## The Hidden Logic of Sudoku## Denis Berthier## Online Supplements## Braids |

© 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) NRCZT-BRAIDSOne can define nrczt-nets by generalising nrczt- chains and lassos (or nrczt-whips, in the new perspective) with the possibility of having several right-linking candidates in the same rc-, rn-, cn- or bn- cell as a left-linking one. These nets include grouped nrczt-chains, which are a relatively mild special case. But the most general nrczt-nets are very hard to use, because their structure is complex and the conditions of the associated elimination theorem are very hard to check. In this page, I introduce nrczt-braids, a much milder kind of nets, with an easier elimination theorem. Informally, given a target candidate z, an nrczt-braid built on z is, as an nrczt-whip, a sequence (i.e. a linearly ordered set) of candidates, alternatively called left-linking and right-linking. Its precise definition is obtained from that of an nrczt-whip by merely changing the condition "any left-linking candidate is nrc-linked to the previous right-linking one" by "any left-linking candidate is nrc-linked to a previous right-linking one or to the target". As any nrczt-whip, an nrczt-braid can also be considered as a sequence of rc, rn, cn or bn cells. And, still as any nrczt-whip, an nrczt-braid produces a contradiction on its target when its last cell has no right-linking candidate compatible the previous ones and with the target. The condition that we have a sequence of candidates is the trace of the idea that, in a chain, we are following "a single line of reasoning". Now the formal definition. Remember that: 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: an nrczt-braid of length n for a target z is a sequence of candidates L1, R1, L2, R2, .... Ln-1, Rn-1, Ln, alternatively called left-linking and right-linking, such that: - for any 1<= k <= n, Lk is nrc-linked to the target or to a previous right-linking candidate (some Ri, for i < k); this is the first part of the braid condition and the only difference with whips, - for each 1 <= k < n, there is one of the rc- (resp. rn-, cn- or bn-) cells containing Lk such that Rk is in this cell and it is the only candidate in this cell compatible with the target and the previous right-linking candidates; this is the second part of the braid 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. Notice that, as nrczt-whips, nrczt-braids still have one and only one right-linking candidate per cell but the last one, but they may have several left-linking candidates branching off the target or off any right-linking candidate - with the restrictive condition that the set of candidates must be totally ordered. Their branching structure is thus severely restricted. There are therefore two complementary views of nrczt-braids: sequential (a sequence of candidates) or as a net with a restricted form of branching. Both of them are useful. The following definitions are used: - an additional z-candidate in a cell is defined as usual: it is nrc-linked to the target); - an additional t-candidate in a cell is defined as usual: it is nrc-linked to a previous right-linking candidate (one could add: in the same branch OR in any other branch; but such a distinction is pointless as "previous" refers to the total ordering of the right-linking candidates induced by the total ordering of all the candidates; - a terminal right-linking candidate is one which sprouts no left linking candidate. Pruning useless parts of branches: we may add the condition that the any terminal right-linking candidate is nrc-linked to a t-candidate in a cell after his own (in the global ordering). Otherwise, its cell can merely be eliminated from the nrczt-braid, thus leaving a smaller one with the same target. Doing this as many times as necessary, we can reach an nrczt-braid with no useless cell; we'll call this a minimal nrczt-braid. Unless otherwise stated, all nrczt-braids will be minimal (but this is only for efficiency reasons). Any branch of an nrczt-braid starting from the first llc and ending with the last llc is thus almost an nrczt-whip; it only has a few additional candidates (other than the usual t- or z-) in some of its cells, candidates which must be dealt with by considering additional branches. (Notice that this is nevertheless very different from accepting to take left-linking candidates among already deleted candidates: when we progressively build our nrczt-braid, we don't have to consider all the missing candidates anywhere in the grid as potential left-linking ones for the next cell. We can only consider those that are nrc-linked to an already chosen previous right-linking candidate. Said otherwise, this definition sticks to the idea that deleted candidates play no role in the rules.) It is then easy to prove the nrczt-braid elimination theorem: given an nrczt-braid, its target can be eliminated. Proof: exactly the same as for nrczt-whips, following the total ordering of the candidates, until the last left-linking candidate, with no compatible right-linking one, is reached. The proof shows progressively that, if the target was true, then all the left-linking candidates would be false and all the right-linking candidates would be true. When the final Ln is reached, we get a cell with no possible candidate, which contradicts the possibility of the target being true. Finally, it may be useful to give an elementary example of something which cannot be the beginning of any nrczt-braid (it is impossible to define any total ordering