> 1
nrczt-whip[10]  b8n7{r8c4 r7c5} - c2n7{r7 r6} - c8n7{r6 r3} - c9n7{r2 r5} - r5n2{c9 c1} - b4n1{r5c1 r6c1} - b6n1{r6c8 r5c7} - b3n1{r3c7 r2c9} - r8n1{c9 c6} - c5n1{r9 .} ==> r1c4 <> 7
nrczt-whip[10]  r5c7{n1 n9} - r5c3{n9 n7} - r5c9{n7 n2} - r8n2{c9 c4} - r8n7{c4 c1} - c2n7{r7 r1} - r1c5{n7 n9} - r1c8{n9 n5} - r2c7{n5 n8} - r3c7{n8 .} ==> r6c7 <> 1
nrczt-whip[12]  r9c5{n1 n9} - r1c5{n9 n7} - r7c5{n7 n2} - c5n1{r7 r2} - r2n2{c5 c4} - r8n2{c4 c9} - c9n1{r8 r5} - r5c7{n1 n9} - b9n9{r8c7 r7c8} - c8n1{r7 r3} - b3n9{r3c8 r2c9} - b3n7{r2c9 .} ==> r9c6 <> 1
nrczt-whip[13]  r5c7{n1 n9} - r5c3{n9 n7} - r5c9{n7 n2} - r8n2{c9 c4} - r8n7{c4 c1} - c2n7{r7 r1} - r1c5{n7 n9} - r1c8{n9 n5} - r4c8{n5 n6} - r9c8{n6 n9} - r3n9{c8 c3} - c1n9{r2 r4} - r4n2{c1 .} ==> r6c8 <> 1
interaction row r6 with block b4 ==> r5c1 <> 1
nrczt-whip[9]  c8n1{r7 r3} - c9n1{r2 r5} - r5c7{n1 n9} - r5c3{n9 n7} - r3n7{c3 c4} - r8n7{c4 c1} - r2n7{c1 c9} - b3n9{r2c9 r1c8} - r1c5{n9 .} ==> r8c7 <> 1, r7c7 <> 1
nrczt-whip[13]  r8c7{n4 n9} - r5c7{n9 n1} - r3c7{n1 n8} - r3c4{n8 n7} - r1c5{n7 n9} - r9c5{n9 n1} - r8c6{n1 n8} - r8c3{n8 n7} - r5c3{n7 n9} - r3n9{c3 c8} - r2n9{c9 c1} - r2n7{c1 c9} - b3n1{r2c9 .} ==> r8c4 <> 4
nrczt-whip[16]  r5n2{c1 c9} - b9n2{r8c9 r7c8} - b8n2{r7c5 r8c4} - r8n7{c4 c3} - c2n7{r7 r1} - r1c5{n7 n9} - r9c5{n9 n1} - c8n1{r9 r3} - r7n1{c8 c2} - r8n1{c1 c9} - r5n1{c9 c7} - r5n9{c7 c3} - c2n9{r4 r9} - c1n9{r9 r2} - r3n9{c3 c7} - b9n9{r8c7 .} ==> r5c1 <> 7
nrczt-whip[16]  r8n2{c9 c4} - r7n2{c5 c8} - r4n2{c8 c1} - r5c1{n2 n9} - r5c3{n9 n7} - r8n7{c3 c1} - c2n7{r7 r1} - r1c5{n7 n9} - r9c5{n9 n1} - c8n1{r9 r3} - b9n1{r7c8 r8c9} - b7n1{r8c1 r7c2} - c2n9{r7 r9} - c8n9{r9 r4} - c9n9{r5 r2} - b3n7{r2c9 .} ==> r5c9 <> 2
hidden-single-in-row r5 ==> r5c1 = 2
nrczt-whip[17]  r5c3{n7 n9} - r5c7{n9 n1} - r5c9{n1 n7} - c8n7{r6 r1} - r1c5{n7 n9} - r1c2{n9 n8} - r4c2{n8 n5} - b7n5{r9c2 r7c3} - c3n4{r7 r8} - r8c7{n4 n9} - r3c7{n9 n8} - r3c4{n8 n4} - r3c6{n4 n1} - r8c6{n1 n8} - c4n8{r9 r2} - r2n7{c4 c5} - r2n2{c5 .} ==> r3c3 <> 7
nrczt-whip[4]  c8n1{r7 r3} - r3n7{c8 c4} - r1c5{n7 n9} - r9c5{n9 .} ==> r9c9 <> 1
nrczt-whip[9]  r9c5{n1 n9} - r1c5{n9 n7} - r3n7{c4 c8} - c8n1{r3 r7} - c5n1{r7 r2} - r3n1{c6 c7} - r5c7{n1 n9} - b9n9{r8c7 r8c9} - b9n2{r8c9 .} ==> r9c2 <> 1
nrczt-whip[8]  r8c7{n4 n9} - r9c9{n9 n5} - r6c9{n5 n7} - b4n7{r6c3 r5c3} - r8c3{n7 n8} - r8c6{n8 n1} - r9c5{n1 n9} - r9c2{n9 .} ==> r8c9 <> 4