of the candidates, compatible with the above definition): / {llc1 rlc1 t1#rlc2} / z \ \ {llc2 rlc2 t2#rlc1} each of the 2 lines represents an rc, rn, cn or bn cell, with t2 an additional candidate nrc-linked to rlc1 and t1 an additional candidate nrc-linked to rlc2. From z true, there is no possibility of concluding that rlc1 is true or that rlc2 is true: one can only conclude (rlc1 & rlc2) or (t1 & t2) This example is also an indication on how nrczt-braids could be generalised. But before generalising them, there's a lot more to say about them. Remarks: - As for any chain, lasso or whip in the xy-to-nrczt family, the additional t- or z- candidates are not part of the nrczt-braids. This is not an arbitrary convention; it has practical consequences: if I notice a chain 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 chain. This is extensively used in SudoRules. - More complex types of braids - braids(FP) - are defined in the "zt-ing principle" page. |

## 2) NRCZT-BRAID THEORIESAs in the case of whips, one can define a hierarchy of resolution theories based on braids: Definition: let Bn be the following sets of resolution rules: pB-NRCZT0: only elementary constraints propagation (no unsolved puzzle can be solved here) pB-NRCZT1_0: Naked and Hidden Singles pB-NRCZT1: elementary interactions between rows/columns and blocks (equivalent to nrczt-whips[1] or to nrczt-braids[1] - see subsumption page) pB-NRCZTn: nrczt-braids[n], for n>1, where n is the length of the braid. Let Tn be the set of resolution rules at levels from 0 to n, e.g. Tn = pB-NRCZT0 + pB-NRCZT1_0 + pB-NRCZT1 + ... pB-NRCZTn. The Tn form an increasing sequence of resolution theories. One can also define the pB-NRCZT rating of a puzzle as the smallest B-NRCZT theory which can solve it. One can also define the ("not pure") B-NRCZT rating, by allowing Subset rules at levels 2, 3, 4, in a way similar to what was done for whips. Theorem: All the pB-NRCZTn theories have the confluence property. This is proven here. Theorem: All the B-NRCZTn theories have the confluence property. An easy consequence of the previous theorem. |

## 3) ELIMINABLE BY AN NRCZT-BRAID <=> ELIMINABLE BY T&E(ECP+NS+HS)We now give a precise answer to the question: what is the solving potential of nrczt-braids? Given any resolution theory, such a question is generally very hard to answer. E.g., if we considered whips instead of braids, we would have no answer. First, I need non ambiguous definitions. And for this, you must remember my distinction between a resolution rule and a resolution technique. Definition: Given a resolution theory T and a candidate z, T&E(T, z) is the following resolution technique: - start a new grid by copying the current PM; add z to this new grid as a decided value; apply to this new PM all the rules in T; - if a contradiction is obtained in this new PM, then delete z from the original one; if no contradiction is obtained, then merely go back to the original PM. I think this is what everyone means by "using T&E(T) to eliminate candidate z". Notice that no "guessing" is allowed. Definition: Given a resolution theory T, T&E(T) is the following resolution technique (this is only conceptual, any implementation should optimise this): - a) define phase = 1; - b) apply all the rules in T; - c) set changed-PM = false; mark all the remaining candidates as "not tried"; - d) if there is at least one "not-tried" candidate, then choose any one of them, say z; apply T&E(z); if it eliminates z, then set changed-PM = true and apply all the rules in T, otherwise un-mark z; in both cases, goto d; - e) if there is no not-tried candidate: ----------- if changed-PM = true, then set phase = phase +1 and goto c); ----------- if changed-PM = false, then stop. Notice that this procedure allows considering the same candidate z several times. This is reasonable, because any elimination on another candidate z' by T&E(z') may entail a new possibility of a contradiction appearing in a new application of T&E(z). Definition: a puzzle P is solvable by T&E(T) if the above procedure leads to a solution. We'll be interested mainly in T= ECP+NS+HS (with ECP = the rules for the elementary constraints propagation). Notation: T&E designates the simpler T&E(ECP+NS+HS) Theorem: Any elimination that can be done by an nrczt-braid with target z can be done by T&E(ECP+NS+HS, z). Corollary: Any puzzle that can be solved by with only nrczt-braids (+ of course ECP, NS and HS) can be solved by T&E(ECP+NS+HS). Proof: obvious. What is interesting is that the converse is true: Theorem: Any elimination of a candidate z that can be done by ordinary Trial-and-Error, i.e. by T&E(ECP+NS+HS, z), can be done by an nrczt-braid with target z. Corollary: Any puzzle that can be solved by ordinary Trial-and-Error, i.e. by T&E(ECP+NS+HS), can be solved by nrczt-braids. Notice that puzzles that can't be solved by T&E(ECP+NS+HS) are extremely rare, as will be shown in the sequel. Proof: We shall show that any elimination done by T&E, at any step in the resolution process, could have been done by an nrczt-braid. Consider the PM generated by the T&E hypothesis z=n0r0c0 and take n0r0c0 as the target of the nrczt-braid we're going to build. In the auxiliary PM generated by the T&E hypothesis, the T&E procedure is a well defined sequence of deletions of candidates based on ECP and of assertions of values based on NS or HS, until a contradiction is reached (if one is reached). 1) Consider the first step of T&E that is not an elimination by ECP. Suppose first that the assertion made by this step relies on NS: say r'c'=n' If this assertion can be made, it can only be because: - n'r'c' is not nrc-linked to n0r0c0 (otherwise it would have been eliminated by ECP); - all the other candidates for cell r'c' have been eliminated by the assertion of n0r0c0, which supposes that they are nrc-linked to n0r0c0, because only ECP can produce eliminations. Take any of the candidates in cell r'c', different from n'r'c', say n1r'c'. Then {n1r'c' n'r'c'} is the first cell of our nrczt-braid (and all the other candidates in cell r'c' are z-candidates). The same reasoning applies if this step is an assertion by HS(row): instead of considering the rc-cell r'c', we consider the rn-cell r'n'. The same reasoning applies if this step is an assertion by HS(col): instead of considering the rc-cell r'c', we consider the cn-cell c'n'. The same reasoning applies if this step is an assertion by HS(blk): instead of considering the rc-cell r'c', we consider the bn-cell b'n'. 2) The sequel is done by recursion. If we have been able to replace the T&E procedure upto the nth assertion step, then we can replace it by a longer nrczt-braid upto the (n+1)th assertion step. Name n'r'c' the nth candidate asserted. Suppose first that the (n+1)th assertion made by T&E relies on NS: say r'c'=n' If this assertion can be made, it can only be because: - n'r'c' is not nrc-linked to n0r0c0 or to any of the previous candidates asserted in any of the previous steps, i.e. to any of the right-linking candidates of the partial nrczt-braid we've already built (otherwise it would have been eliminated by ECP); - all the other candidates for cell r'c' have been eliminated by the assertions of n0r0c0 and of all the previously asserted candidates, which supposes that each of them is individually nrc-linked to the target or to some of the previous right-linking candidates; - there is a candidate n1r'c' in cell r'c' which is nrc-linked to n0r0c0 OR to a previous right-linking candidate n2r2c2. Take {n1r'c' n'r'c'} as the next cell of our braid. NOTICE THAT IT WOULD BLOCK HERE IF WE CONSIDERED WHIPS INSTEAD OF BRAIDS. The new cell of our nrczt-braid, {n1r'c' n'r'c'}, is appended to the right of n2r2c2 (all the other candidates in cell r'c' are z- or t- candidates with respect to already existing branches of the net). The same reasoning applies if this step is an assertion by HS(row): instead of considering the rc-cell r'c', we consider the rn-cell r'n'. The same reasoning applies if this step is an assertion by HS(col): instead of considering the rc-cell r'c', we consider the cn-cell c'n'. The same reasoning applies if this step is an assertion by HS(blk): instead of considering the rc-cell r'c', we consider the bn-cell b'n'. 3) End: a contradiction is detected by T&E when a cell (rc-, rn-, cn- or bn-) has no candidate left. Suppose it is an rc-cell. How can that happen? Only via an ECP step. But it means then that the last asserted value is nrc-linked to a previous right-linking candidate or to the target, i.e. we have an nrczt-braid. q.e.d. Remark: we have thus solved a problem that had been pending for years: for the first time, we have a set of resolution rules (the set of nrczt-braid[n] rules, for any length n, together with ECP, NS and HS) which is equivalent to T&E. Remark: If we add recursion in our procedure (but still no guessing), we get: Theorem: T&E(NS+HS, n) = T&E(B-NRCZT, n-1) Said otherwise, any puzzle that can be solved by ordinary T&E (i.e. using only NS and HS to prune the search) with at most n hypotheses, can be solved with only n-1 hypotheses if rules from B-NRCZT are used to prune the search. More formally: B-NRCZT is defined as for the other resolution theories I've already introduced: BSRT = {ECP, NS} L1_0 = BSRT + HS B-NRCZT1 = L1_0 + B-NRCZT1 = NRCZT1 B-NRCZTn+1 = B-NRCZTn + rule for nrczt-braids of length n. B-NRCZT = union of all the B-NRCZTn {= B-NRCZT729, as no braid can have a longer length} For any resolution theory T, T&E(T, n) is to be understood as the extension of the T&E(T) procedure defined above, obtained by allowing it to generate auxiliary grids at any depth ≤ n. Proof: this is an obvious corollary of the previous theorem. It is also obvious that other theorems can be obtained by projecting everything into the rc-space (or any of the rn, cn or bn spaces), e.g. Theorem: T&E(NS, n) = T&E(B-XYZT, n-1) where an xyzt-braid of length k is defined in an obvious way as the 2D counterpart of an nrczt-braid of length k, in rc-space. |

## 4) THE SCOPE OF NRCZT-BRAIDS (CONCRETE RESULTS)There are several ways one can use the theorem saying that nrczt-braids make the same eliminations and can solve the same puzzles as ordinary T&E (with no guessing). As a player, if we eliminate a candidate by T&E in an auxiliary grid where we use only ECP, NS and HS, then we know that we could justify this elimination by an nrczt-braid. Of course, knowing that something is possible is not the same thing as seeing it done in real life. Otherwise, why should we solve puzzles: for all the puzzles we try to solve, we generally know in advance that they have a solution. The equivalence theorem can also be used to answer the following question: what is the scope of nrczt-braids? In this case, we only want to know whether a puzzle is solvable by braids, without knowing exactly by which braids or at what level it is in the B-NRCZT rating (i.e. the length of the longest braids necessary to solve it). In this case, we can merely answer "all the puzzles that can be solved by T&E" and use this much faster procedure in all our computations. That's what I've done below. 