nrczt-whip[9]  r8n2{c9 c4} - r7n2{c5 c8} - b9n1{r7c8 r9c8} - r9c5{n1 n9} - r1c5{n9 n7} - b8n7{r7c5 r7c4} - c2n7{r7 r6} - c2n1{r6 r7} - r7c5{n1 .} ==> r8c9 <> 9
nrczt-whip[8]  r9c5{n1 n9} - r1c5{n9 n7} - r3n7{c4 c8} - c8n1{r3 r7} - r8n1{c9 c6} - r3n1{c6 c7} - r5c7{n1 n9} - b9n9{r7c7 .} ==> r9c1 <> 1
nrczt-whip[7]  b7n1{r8c1 r7c2} - c5n1{r7 r2} - c5n2{r2 r7} - c5n7{r7 r1} - c2n7{r1 r6} - b6n7{r6c9 r5c9} - c9n1{r5 .} ==> r8c6 <> 1
nrczt-whip[7]  r8n1{c1 c9} - c8n1{r9 r3} - r3n7{c8 c4} - r8n7{c4 c3} - r5c3{n7 n9} - c2n9{r4 r1} - r1c5{n9 .} ==> r8c1 <> 9
nrczt-whip[8]  b4n9{r4c1 r5c3} - r8n9{c3 c6} - r3n9{c6 c8} - r7n9{c8 c2} - c2n1{r7 r6} - c2n7{r6 r1} - c8n7{r1 r6} - b4n7{r6c3 .} ==> r4c7 <> 9
nrczt-whip[9]  r4n2{c8 c9} - r8c9{n2 n1} - b7n1{r8c1 r7c2} - c8n1{r7 r3} - r3n7{c8 c4} - r1c5{n7 n9} - c2n9{r1 r9} - c9n9{r9 r2} - c1n9{r2 .} ==> r4c8 <> 9
nrct-chain[5]  r5c3{n7 n9} - r4n9{c1 c9} - c9n2{r4 r8} - r8n1{c9 c1} - c2n1{r7 r6} ==> r6c2 <> 7
nrct-chain[3]  c2n7{r1 r7} - b8n7{r7c4 r8c4} - r3n7{c4 c8} ==> r1c8 <> 7
nrct-chain[4]  c1n1{r8 r6} - c2n1{r6 r7} - c2n7{r7 r1} - c1n7{r2 r8} ==> r8c1 <> 8, r8c1 <> 4
nrc-chain[6]  r1c8{n5 n9} - r1c5{n9 n7} - c2n7{r1 r7} - b7n1{r7c2 r8c1} - r8c9{n1 n2} - c8n2{r7 r4} ==> r4c8 <> 5
nrct-chain[8]  r3n7{c8 c4} - c5n7{r1 r7} - c2n7{r7 r1} - r2n7{c3 c9} - b6n7{r5c9 r6c8} - c1n7{r6 r8} - r8n1{c1 c9} - c8n1{r9 r3} ==> r3c8 <> 9
nrczt-whip[7]  c8n9{r7 r1} - r1c5{n9 n7} - c2n7{r1 r7} - c2n9{r7 r4} - c1n9{r4 r2} - c1n7{r2 r6} - r5c3{n7 .} ==> r9c9 <> 9
xyt-chain[5]  r9c9{n5 n4} - r8c7{n4 n9} - r5c7{n9 n1} - r3c7{n1 n8} - r2c7{n8 n5} ==> r7c7 <> 5, r2c9 <> 5
nrczt-whip[8]  b9n2{r7c8 r8c9} - b9n1{r8c9 r9c8} - r9c5{n1 n9} - c8n9{r9 r1} - r1c5{n9 n7} - r1c2{n7 n8} - r9c2{n8 n5} - b9n5{r9c9 .} ==> r7c8 <> 6
nrczt-whip[8]  c2n1{r6 r7} - r8n1{c1 c9} - b9n2{r8c9 r7c8} - r7n5{c8 c3} - r9c2{n5 n9} - r1c2{n9 n7} - r1c5{n7 n9} - c8n9{r1 .} ==> r6c2 <> 8
nrczt-whip[9]  c2n7{r7 r1} - r1c5{n7 n9} - r9c5{n9 n1} - r7n1{c6 c8} - r8n1{c9 c1} - c1n7{r8 r6} - r5c3{n7 n9} - c2n9{r4 r9} - c8n9{r9 .} ==> r7c2 <> 5
nrc-chain[4]  r7n5{c3 c8} - r1c8{n5 n9} - r1c5{n9 n7} - c2n7{r1 r7} ==> r7c3 <> 7
nrczt-whip[6]  r7n5{c3 c8} - b3n5{r1c8 r2c7} - c7n8{r2 r3} - r3c3{n8 n4} - b7n4{r8c3 r9c1} - r9c9{n4 .} ==> r7c3 <> 9
nrczt-whip[6]  c7n8{r3 r2} - b3n5{r2c7 r1c8} - r7n5{c8 c3} - c3n4{r7 r8} - r8c7{n4 n9} - c8n9{r9 .} ==> r3c3 <> 8
nrczt-whip[7]  b9n2{r7c8 r8c9} - b9n1{r8c9 r9c8} - r9c5{n1 n9} - r1c5{n9 n7} - c2n7{r1 r7} - r7n9{c2 c7} - b9n6{r7c7 .} ==> r7c8 <> 5