4.1) Random collections All the 10,000 minimal puzzles in the sudogen0 collection randomly generated by suexg are solvable by braids; that's no fresh news, as they were already all solvable by the simpler nrczt-whips (chains and lassos if you prefer the old style). I've therefore tested 1,000,000 minimal puzzles generated with the suexg generator: all of them are solvable by nrczt-braids. Open question: are all of them solvable by nrczt-whips (as is the case for the first 10,000)? The answer is now positive: see the classification page. 4.2) Relationship with the Sudoku Explainer rating (SER) We know that puzzles with high SER are very rare and are difficult to obtain with a random generator. I therefore tried another collection biased towards high SER. We already knew that nrczt-whips can solve almost any puzzle upto SER 9.3 and some puzzles at SER 9.4. With braids, we can go a little further. Computations on gsf's collection of 8152 hardest show that: - nrczt-braids can solve almost any puzzle in it upto SER 9.8; - the proportion of solvable puzzles decreases as SER increases upto 10.2. Additional results for this collection can be found below. 4.3) Experimental results for T&E(B-NRCZT) I've already shown that almost all the puzzles can be solved by ordinary T&R or, equivalently, by braids. Let's now concentrate on the hardest. Using the equivalence between T&E(B-NRCZT) and T&E(ECP+NS+HS, 2) [i.e. T&E at depth 2 pruned by only ECP, NS and HS], which is an obvious corollary of my T&E theorem, I obtain the following experimental results. All the known puzzles can be solved by T&E(B-NRCZT), i.e. by T&E at depth one (a single hypothesis at a time) pruned by rules for nrczt-braids. Some time ago, I conjectured that any puzzle could be solved at depth one of T&E if nrczt-whips were used to prune the search; and I showed that this was true for EasterMonster. I still don't know whether this conjecture is true. But if we take braids instead of whips, then it becomes experimentally true (I have no formal proof though). This result can be formulated in a different, though equivalent, form: Any of the known puzzles can be solved by ECP+NS+HS or by ordinary T&E or by ordinary T&E iterated to depth 2 (i.e. considering 2 hypotheses at a time). Notice that I have no formal proof that there can't be a puzzle that requires T&E at depth 3. But if such cases existed, they would be utterly rare. 4.4) Backdoor-size vs depth of T&E From the above results, all the known puzzles can be solved with T&E(ECP+NS+HS, d) where: - d = 0, no T&E <==> solvable by NS+HS - d = 1, one hypothesis at a time <==> solvable by nrczt-braids - d = 2, two hypotheses at a time <==> solvable by T&E(nrczt-braids), i.e. with only one level of T&E pruned by nrczt-braids. Two intrinsic constants can be associated with any puzzle: - its (NS+HS)-backdoor size b: b= 0, 1, 2 or 3 for all the known puzzles; - the depth of T&E(ECP+NS+HS) necessary to solve it: d= 0, 1 or 2 for all the known puzzles. A natural question is: is there a relationship between b and d? The answer is not obvious because the backdoor-size is based on guessing (without the constraint of having to justify the elimination of other candidates in the same rc- rn- cn- or bn- cells as the backdoors), whereas depth of T&E is based on proving contradictions in hypotheses. None of the relations d ≤ b or b ≤ d is obvious. d ≤ b? If a puzzle can be solved by T&E at depth d, it doesn't mean that one can choose d hypotheses to generate all the auxiliary grids necessary to the T&E procedure. But I currently have no counter-example to d ≤ b. Notice that, if we consider gsf's FN-1 list of 1,183 puzzles with NS+HS backdoor-size 1 (b=1) or his list FN-2 of 28,948 puzzles with NS+HS backdoor-size 2 (b=2), all of them can be solved by ordinary T&E (d=1). For b ≤ d, it is easy to find counter-examples. If we consider gsf's list FN-2 of 28,948 puzzles with (NS+HS) backdoor-size 2 (b=2), all of them can be solved by ordinary T&E (d=1). Even for b ≤ d+1, there are counter-examples. If we consider gsf's list of 14 puzzles with (NS+HS) backdoor-size 3 (b=3) (AFAIK, the only such puzzles known as of today), with their names taken from gsf's hardest list and their SER (new version) computed: 100000002090400050006000700050903000000070000000850040700000600030009080002000001 Easter-Monster SER= 11.6 900000005040300060002000100080740000000020000000806070100000900030007040005000002 tarek-ultra-0300 SER= 11.3 700000004020600010005000800030910000000050000000203090800000700060009020004000005 tarek-ultra-0301 SER= 11.3 100000089000009102000000400007600000030040000900002005004070000500008010060300000 tarek-4/08 SER= 11.5 100000002003400050060000700000890040000306000009040000020000100700000006005080030 jpf-04/14/08 SER= 11.2 500000003020600010008000900040701000000030000000420070900000500010007020003000008 tarek-ultra-0302 SER= 11.2 100000002003400050060000700000050040000301000008940000020000100700000006005090030 jpf-04-10 SER= 11.2 100000006020500040003000700040850000000010000000024080007000300050009020600000001 coloin SER= 11.3 100000006020500040003000700040890000000204000000015080007000300050009020600000001 coloin-05/11/01 SER= 11.4 001000200030000040500030006000107000040000080000902000300000008060050030002000700 ocean-2007-05-29-1 SER= 9.4 300000200000540000000600000102003000000000064800000000090700050000000108050060000 gfroyle-2007-05-30-4 SER= 3.6 080090000300000060000300040000010005002000900007000800650000100000207000000004000 gfroyle-2007-05-30-3 SER= 4.2 000020580040300000010000000000600071500000200000400000200059000000000306700000000 gfroyle-2007-05-30-2 SER= 5.7 000500001308000000400003000000610000900000800000050000060700020010000300000000490 gfroyle-2007-05-30-1 SER= 6.6 then four of them can be solved by ordinary T&E (d=1): #10 (SER= 9.4), #11 (SER=3.6), #12 (SER= 4.2), #13 (SER= 5.7) and #14 (SER= 6.6). Remark: The scope of more complex zt-braids(FP) is studied in the "zt-principle" page. |

## 5) EXAMPLES OF NRCZT-BRAIDS5.1) First examples WARNINGS: As almost any puzzle with SER ≤ 9.3 can be solved with nrczt-whips (i.e. chains or lassos) and as I'm very reluctant to rely on nets (even braids, which are a very special form of nets) when a chain is available, any interesting example of a braid will be obtained for a puzzle with SER > 9.3. As a result, the solution to such a puzzle will require long chains and braids and it will seem very complex. In the following, one should therefore not forget that any such puzzle is likely to be beyond normal human solving. I'm just giving an example of braids, I'm not stating that such complex puzzles should be proposed for human solving. This was before nrczt-braids were coded in SudoRules (and also before chains and lassos were replaced by whips). This is superseded by the next examples, produced by SudoRules after these changes. But I keep it here to illustrate the difference between having any solution using braids and having a solution using the shortest available braids. How did I proceed for this example? As rules for braids were not yet coded in SudoRules, I used the following procedure (half manual): Input puzzle Loop until solution found: Run SudoRules from the current state with the usual rules for chains and lassos. When no rule is applicable, make ONE elimination with a braid. End loop Contrary to the solutions I usually give with chains, I can't guarantee here that the braids used are the shortest ones available. (The contrary is very likely, as I didn't even try to optimise their length.) The only purpose is to show a few cases of what a braid looks like. The puzzle I shall use is #3263 in gsf's list of 8152 hardest: 20627, 094, 0640, 100800002003400050060005700000090040000006000009040000020000100700000006005080030, gsf-2007-05-24-0753, 0, 49.00s, C21.m/F10135.16143/N12760.27297/P3.33.6422.13.22.20618.18.4.2.230/M2.69.190/V2, C21.m/F15.57/N10.22/B8.18.18/H2.4.2/X2.3/Y1.30/K1.1.8.0.0.1/O1.1/G11.0.1/M1.27.1 100800002 003400050 060005700 000090040 000006000 009040000 020000100 700000006 005080030 ***** SudoRules version 13.7w ***** 100800002003400050060005700000090040000006000009040000020000100700000006005080030 hidden-single-in-a-row ==> r1c2 = 5 interaction column c4 with block b8 for number 6 ==> r7c5 <> 6 hidden-pairs-in-a-block {n3 n4}{r1c7 r3c9} ==> r3c9 <> 9, r3c9 <> 8, r3c9 <> 1, r1c7 <> 9, r1c7 <> 6 At this point, the PM is: +-----------------------------+-------------------------------+-------------------------------+ | 1 5 47 | 8 367 379 | 34 69 2 | | 289 789 3 | 4 1267 1279 | 689 5 189 | | 2489 6 248 | 1239 123 5 | 7 189 34 | +-----------------------------+-------------------------------+-------------------------------+ | 23568 1378 12678 | 12357 9 12378 | 23568 4 1357 | | 23458 13478 12478 | 12357 12357 6 | 23589 12789 135789 | | 23568 1378 9 | 12357 4 12378 | 23568 12678 13578 | +-----------------------------+-------------------------------+-------------------------------+ | 34689 2 468 | 35679 357 3479 | 1 789 45789 | | 7 13489 148 | 12359 1235 12349 | 24589 289 6 | | 469 149 5 | 12679 8 12479 | 249 3 479 | +-----------------------------+-------------------------------+-------------------------------+ Now comes a special case of an nrczt-braid, in rc-space: an yxzt-braid. For easier reading, all the cells are numbered: C1 to C18. Each branch is written in a different line. The cell of the branching points (always an rlc or *) are recalled before the links (as C5 in "C5 ------- C8:{n3 n7 n6#1}r1c5"); "*" is the target. As usual, #k after a candidate means it is justified by the rlc of cell Ck; * means it is justified by the target. Remember that the ordering of the candidates is essential and that, in a braid, any t-candidate is still justified by the target or a PREVIOUS right-linking candidate (rlc) with respect to this ordering. xyzt-braid[18] * ------- C1:{n9 n6}r1c8 - C2:{n6 n8 n9*}r2c7 - C3:{n8 n7 n9*}r2c2 - C4:{n7 n4}r1c3 - C5:{n4 n3}r1c7 - C6:{n3 n4}r3c9 - C7:{n4 n7 n9*}r9c9 - C5 ------- C8:{n3 n7 n6#1}r1c5 - * ------- C9:{n9 n2 n8#2}r2c1 - C10:{n2 n8 n4#6}r3c3 - C11:{n8 n1 n4#4}r8c3 - C12:{n4 n6 n8#10}r7c3 - C10 ------- C13:{n8 n9 n2#9 n4#6}r3c1 - C13:{n9 n4 n6#12}r9c1 - C15:{n4 n9 n1#11}r9c2 - C16:{n9 n2 n4#14}r9c7 - C17:{n2 n1 n4#14 n7#7 n9#15}r9c6 - C18{n1 . n2#9 n6#1 n7#8}r2c6 ==> r2c9 <> 9 As in any chain, the llc of C1 is nrc-linked to the target. But, contrary to a chain: - the left-linking candidate of C9 is linked to the target instead of to the right-linking candidate of C8, - the left-linking candidate of C8 is linked to the right-linking candidate of C5 instead of to the right-linking candidate of C7, - the left-linking candidate of C13 is linked to the right-linking candidate of C10 instead of to the right-linking candidate of C12. nrczt-whip-rn[11] n9{r1c8 r2c7} - n9{r2c2 r9c2} - n9{r7c1 r3c1} - n9{r3c4 r7c4} - n6{r7c4 r9c4} - n1{r9c4 r9c6} - n7{r9c6 r9c9} - {n7 n8}r7c8 - n8{r3c8 r2c9} - {n6 n1}r2c5 - {n6r2c5 .