hidden-single-in-row r7 ==> r7c3 = 5
nrc-chain[4]  b4n5{r4c2 r6c2} - c2n1{r6 r7} - r8n1{c1 c9} - c9n2{r8 r4} ==> r4c9 <> 5
nrczt-whip[5]  r1c5{n9 n7} - r1c2{n7 n8} - r9c2{n8 n9} - c8n9{r9 r7} - r8n9{c7 .} ==> r1c6 <> 9
nrczt-whip[5]  r5c7{n1 n9} - r8c7{n9 n4} - c3n4{r8 r3} - r3n9{c3 c6} - r3n1{c6 .} ==> r2c7 <> 1
nrct-chain[6]  b7n4{r9c1 r8c3} - r8c7{n4 n9} - c8n9{r7 r1} - r1c5{n9 n7} - r1c2{n7 n8} - r9c2{n8 n9} ==> r9c1 <> 9
nrczt-whip[6]  r8c7{n9 n4} - c3n4{r8 r3} - r3n9{c3 c6} - b8n9{r9c6 r9c5} - r9n1{c5 c8} - b9n6{r9c8 .} ==> r7c7 <> 9
nrczt-whip[6]  r8c7{n9 n4} - c3n4{r8 r3} - r3n9{c3 c6} - r1c5{n9 n7} - r3c4{n7 n8} - c7n8{r3 .} ==> r2c7 <> 9
xyt-chain[4]  r2c7{n8 n5} - r1c8{n5 n9} - r1c5{n9 n7} - r1c2{n7 n8} ==> r2c3 <> 8, r2c1 <> 8
interaction block b1 with row r1 for number 8 ==> r1c6 <> 8, r1c4 <> 8
nrczt-whip[6]  r1n8{c1 c2} - r9c2{n8 n9} - c8n9{r9 r7} - b9n2{r7c8 r8c9} - b9n1{r8c9 r9c8} - r9c5{n1 .} ==> r1c1 <> 9
nrczt-whip[6]  r5c3{n7 n9} - c1n9{r4 r2} - r2c9{n9 n1} - r3n1{c8 c6} - b2n9{r3c6 r1c5} - r1n7{c5 .} ==> r2c3 <> 7
nrczt-whip[6]  r2c3{n6 n9} - r2c1{n9 n7} - r2c9{n7 n1} - r3n1{c8 c6} - b2n9{r3c6 r1c5} - r1n7{c5 .} ==> r1c1 <> 6
interaction row r1 with block b2 for number 6 ==> r2c6 <> 6, r2c4 <> 6
nrct-chain[6]  r8c7{n4 n9} - c8n9{r9 r1} - r1c5{n9 n7} - r1c2{n7 n8} - r1c1{n8 n4} - b7n4{r9c1 r8c3} ==> r8c6 <> 4
nrct-chain[4]  r2c7{n8 n5} - r1c8{n5 n9} - b9n9{r9c8 r8c7} - r8c6{n9 n8} ==> r2c6 <> 8
nrczt-whip[4]  r9n3{c4 c6} - r9n6{c6 c8} - r7c7{n6 n4} - b8n4{r7c4 .} ==> r9c4 <> 8, r9c6 <> 8
interaction row r9 with block b7 for number 8 ==> r8c3 <> 8
interaction column c3 with block b4 for number 8 ==> r6c1 <> 8, r4c2 <> 8, r4c1 <> 8
hidden-pairs-in-a-block b4{r4c3 r6c3}{n3 n8} ==> r6c3 <> 7, r6c3 <> 6, r4c3 <> 9, r4c3 <> 6
hidden-single-in-column c3 ==> r2c3 = 6
hidden-pairs-in-a-column c1{n4 n8}{r1 r9} ==> r1c1 <> 7
nrc-chain[4]  r8c9{n2 n1} - c1n1{r8 r6} - c1n6{r6 r4} - r4c8{n6 n2} ==> r7c8 <> 2
singles ==> r8c9 = 2, r4c8 = 2, r8c1 = 1, r6c2 = 1, r4c2 = 5
interaction block b9 with column c8 for number 1 ==> r3c8 <> 1
naked-single ==> r3c8 = 7
nrc-chain[3]  r3c4{n8 n4} - c3n4{r3 r8} - r8n7{c3 c4} ==> r8c4 <> 8
singles
GRID 0 SOLVED. LEVEL = NRCZT17, MOST COMPLEX RULE = NRCZT17
481675293
736829541
529431876
953187624
247563189
618942357
375296418
164758932
892314765


6) A STILL HARDER PUZZLE WITH SER = 9.3 AND NRCZT = 21



The second hard example has SER 9.3 and NRCZT = 21.
It is #5287 in gsf's list of 8152 hardest.


100
040
008
000
602
000
000
000
500
090
000
000
300
070
209
020
900
060
500
060
007
000
004
000
001
090
008


***** SudoRules version 13.7wbis *****
100000000040602000008000500090300020000070900000209060500000001060004090007000008
interaction column c9 with block b3 for number 6 ==> r1c7 <> 6
hidden-pairs-in-a-block b7{r7c3 r9c1}{n4 n9} ==> r9c1 <> 3, r9c1 <> 2, r7c3 <> 3, r7c3 <> 2
nrczt-whip-bn[5]  n6{r1c3 r3c1} - n6{r3c9 r1c9} - n2{r1c9 r3c9} - n9{r3c9 r2c9} - b1n9{r2c3 .} ==> r1c3 <> 2
nrczt-whip-rn[5]  n6{r3c9 r1c9} - n2{r1c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - r3n6{c1 .} ==> r3c9 <> 9
nrct-chain[4]  n6{r1c3 r3c1} - n6{r3c9 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} ==> r1c3 <> 5, r1c3 <> 3
nrczt-whip-rn[5]  n6{r3c9 r1c9} - n2{r1c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - r3n6{c1 .} ==> r3c9 <> 7, r3c9 <> 4, r3c9 <> 3
nrczt-whip-rn[6]  n2{r8c3 r5c3} - n2{r5c1 r3c1} - n2{r3c9 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - r1n6{c3 .} ==> r8c7 <> 2, r8c5 <> 2
nrczt-whip-rn[6]  n2{r5c3 r8c3} - n2{r8c1 r3c1} - n2{r3c9 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - r1n6{c3 .} ==> r5c2 <> 2
nrczt-whip-cn[6]  n9{r2c9 r1c9} - n9{r1c3 r3c1} - n6{r3c1 r3c9} - n2{r3c9 r8c9} - n2{r8c3 r5c3} - c1n2{r5 .} ==> r2c5 <> 9
nrczt-whip-rc[11]  r8c7{n3 n7} - r7c8{n7 n4} - r9c8{n4 n5} - r8c9{n5 n2} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n2{r3c9 r1c7} - n4{r1c7 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - r7c3{n9 .} ==> r9c7 <> 3
nrczt-whip-rc[11]  r8c7{n7 n3} - r7c8{n3 n4} - r9c8{n4 n5} - r8c9{n5 n2} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n2{r3c9 r1c7} - n4{r1c7 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - r7c3{n9 .} ==> r7c7 <> 7
nrczt-whip-rc[11]  r8c7{n3 n7} - r7c8{n7 n4} - r9c8{n4 n5} - r8c9{n5 n2} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n2{r3c9 r1c7} - n4{r1c7 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - r7c3{n9 .} ==> r7c7 <> 3
nrczt-whip-rc[5]  n4{r4c9 r1c9} - n6{r1c9 r3c9} - n2{r3c9 r8c9} - {n2 n6}r9c7 - {n6r7c7 .} ==> r4c7 <> 4, r6c7 <> 4
hidden-triplets-in-a-column {n2 n4 n6}{r7 r1 r9}c7 ==> r1c7 <> 8, r1c7 <> 7, r1c7 <> 3
nrczt-whip[21]  b4n2{r5c3 r5c1} - c1n6{r5 r3} - r3c9{n6 n2} - r8n2{c9 c3} - b7n1{r8c3 r9c2} - c2n2{r9 r1} - b1n5{r1c2 r2c3} - c3n3{r2 r6} - c3n1{r6 r4} - c3n4{r4 r7} - r9c1{n4 n9} - r9c4{n9 n5} - c8n5{r9 r5} - r5c2{n5 n8} - b7n8{r7c2 r8c1} - c1n3{r8 r2} - c7n3{r2 r8} - r8c5{n3 n1} - r2c5{n1 n8} - r6n8{c5 c7} - r6n1{c7 .} ==> r5c3 <> 6
nrczt-whip-rn[18]  n2{r7c5 r9c5} - n6{r9c5 r4c5} - n6{r4c3 r1c3} - n9{r1c3 r2c3} - n9{r2c9 r1c9} - n6{r1c9 r3c9} - n2{r3c9 r8c9} - n5{r8c9 r9c8} - r9c4{n5 n1} - r9c2{n1 n3} - n2{r9c2 r7c2} - n2{r3c2 r3c1} - n3{r3c1 r2c1} - r2c9{n3 n7} - n7{r3c8 r7c8} - n7{r8c9 r8c4} - n5{r8c4 r8c5} - r2n5{c5 .} ==> r7c5 <> 9