} ==> r8c8 <> 9 Alternative path: the above braid and whip could have been replaced by the two whips below: nrczt-rl-lasso[12] n9{r1c8 r1c6} - n9{r3c4 r3c1} - n9{r2c2 r9c2} - n9{r9c4 r7c4} - n6{r7c4 r9c4} - n1{r9c4 r9c6} - n7{r9c6 r9c9} - {n7 n8}r7c8 - n8{r3c8 r3c3} - n2{r3c3 r2c1} - {n2 n7}r2c6 - {n7 n9}r2c2 ==> r8c8 <> 9 nrczt-rl-lasso[14] n1{r2c9 r3c8} - n8{r3c8 r2c7} - n6{r2c7 r1c8} - n9{r1c8 r1c6} - n9{r3c4 r3c1} - n8{r3c1 r3c3} - n4{r3c3 r1c3} - {n4 n6}r7c3 - n6{r7c4 r9c4} - {n6 n4}r9c1 - {n4 n1}r8c3 - n1{r9c2 r9c6} - n1{r2c6 r2c5} - n6{r2c5 r2c7} ==> r2c9 <> 9 But I wanted to give an example of an xyzt-braid. End alternative path. At this point, the PM is: +-----------------------------+-------------------------------+-------------------------------+ | 1 5 47 | 8 367 379 | 34 69 2 | | 289 789 3 | 4 1267 1279 | 689 5 18 | | 2489 6 248 | 1239 123 5 | 7 189 34 | +-----------------------------+-------------------------------+-------------------------------+ | 23568 1378 12678 | 12357 9 12378 | 23568 4 1357 | | 23458 13478 12478 | 12357 12357 6 | 23589 12789 135789 | | 23568 1378 9 | 12357 4 12378 | 23568 12678 13578 | +-----------------------------+-------------------------------+-------------------------------+ | 34689 2 468 | 35679 357 3479 | 1 789 45789 | | 7 13489 148 | 12359 1235 12349 | 24589 28 6 | | 469 149 5 | 12679 8 12479 | 249 3 479 | +-----------------------------+-------------------------------+-------------------------------+ Now comes our second braid, an nrczt-braid which uses the four types of 2D cells (rc, rn, cn and bn): nrczt-braid-cn[14] * ------- C1:n8r3{c1 c8 c3*} - C2:{n8 n2}r8c8 - * ------- C3:n7{r2c2 r1c3} - C4:n4r1{c3 c7} - C5:{n4 n9 n2#2}r9c7 - C6:{n9 n7 n8#1}r7c8 - C7:{n7 n4 n9#5}r9c9 - C8:{n4 n1 n9#5}r9c2 - C7 ------- C9:{n4 n6 n9#5}r9c1 - C2 ------- C10:{n8 n6 n9#5}r2c7 - C11:{n6 n9}r1c8 - C12:{n9 n1 n2#2 n7#6 n8#1}r5c8 - C13:n1{r5c3 r4c3 r4c2#8 r5c2#8 r6c2#8} - C14:n6{r4 . r7#9}c3 ==> r2c2 <> 8 Here again, we have a non-first left-linking candidate (in C3) which is linked to the target instead of the pevious right-linking candidate; and two left-linking candidates (in C9 and C10) which are linked to a right-linking candidate that is not the immediately previous one. nrczt-whip-rc[11] {n4 n7}r1c3 - {n7 n9}r2c2 - {n9 n1}r9c2 - {n1 n8}r8c3 - {n8 n2}r3c3 - n4{r3c3 r3c1} - n8{r3c1 r3c8} - n9{r3c8 r1c8} - n6{r1c8 r6c8} - n1{r5c3 r5c8} - {n1r5c3 .} ==> r7c3 <> 4 nrczt-whip-rc[14] n8{r2c1 r3c3} - {n8 n6}r7c3 - n6{r7c4 r9c4} - n6{r9c1 r6c1} - n5{r6c1 r5c1} - n3{r5c1 r7c1} - n8{r7c1 r8c2} - {n8 n2}r8c8 - n2{r9c7 r9c6} - n1{r9c6 r9c2} - n9{r9c2 r9c1} - n4{r9c1 r8c3} - n4{r1c3 r1c7} - {n4r9c7 .} ==> r4c1 <> 8 nrczt-whip-rc[14] n8{r2c1 r3c3} - {n8 n6}r7c3 - n6{r7c4 r9c4} - n6{r9c1 r4c1} - n5{r4c1 r5c1} - n3{r5c1 r7c1} - n8{r7c1 r8c2} - {n8 n2}r8c8 - n2{r9c7 r9c6} - n1{r9c6 r9c2} - n9{r9c2 r9c1} - n4{r9c1 r8c3} - n4{r1c3 r1c7} - {n4r9c7 .} ==> r6c1 <> 8 At this point, the PM is: +----------------------------+-----------------------------+-----------------------------+ | 1 5 47 | 8 367 379 | 34 69 2 | | 289 789 3 | 4 1267 1279 | 689 5 18 | | 2489 6 248 | 1239 123 5 | 7 189 34 | +----------------------------+-----------------------------+-----------------------------+ | 2356 1378 12678 | 12357 9 12378 | 23568 4 1357 | | 23458 13478 12478 | 12357 12357 6 | 23589 12789 135789 | | 2356 1378 9 | 12357 4 12378 | 23568 12678 13578 | +----------------------------+-----------------------------+-----------------------------+ | 34689 2 68 | 35679 357 3479 | 1 789 45789 | | 7 13489 148 | 12359 1235 12349 | 24589 28 6 | | 469 149 5 | 12679 8 12479 | 249 3 479 | +----------------------------+-----------------------------+-----------------------------+ Now comes a braid with two left-linking candidates (in C15 and C19) branching off the same right-linking candidate (in C12). nrczt-braid-cn[23] * ------- C1:{n9 n7}r2c2 - C2:{n7 n4}r1c3 - C3:{n4 n3}r1c7 - C4:{n3 n4}r3c9 - * ------- C5:n3{r8c2 r7c1} - C6:n4r7{c1 c6 c9#4} - C7:n4r8{c6 c7 c2*} - C8:n5{r8c7 r7c9} - C9:{n5 n7 n3#5}r7c5 - C10:{n7 n6 n3#3}r1c5 - C11:{n6 n9}r1c8 - C12:{n9 n8 n7#9}r7c8 - C13:{n8 n2}r8c8 - C14:{n2 n9 n4#7}r9c7 - C12 ------- C15:{n8 n1 n9#11}r3c8 - C16:{n1 n8}r2c9 - C12 ------- C17:{n8 n6}r7c3 - C18:{n6 n4 n9#14}r9c1 - C19:{n4 n1 n9#14}r9c2 - C20:{n1 n8 n4#18}r8c3 - C21:{n8 n2 n4#4}r3c3 - C22:{n2 n9 n8#16}r2c1 - C23:n9{r2 . r1#11 r8* r9#14}c6 ==> r8c2 <> 9 At this point, the PM is: +-----------------------------+-------------------------------+-------------------------------+ | 1 5 47 | 8 367 379 | 34 69 2 | | 289 789 3 | 4 1267 1279 | 689 5 18 | | 2489 6 248 | 1239 123 5 | 7 189 34 | +-----------------------------+-------------------------------+-------------------------------+ | 2356 1378 12678 | 12357 9 12378 | 23568 4 1357 | | 23458 13478 12478 | 12357 12357 6 | 23589 12789 135789 | | 2356 1378 9 | 12357 4 12378 | 23568 12678 13578 | +-----------------------------+-------------------------------+-------------------------------+ | 34689 2 68 | 35679 357 3479 | 1 789 45789 | | 7 1348 148 | 12359 1235 12349 | 24589 28 6 | | 469 149 5 | 12679 8 12479 | 249 3 479 | +-----------------------------+-------------------------------+-------------------------------+ nrczt-whip-rn[9] n9{r9c2 r2c2} - n9{r3c1 r3c8} - {n9 n6}r1c8 - n6{r2c7 r2c5} - n7{r2c5 r2c6} - n7{r9c6 r9c9} - {n7 n8}r7c8 - {n8 n6}r7c3 - {n6r9c1 .} ==> r9c4 <> 9 nrczt-whip-rc[11] n3{r3c9 r1c7} - n4{r1c7 r1c3} - n7{r1c3 r2c2} - n9{r2c2 r9c2} - n9{r9c9 r7c9} - n5{r7c9 r8c7} - n4{r8c7 r9c7} - {n4 n7}r9c9 - {n7 n8}r7c8 - n6{r9c1 r7c3} - {n6r9c1 .} ==> r5c9 <> 3 nrczt-whip-rc[12] n9{r3c8 r2c7} - n9{r2c1 r3c1} - n9{r9c1 r9c2} - {n9 n7}r2c2 - {n7 n4}r1c3 - n4{r1c7 r3c9} - {n4 n7}r9c9 - {n7 n8}r7c8 - {n8 n2}r8c8 - {n2 n4}r9c7 - {n4 n6}r9c1 - {n6r7c3 .} ==> r5c8 <> 9 nrczt-whip-rc[12] n9{r1c6 r1c8} - n9{r3c8 r3c1} - n9{r9c1 r9c2} - {n9 n7}r2c2 - {n7 n4}r1c3 - n4{r1c7 r3c9} - {n4 n7}r9c9 - {n7 n8}r7c8 - {n8 n2}r8c8 - {n2 n4}r9c7 - {n4 n6}r9c1 - {n6r7c3 .} ==> r2c6 <> 9 nrczt-whip-rc[13] n9{r1c6 r1c8} - n6{r1c8 r1c5} - n3{r1c5 r1c7} - {n3 n4}r3c9 - n4{r1c7 r1c3} - n7{r1c3 r2c2} - n9{r2c2 r9c2} - {n9 n7}r9c9 - {n7 n8}r7c8 - {n8 n2}r8c8 - {n2 n4}r9c7 - {n4 n6}r9c1 - {n6r7c3 .} ==> r1c6 <> 7 nrczt-whip-rn[11] {n3 n4}r3c9 - n4{r1c7 r1c3} - n7{r1c3 r1c5} - {n7 n5}r7c5 - n5{r7c9 r8c7} - n4{r8c7 r9c7} - n2{r9c7 r8c8} - {n2 n1}r8c5 - n1{r8c2 r9c2} - {n7 n9}r2c2 - {n7r2c2 .} ==> r3c5 <> 3 Nothing remarkable in the sequel. 5.2) A better example of NRCZT-braids I've had a hard time coding braids in SudoRules in a way that doesn't lead to memory overflow problems. I can't yet say I've done it in a fully satisfactory way (computation times remain very long and lots of optimisations are still possible, but my purpose here is not to speak of implementation details in CLIPS). At least, SudoRules can now find braids of decent length. Once introduced in SudoRules, braids blend perfectly with whips, as shown by the following example. I had to make some choice about where to put braids in my complexity hierarchy. As shown in the "rating" page, length remains the best criterion from a statistical POV. I have therefore extended SudoRules default strategy so as to include braids of length n just after whips of length n and before any chain of length n+1. (As braids of length n subsume all the chains of the same length in the xy-to-nrczt family, this is a natural choice). As for the rest of SudoRules, the way braids are coded would allow me to change this strategy in a very straightforward way if an alternative one was needed. But this is beyond the purpose of this example, which is only to show how resolution rules for braids (here, only the most elementary ones - nrczt-braids) can be used effectively as any other resolution rule. The example I shall consider now is # 89 (SER = 9.3) in the Magictour top1465 collection. 000300500 050010030 007004001 200000400 060090000 001006002 800700200 090080050 005009007 1) # 89 is one of the 5 puzzles in the top1465 collection that cannot be solved by nrczt-whips only. (Notice that this is very rare for a puzzle with SER = 9.3). More specifically, if I run SudoRules with the usual rules for nrczt-whips (no braids), I get the following resolution path. Some eliminations are preceded by a capital letter at the beginning of their line. Corresponding eliminations in the resolution path with braids will be indicated by the same capital letter in the path with braids; they will be primed if the justifying pattern is different. ***** SudoRules version 13.7wbis ***** ...3..5...5..1..3...7..4..12.....4...6..9......1..6..28..7..2...9..8..5...5..9..7 hidden-single-in-a-row ==> r8c1 = 7 nrczt-whip-rn[2] n7{r5c8 r5c6} - {n7r2c6 .} ==> r6c7 <> 7 swordfish-in-columns n2{r3 r9 r1}{c2 c5 c8} ==> r9c4 <> 2, r3c4 <> 2, r1c6 <> 2, r1c3 <> 2 nrc-chain[3] n5{r4c9 r5c9} - n5{r5c1 r6c1} - n9{r6c1 r4c3} ==> r4c9 <> 9 nrczt-whip-cn[3] n9{r4c3 r4c8} - n9{r6c7 r3c7} - {n9r3c4 .} ==> r2c3 <> 9 nrc-chain[4] n9{r4c3 r6c1} - n5{r6c1 r5c1} - n5{r5c9 r4c9} - n6{r4c9 r4c8} ==> r4c8 <> 9 hidden-single-in-a-row ==> r4c3 = 9 nrczt-whip-rn[7] n9{r7c9 r7c8} - n9{r1c8 r1c1} - n1{r1c1 r1c2} - n1{r7c2 r7c6} - n5{r7c6 r7c5} - n5{r3c5 r3c4} - {n9r3c4 .} ==> r2c9 <> 9 ;;; end common part A: nrczt-whip-rn[9] n7{r2c7 r5c7} - n7{r6c8 r1c8} - n9{r1c8 r1c1} - n1{r1c1 r9c1} - n1{r9c7 r8c7} - n1{r8c6 r7c6} - n5{r7c6 r7c5} - n5{r3c5 r3c4} - {n9r3c4 .} ==> r2c7 <> 9 B: nrczt-whip-cn[11] n7{r2c7 r1c8} - {n7 n8}r1c6 - n8{r1c3 r5c3} - n8{r5c9 r4c9} - n8{r6c8 r9c8} - n4{r9c8 r7c8} - n9{r7c8 r7c9} - n9{r1c9 r1c1} - n1{r1c1 r1c2} - n3{r8c3 r7c2} - {n3r8c3 .} ==> r2c7 <> 8 C: nrczt-whip-cn[18] {n8 n7}r1c6 - n7{r2c6 r2c7} - n7{r5c7 r5c8} - {n7 n9}r6c8 - n9{r7c8 r7c9} - n9{r1c9 r3c7} - n9{r1c8 r1c1} - n1{r1c1 r1c2} - n2{r1c2 r1c5} - n2{r2c6 r2c3} - n8{r2c3 r5c3} - n8{r5c9 r4c9} - n8{r6c7 r9c7} - n6{r9c7 r8c7} - n3{r8c7 r8c9} - n3{r8c3 r7c3} - n6{r7c3 r9c1} - {n1r9c1 .} ==> r1c8 <> 8 GRID 89 NOT SOLVED. 55 VALUES MISSING. Notice that the presence of the final very long whip is, as very often, an indication that this puzzle has few nrczt-chains. Puzzles with SER 9.3 can usually be solved with shorter whips. For examples of such resolution paths for the hardest puzzles in the Magictour/top1465 collection, see the top1465 solutions page. 2) Using an easy implementation of the T&E procedure (independent of SudoRules), it can be checked that # 89 is solvable by T&E = T&E(ECP+NS+HS). (Indeed, # 89 is one of the 2 puzzles (together with #187, SER 9.4) in the top1465 collection that can't be solved by nrczt-whips but can be solved by nrczt-braids.) Thanks to the T&E vs braids theorem (see the top of this page), we know that there must be a resolution path with nrczt-braids, but we don't yet have such a path. We could get one from the output of the T&E procedure but, as I explained elsewhere, it would be hard manual work, the path wouldn't be optimised and we wouldn't be able to guarantee any minimality of the braids it uses. 3) Resolution path with nrczt-braids ***** SudoRules version 13.7wbis-B2 ***** ;;; Path unchanged down to "end common part" ;;; The next two eliminations are done by slightly shorter braids A': nrczt-braid-bn[8] n9{r1c8 r1c1} - n9{r3c1 r3c4} - n5{r3c4 r3c5} - n5{r7c5 r7c6} - n1{r1c1 r9c1} - n7{r2c7 r5c7} - n1{r5c7 r8c7} - {n1r9c4 .