nrczt-whip-bn[17]  n9{r7c3 r7c4} - n9{r3c4 r3c5} - n9{r1c5 r1c9} - n6{r1c9 r3c9} - n2{r3c9 r8c9} - n5{r8c9 r9c8} - r9c4{n5 n1} - n1{r9c2 r8c3} - n2{r8c3 r5c3} - n3{r5c3 r6c3} - n5{r6c3 r4c3} - n5{r2c3 r2c5} - n5{r9c5 r8c4} - n7{r8c4 r8c7} - n3{r8c7 r2c7} - r2c9{n3 n7} - b6n7{r6c9 .} ==> r2c3 <> 9
nrc-chain[5]  r1c7{n4 n2} - r3c9{n2 n6} - n6{r1c9 r1c3} - n9{r1c3 r7c3} - r9c1{n9 n4} ==> r9c7 <> 4
nrct-chain[7]  r1c7{n4 n2} - n2{r9c7 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n6{r3c1 r1c3} - n9{r1c3 r2c1} - n9{r2c9 r1c9} ==> r1c9 <> 4
interaction column c9 with block b6 ==> r5c8 <> 4
nrczt-whip-cn[7]  n6{r3c1 r3c9} - n6{r1c9 r1c3} - n9{r1c3 r2c1} - n9{r2c9 r1c9} - n2{r1c9 r8c9} - n2{r8c3 r5c3} - c1n2{r5 .} ==> r3c1 <> 7, r3c1 <> 3
nrczt-whip-cn[7]  n6{r1c9 r1c3} - n6{r3c1 r3c9} - n2{r3c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n9{r3c1 r2c1} - c9n9{r2 .} ==> r1c9 <> 3
jellyfish-in-columns n3{c1 c9 c3 c7}{r8 r6 r5 r2} ==> r8c5 <> 3, r6c2 <> 3, r5c8 <> 3, r5c2 <> 3, r2c8 <> 3, r2c5 <> 3
nrczt-whip-cn[7]  n6{r1c9 r1c3} - n6{r3c1 r3c9} - n2{r3c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n9{r3c1 r2c1} - c9n9{r2 .} ==> r1c9 <> 7
nrczt-whip-rn[11]  n2{r5c3 r8c3} - n1{r8c3 r9c2} - r5c2{n1 n8} - n5{r5c2 r1c2} - n5{r2c3 r2c5} - n5{r6c5 r4c6} - n5{r4c9 r6c9} - r5c8{n5 n1} - n1{r5c6 r3c6} - n1{r3c4 r8c4} - r8n5{c4 .} ==> r5c3 <> 5
nrczt-whip-rn[13]  n2{r9c5 r7c5} - n6{r7c5 r4c5} - n6{r4c3 r1c3} - n9{r1c3 r7c3} - r9c1{n9 n4} - r9c8{n4 n3} - n5{r9c8 r5c8} - n5{r6c9 r8c9} - n2{r8c9 r9c7} - r9c2{n2 n1} - r5c2{n1 n8} - r7c2{n8 n3} - r8n3{c3 .} ==> r9c5 <> 5
nrczt-whip-cn[14]  n2{r7c5 r9c5} - n6{r9c5 r4c5} - n6{r4c3 r1c3} - n6{r3c1 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - n8{r5c4 r1c4} - n8{r8c4 r8c1} - n2{r8c1 r8c9} - r1c9{n2 n9} - n9{r1c5 r3c5} - n4{r3c5 r1c5} - c5n3{r1 .} ==> r7c5 <> 8
nrczt-whip-rn[14]  n4{r9c1 r7c3} - n9{r7c3 r1c3} - n6{r1c3 r3c1} - r3c9{n6 n2} - n2{r3c2 r1c2} - r1c7{n2 n4} - n4{r7c7 r9c8} - n5{r9c8 r5c8} - n5{r5c2 r6c2} - n7{r6c2 r4c1} - r4c9{n7 n4} - n4{r4c5 r3c5} - n9{r3c5 r3c4} - r7n9{c4 .} ==> r6c1 <> 4
nrczt-whip-rn[14]  n5{r2c3 r2c5} - n5{r6c5 r6c9} - n5{r8c9 r8c4} - n5{r9c6 r5c6} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - n4{r6c5 r6c3} - n1{r6c3 r8c3} - r8c5{n1 n8} - n8{r6c5 r4c6} - n8{r4c1 r6c1} - r6n3{c1 .} ==> r4c3 <> 5
nrczt-whip-bn[17]  n9{r7c4 r7c3} - n9{r1c3 r1c9} - n9{r2c9 r2c1} - n9{r9c1 r9c5} - n2{r9c5 r7c5} - n6{r7c5 r4c5} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - r2c9{n3 n7} - n7{r3c8 r7c8} - r7c4{n7 n8} - r7c2{n8 n3} - r8c3{n3 n1} - r8c5{n1 n5} - n5{r2c5 r2c3} - b1n3{r2c3 .} ==> r3c4 <> 9
nrczt-whip-rn[18]  n2{r9c5 r7c5} - n2{r7c7 r8c9} - r9c7{n2 n6} - n6{r9c5 r4c5} - n6{r4c3 r1c3} - r1c9{n6 n9} - n9{r2c9 r2c1} - r3c1{n9 n2} - n6{r3c1 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - r2c9{n3 n7} - n7{r3c8 r7c8} - r8c7{n7 n3} - n3{r2c7 r2c3} - n5{r2c3 r2c5} - n5{r8c5 r8c4} - r8n7{c4 .} ==> r9c2 <> 2
nrczt-whip-rc[7]  r9c2{n3 n1} - r8c3{n1 n2} - r7c2{n2 n8} - r5c2{n8 n5} - n5{r5c8 r9c8} - r8c9{n5 n7} - r8c7{n7 .} ==> r8c1 <> 3
nrczt-whip-rn[7]  n5{r9c8 r5c8} - n5{r6c9 r8c9} - n3{r8c9 r8c3} - r9c2{n3 n1} - r5c2{n1 n8} - n8{r7c2 r8c1} - r8n2{c1 .} ==> r9c8 <> 3
nrczt-whip-rc[10]  n2{r9c5 r9c7} - n6{r9c7 r9c6} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n2{r8c3 r8c1} - n8{r8c1 r7c2} - r7c6{n8 n7} - r7c4{n7 n9} - n9{r9c5 r9c1} - r3c1{n9 .} ==> r9c5 <> 3
nrczt-whip-cn[10]  n3{r9c2 r9c6} - n3{r7c6 r7c8} - n3{r7c5 r3c5} - n9{r3c5 r3c1} - r9c1{n9 n4} - n4{r9c8 r7c7} - n6{r7c7 r9c7} - n2{r9c7 r8c9} - n2{r8c3 r5c3} - c1n2{r5 .} ==> r1c2 <> 3
nrczt-whip-rn[11]  n2{r9c5 r7c5} - n6{r7c5 r4c5} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n2{r8c3 r8c1} - r3c1{n2 n9} - n9{r3c5 r1c5} - n3{r1c5 r3c5} - n4{r3c5 r6c5} - n4{r5c4 r5c9} - r5n3{c9 .} ==> r9c5 <> 1
nrczt-whip-bn[13]  n3{r9c6 r9c2} - n1{r9c2 r9c4} - r8c5{n1 n8} - n5{r8c5 r8c9} - n5{r4c9 r4c5} - n5{r5c4 r1c4} - n9{r1c4 r7c4} - n9{r7c3 r1c3} - n6{r1c3 r1c9} - r3c9{n6 n2} - r3c2{n2 n7} - n7{r3c6 r1c6} - b2n8{r1c6 .} ==> r9c6 <> 5
nrczt-whip-rc[11]  r9c2{n1 n3} - r9c6{n3 n6} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - r3c4{n4 n7} - n7{r8c4 r7c6} - n3{r7c6 r7c5} - n2{r7c5 r9c5} - r9c7{n2 .} ==> r9c4 <> 1
hidden-pairs-in-a-row r9{n1 n3}{c2 c6} ==> r9c6 <> 6, r9c5 <> 9
interaction column c5 with block b2 ==> r1c4 <> 9
nrczt-whip-cn[9]  n9{r3c5 r1c5} - n3{r1c5 r7c5} - n2{r7c5 r9c5} - n6{r9c5 r4c5} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - c5n4{r6 .} ==> r3c5 <> 1
nrczt-whip-cn[9]  n9{r1c5 r3c5} - n3{r3c5 r7c5} - n2{r7c5 r9c5} - n6{r9c5 r4c5} - n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - c5n4{r6 .} ==> r1c5 <> 8
nrczt-whip-rn[9]  n9{r1c5 r3c5} - n3{r3c5 r7c5} - n2{r7c5 r9c5} - n6{r9c5 r4c5} - n6{r4c3 r1c3} - r3c1{n6 n2} - r1c2{n2 n7} - r3c2{n7 n3} - r9n3{c2 .} ==> r1c5 <> 5
nrczt-whip-cn[12]  n2{r5c1 r5c3} - n4{r5c3 r7c3} - n4{r7c7 r1c7} - n4{r1c4 r3c4} - n4{r3c8 r9c8} - n5{r9c8 r5c8} - n5{r6c9 r8c9} - n2{r8c9 r8c1} - n8{r8c1 r7c2} - r5c2{n8 n1} - n1{r9c2 r8c3} - c4n1{r8 .} ==> r5c1 <> 4
nrczt-whip-rc[10]  n7{r6c2 r4c1} - n4{r4c1 r9c1} - n9{r9c1 r7c3} - r1c3{n9 n6} - n6{r4c3 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r4c9} - n5{r4c9 r8c9} - r9c8{n5 .} ==> r6c9 <> 7
nrczt-whip-bn[12]  r8c7{n3 n7} - n7{r4c7 r4c9} - r2c9{n7 n9} - r2c1{n9 n7} - n3{r2c1 r3c2} - r2c3{n3 n5} - r1c2{n5 n2} - n7{r1c2 r6c2} - n5{r6c2 r5c2} - n5{r5c9 r6c9} - n4{r6c9 r5c9} - b6n3{r5c9 .} ==> r2c7 <> 3
nrct-chain[6]  n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n3{r6c7 r8c7} - r8c3{n3 n1} - n1{r9c2 r9c6} ==> r5c6 <> 1
nrczt-whip-rn[9]  n9{r2c9 r2c1} - n3{r2c1 r2c3} - n5{r2c3 r2c5} - n5{r1c6 r1c2} - n7{r1c2 r3c2} - n7{r3c8 r7c8} - n7{r8c9 r8c4} - n5{r8c4 r9c4} - r9n9{c4 .} ==> r2c9 <> 7
nrct-chain[7]  r1c7{n4 n2} - r3c9{n2 n6} - r1c9{n6 n9} - r2c9{n9 n3} - n3{r3c8 r7c8} - n3{r7c6 r9c6} - n3{r1c6 r1c5} ==> r1c5 <> 4
nrczt-whip-cn[8]  n2{r5c3 r5c1} - n3{r5c1 r5c9} - n4{r5c9 r5c4} - n4{r1c4 r3c5} - n9{r3c5 r1c5} - n3{r1c5 r7c5} - r9c6{n3 n1} - c2n1{r9 .} ==> r5c3 <> 1
nrct-chain[9]  n6{r5c6 r5c1} - n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - n4{r1c4 r3c5} - n9{r3c5 r1c5} - n3{r1c5 r7c5} - n2{r7c5 r9c5} - n6{r9c5 r4c5} ==> r4c6 <> 6
nrczt-whip-rn[9]  n2{r5c1 r5c3} - n3{r5c3 r5c9} - n4{r5c9 r5c4} - n4{r1c4 r3c5} - n9{r3c5 r1c5} - n3{r1c5 r7c5} - n2{r7c5 r9c5} - n6{r9c5 r4c5} - r5n6{c6 .} ==> r5c1 <> 8
nrczt-whip-rc[10]  n6{r5c6 r4c5} - r9c5{n6 n2} - r7c5{n2 n3} - r9c6{n3 n1} - n1{r9c2 r8c3} - n1{r4c3 r4c7} - r5c8{n1 n5} - r9c8{n5 n4} - n4{r9c1 r7c3} - r4c3{n4 .} ==> r5c6 <> 8
nrczt-whip-rc[8]  n4{r4c1 r9c1} - n9{r9c1 r7c3} - r1c3{n9 n6} - n6{r4c3 r5c1} - r5c6{n6 n5} - r4c6{n5 n1} - n1{r9c6 r9c2} - r5c2{n1 .} ==> r4c1 <> 8
nrczt-whip-cn[11]  n7{r2c8 r2c1} - n9{r2c1 r2c9} - n3{r2c9 r3c8} - r7c8{n3 n4} - n4{r9c8 r9c1} - r4c1{n4 n6} - n6{r4c3 r1c3} - n9{r1c3 r1c5} - n3{r1c5 r7c5} - n6{r7c5 r9c5} - c5n2{r9 .} ==> r1c8 <> 7
nrczt-whip-rn[10]  n3{r9c6 r9c2} - n1{r9c2 r9c6} - r3c6{n1 n7} - r3c2{n7 n2} - r7c2{n2 n8} - n8{r7c6 r4c6} - n8{r5c4 r5c8} - n8{r1c8 r1c4} - n5{r1c4 r1c2} - r1n7{c2 .} ==> r1c6 <> 3
nrczt-whip-rn[11]  n5{r9c8 r9c4} - n9{r9c4 r9c1} - n9{r3c1 r3c5} - r1c5{n9 n3} - r1c8{n3 n8} - n4{r1c8 r1c7} - r1c4{n4 n7} - n7{r8c4 r7c6} - r7c8{n7 n3} - n3{r3c8 r2c9} - r2n9{c9 .} ==> r9c8 <> 4
naked and hidden singles ==> r9c8 = 5, r9c4 = 9, r9c1 = 4, r7c3 = 9, r1c3 = 6, r3c9 = 6
interaction row r3 with block b1 ==> r1c2 <> 2
nrc-chain[3]  n3{r1c8 r1c5} - n9{r1c5 r1c9} - r2c9{n9 n3} ==> r3c8 <> 3
xyt-chain[4]  r1c2{n7 n5} - r2c3{n5 n3} - r2c9{n3 n9} - r2c1{n9 n7} ==> r3c2 <> 7
nrc-chain[5]  n3{r7c8 r1c8} - r1c5{n3 n9} - r1c9{n9 n2} - r1c7{n2 n4} - n4{r7c7 r7c8} ==> r7c8 <> 7
interaction column c8 with block b3 ==> r2c7 <> 7
interaction row r7 with block b8 ==> r8c4 <> 7
nrczt-whip-bn[6]  n7{r6c2 r1c2} - n5{r1c2 r5c2} - r5c6{n5 n6} - n6{r5c1 r4c1} - n7{r4c1 r6c1} - b4n8{r6c1 .} ==> r6c2 <> 1
nrczt-whip-rn[5]  n8{r4c5 r4c7} - r2c7{n8 n1} - r2c5{n1 n5} - n5{r2c3 r6c3} - r6n1{c3 .} ==> r6c5 <> 8
nrct-chain[6]  r9c6{n3 n1} - n1{r9c2 r5c2} - n1{r5c4 r3c4} - n1{r3c8 r2c8} - n7{r2c8 r3c8} - r3c6{n7 n3} ==> r7c6 <> 3
nrct-chain[7]  n3{r9c6 r7c5} - n2{r7c5 r9c5} - n6{r9c5 r7c6} - n6{r5c6 r5c1} - r4c1{n6 n7} - n7{r6c2 r1c2} - n7{r1c6 r3c6} ==> r3c6 <> 3
naked and hidden singles ==> r9c6 = 3, r9c2 = 1
naked-pairs-in-a-block b8{r7c5 r9c5}{n2 n6} ==> r7c6 <> 6
hidden-singles ==> r5c6 = 6, r4c1 = 6
interaction row r4 with block b6 ==> r6c7 <> 7
naked-pairs-in-a-block b8{r7c4 r7c6}{n7 n8} ==> r8c5 <> 8, r8c4 <> 8
hidden-single-in-a-row ==> r8c1 = 8
hidden-pairs-in-a-column c5{n3 n9}{r1 r3} ==> r3c5 <> 4
interaction column c5 with block b5 ==> r5c4 <> 4
nrc-chain[2]  n1{r3c6 r4c6} - n1{r5c4 r5c8} ==> r3c8 <> 1
interaction row r3 with block b2 ==> r2c5 <> 1
naked-triplets-in-a-row r5{c2 c4 c8}{n8 n5 n1} ==> r5c9 <> 5
nrc-chain[2]  n5{r1c6 r4c6} - n5{r5c4 r5c2} ==> r1c2 <> 5
singles
GRID 5287 SOLVED. LEVEL = NRCZT21, MOST COMPLEX RULE = NRCZT21
176435289
945682173
238197546
694351827
352876914
781249365
529768431
863514792
417923658



Remember however that all the examples found in collections of 10,000,000 randomly generated minimal puzzles (by various random generators) can be solved with much shorter whips.

Indeed, this puzzle is very exceptional in many respects.
If nrczt-braids are allowed, the whip[21] is not necessary. Instead a much shorter braid appears at the same place in the resolution path (and the previous whips[11] are replaced by braids[7]):
interaction column c9 with block b3 ==> r1c7 <> 6
hidden-pairs-in-a-block {n4 n9}{r7c3 r9c1} ==> r9c1 <> 3, r9c1 <> 2, r7c3 <> 3, r7c3 <> 2
nrczt-whip-bn[5]  n6{r1c3 r3c1} - n6{r3c9 r1c9} - n2{r1c9 r3c9} - n9{r3c9 r2c9} - {n9r2c3 .} ==> r1c3 <> 2
nrczt-whip-rn[5]  n6{r3c9 r1c9} - n2{r1c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - {n6r3c1 .} ==> r3c9 <> 9
nrct-chain[4]  n6{r1c3 r3c1} - n6{r3c9 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} ==> r1c3 <> 5, r1c3 <> 3
nrczt-whip-rn[5]  n6{r3c9 r1c9} - n2{r1c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - {n6r3c1 .} ==> r3c9 <> 7, r3c9 <> 4, r3c9 <> 3
nrczt-whip-rn[6]  n2{r8c3 r5c3} - n2{r5c1 r3c1} - n2{r3c9 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - {n6r1c3 .} ==> r8c7 <> 2, r8c5 <> 2
nrczt-whip-rn[6]  n2{r5c3 r8c3} - n2{r8c1 r3c1} - n2{r3c9 r1c9} - n9{r1c9 r2c9} - n9{r2c3 r1c3} - {n6r1c3 .} ==> r5c2 <> 2
nrczt-whip-cn[6]  n9{r2c9 r1c9} - n9{r1c3 r3c1} - n6{r3c1 r3c9} - n2{r3c9 r8c9} - n2{r8c3 r5c3} - {n2r5c1 .} ==> r2c5 <> 9
;;; diverges here from nrczt-whips resolution path ...
nrczt-braid-bn[7]  n6{r1c9 r1c3} - n6{r3c1 r3c9} - n2{r3c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n9{r1c9 r2c9} - {n9r3c1 .} ==> r1c9 <> 3, r1c9 <> 4
interaction column c9 with block b6 for number 4 ==> r6c7 <> 4, r5c8 <> 4, r4c7 <> 4
hidden-triplets-in-a-column {n2 n4 n6}{r9 r1 r7}c7 ==> r9c7 <> 3, r7c7 <> 7, r7c7 <> 3, r1c7 <> 8, r1c7 <> 7, r1c7 <> 3
;;; but same situation  here. Now really diverges:
nrczt-braid-bn[7]  n6{r1c9 r1c3} - n6{r3c1 r3c9} - n2{r3c9 r8c9} - n2{r8c3 r5c3} - n2{r5c1 r3c1} - n9{r1c9 r2c9} - {n9r3c1 .} ==> r1c9 <> 7
.....




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