} ==> r2c7 <> 9 B': nrczt-braid-cn[10] n7{r2c7 r1c8} - {n7 n8}r1c6 - n8{r1c3 r5c3} - n8{r9c7 r9c8} - n4{r1c8 r7c8} - n9{r7c8 r7c9} - n9{r1c8 r1c1} - n1{r1c1 r9c1} - {n1 n3}r7c2 - {n3r8c3 .} ==> r2c7 <> 8 ;;; Now the two paths diverge completely: nrczt-braid-cn[11] n9{r7c9 r1c9} - n9{r3c7 r6c7} - n3{r6c7 r5c7} - n3{r5c3 r8c3} - n2{r8c3 r2c3} - n7{r5c7 r2c7} - {n2 n8}r2c6 - {n8 n7}r1c6 - n7{r5c6 r5c8} - n8{r5c9 r6c8} - {n8r5c9 .} ==> r7c9 <> 3 nrczt-braid-cn[10] n6{r4c8 r4c9} - n5{r4c9 r5c9} - n3{r5c9 r8c9} - n7{r5c8 r5c6} - {n7 n8}r1c6 - n8{r5c9 r2c9} - n8{r3c8 r3c2} - {n8 n3}r4c2 - n3{r8c6 r7c6} - {n3r8c3 .} ==> r4c8 <> 7 nrczt-braid-rn[12] n9{r3c7 r6c7} - n8{r9c7 r9c8} - {n9 n7}r6c8 - {n8 n1}r5c8 - {n8 n3}r5c7 - n3{r8c7 r8c9} - n7{r5c8 r5c6} - {n7 n8}r1c6 - {n8 n2}r2c6 - n3{r8c3 r7c3} - n2{r1c5 r9c5} - {n3r9c5 .} ==> r3c7 <> 8 nrct-chain[8] {n7 n6}r2c7 - {n6 n9}r3c7 - n9{r1c9 r1c1} - n1{r1c1 r9c1} - n6{r9c1 r3c1} - {n6 n4}r2c1 - {n4 n8}r1c3 - {n8 n7}r1c6 ==> r2c6 <> 7 hidden-single-in-a-row ==> r2c7 = 7 nrczt-whip-bn[4] n7{r5c8 r5c6} - {n7 n8}r1c6 - n8{r3c4 r3c2} - {n8r6c2 .} ==> r5c8 <> 8 nrczt-braid-rc[9] {n8 n2}r2c6 - n9{r2c4 r3c4} - {n9 n6}r3c7 - {n8 n4}r2c9 - n2{r2c3 r8c3} - n4{r8c9 r8c4} - n6{r8c4 r9c4} - n6{r9c1 r7c3} - {n8r2c3 .} ==> r2c4 <> 8 nrczt-braid-cn[9] {n2 n8}r2c6 - n9{r2c4 r3c4} - {n9 n6}r3c7 - {n8 n4}r2c9 - {n8 n6}r2c3 - n6{r8c3 r9c1} - n6{r9c4 r8c4} - n4{r8c9 r8c3} - {n2r8c3 .} ==> r2c4 <> 2 nrczt-braid-cn[11] n2{r8c3 r2c3} - {n2 n8}r2c6 - n3{r8c9 r9c7} - n8{r9c7 r9c8} - n8{r3c8 r1c9} - n8{r2c3 r5c3} - {n8 n1}r5c7 - n1{r9c8 r7c8} - {n3 n4}r7c2 - n4{r1c3 r1c8} - {n4r1c3 .} ==> r8c3 <> 3 nrczt-whip-cn[9] {n6 n9}r3c7 - n9{r1c9 r1c1} - n1{r1c1 r9c1} - n6{r9c1 r3c1} - n3{r3c1 r3c2} - n3{r7c2 r7c3} - n6{r7c3 r8c3} - n6{r8c7 r9c7} - {n6r9c4 .} ==> r2c9 <> 6 nrczt-whip-rc[3] n6{r2c3 r2c4} - n9{r2c4 r3c4} - {n9r3c7 .} ==> r3c1 <> 6 nrczt-braid-rc[9] n2{r5c4 r5c6} - n2{r2c6 r2c3} - n7{r5c6 r5c8} - n8{r5c3 r1c3} - n8{r3c4 r3c8} - {n8 n9}r6c8 - {n8 n3}r3c2 - n9{r6c7 r3c7} - {n3r3c1 .} ==> r5c4 <> 8 nrczt-braid-rc[10] n1{r1c1 r9c1} - n6{r9c1 r2c1} - {n6 n8}r1c3 - n9{r2c1 r3c1} - n9{r3c7 r6c7} - n4{r2c3 r2c9} - n8{r2c9 r3c8} - {n9 n7}r6c8 - n7{r5c8 r5c6} - {n8r1c6 .} ==> r1c1 <> 4 nrczt-whip-bn[6] n4{r6c5 r5c4} - n2{r5c4 r5c6} - {n2 n8}r2c6 - {n8 n4}r2c9 - n4{r8c9 r8c3} - {n4r1c3 .} ==> r6c2 <> 4 nrczt-braid-rn[8] n4{r7c2 r1c2} - n3{r7c3 r5c3} - n4{r7c8 r9c8} - n4{r9c5 r6c5} - n3{r6c5 r6c7} - n9{r6c7 r3c7} - n9{r1c9 r1c1} - {n1r1c2 .} ==> r7c3 <> 4 nrczt-braid-rc[9] n6{r2c3 r2c4} - n1{r1c1 r9c1} - {n6 n4}r9c4 - n1{r9c2 r1c2} - n4{r1c2 r7c2} - n4{r9c8 r1c8} - n2{r1c8 r1c5} - {n4 n8}r2c9 - {n8r2c6 .} ==> r1c1 <> 6 nrczt-braid-cn[8] n6{r9c1 r2c1} - n6{r4c8 r4c9} - n6{r1c9 r1c5} - n6{r7c9 r7c3} - n3{r7c3 r5c3} - n3{r5c9 r6c7} - n6{r9c7 r3c7} - {n9r6c7 .} ==> r9c8 <> 6 nrczt-braid-rc[9] n4{r5c1 r5c3} - n4{r2c3 r2c9} - n3{r5c3 r7c3} - n6{r9c1 r8c3} - {n6 n3}r8c9 - {n6 n1}r8c7 - {n1 n8}r9c8 - n8{r3c8 r1c9} - {n8r1c3 .} ==> r9c1 <> 4 nrczt-braid-cn[9] n6{r9c1 r2c1} - n2{r9c5 r9c2} - n6{r9c4 r3c4} - n6{r9c7 r8c7} - {n6 n4}r8c3 - n4{r9c2 r1c2} - n1{r1c2 r1c1} - {n6 n9}r3c7 - {n9r3c1 .} ==> r9c5 <> 6 nrczt-whip-rc[10] {n6 n9}r3c7 - n9{r1c9 r1c1} - n1{r1c1 r9c1} - n6{r9c1 r9c7} - n6{r8c9 r8c3} - n2{r8c3 r2c3} - {n2 n8}r2c6 - {n8 n4}r2c9 - n4{r8c9 r8c4} - {n4r9c4 .} ==> r3c4 <> 6 nrczt-whip-cn[8] n2{r1c8 r3c8} - n6{r3c8 r3c5} - n5{r3c5 r3c4} - n8{r3c4 r3c2} - n8{r6c2 r5c3} - n3{r5c3 r7c3} - n6{r7c3 r7c9} - {n6r9c7 .} ==> r1c8 <> 6 ;;; Here, in order to make computations faster, I have de-activated braids; there is nothing new after this point. nrczt-whip-rc[10] {n6 n9}r3c7 - n9{r1c8 r1c1} - n1{r1c1 r1c2} - n1{r9c2 r9c1} - {n1 n4}r9c4 - n4{r9c2 r7c2} - n4{r7c8 r1c8} - n2{r1c8 r1c5} - {n2 n8}r2c6 - {n8r2c9 .} ==> r9c7 <> 6 nrczt-whip-cn[11] {n8 n7}r1c6 - n7{r5c6 r5c8} - {n7 n9}r6c8 - n9{r6c7 r3c7} - n9{r1c9 r1c1} - n1{r1c1 r1c2} - n2{r1c2 r1c5} - n2{r9c5 r9c2} - n4{r9c2 r7c2} - n6{r8c7 r8c3} - {n6r8c7 .} ==> r1c8 <> 8 nrczt-whip-rc[8] n8{r1c9 r3c8} - n8{r3c4 r6c4} - n8{r6c2 r1c2} - n8{r2c3 r2c6} - n2{r2c6 r2c3} - {n2 n3}r3c2 - {n3 n7}r6c2 - {n7r4c2 .} ==> r4c9 <> 8 nrczt-whip-rn[7] n8{r4c2 r5c3} - n8{r5c9 r2c9} - {n8 n2}r2c6 - n2{r1c5 r1c8} - n4{r1c8 r1c9} - n9{r1c9 r1c1} - {n1r1c1 .} ==> r1c2 <> 8 nrczt-whip-rn[11] n3{r5c3 r7c3} - n3{r9c2 r3c2} - {n3 n9}r3c1 - n9{r3c7 r6c7} - n3{r6c7 r6c5} - n3{r9c5 r9c7} - n8{r9c7 r9c8} - {n8 n7}r6c8 - n7{r5c8 r5c6} - n8{r3c4 r1c6} - {n8r3c4 .} ==> r5c1 <> 3 nrczt-whip-rc[11] {n3 n6}r7c3 - {n6 n1}r9c1 - n1{r9c2 r1c2} - n4{r1c2 r9c2} - n2{r9c2 r3c2} - n2{r3c8 r1c8} - n4{r1c8 r7c8} - {n4 n8}r9c8 - n8{r3c8 r3c4} - n5{r7c5 r3c5} - {n5r7c5 .} ==> r7c2 <> 3 nrczt-whip-cn[7] n9{r6c7 r3c7} - n9{r1c8 r1c1} - n1{r1c1 r9c1} - {n1 n3}r9c7 - n3{r9c1 r7c3} - n6{r7c3 r8c3} - {n6r8c7 .} ==> r6c7 <> 8 nrczt-whip-cn[9] {n3 n9}r6c7 - {n9 n6}r3c7 - {n6 n1}r8c7 - {n1 n8}r5c7 - {n8 n7}r6c8 - n7{r5c8 r5c6} - {n7 n8}r1c6 - n8{r1c9 r2c9} - {n8r2c3 .} ==> r9c7 <> 3 interaction block b9 with row r8 for number 3 ==> r8c6 <> 3 nrct-chain[4] {n6 n3}r7c3 - n3{r7c5 r9c5} - n2{r9c5 r9c2} - n2{r8c3 r2c3} ==> r2c3 <> 6 hidden-pairs-in-a-row {n6 n9}r2{c1 c4} ==> r2c1 <> 4 interaction column c1 with block b4 for number 4 ==> r5c3 <> 4 hidden-pairs-in-a-block {n4 n5}{r5c1 r6c1} ==> r6c1 <> 3 x-wing-in-rows n6{r2 r9}{c1 c4} ==> r8c4 <> 6 nrct-chain[4] n4{r2c9 r2c3} - n2{r2c3 r8c3} - {n2 n1}r8c6 - {n1 n4}r8c4 ==> r8c9 <> 4 nrczt-whip-rn[4] {n1 n2}r8c6 - {n2 n4}r8c4 - {n4 n6}r8c3 - {n6r9c1 .} ==> r9c4 <> 1 xyzt-chain[5] {n6 n3}r8c9 - {n3 n5}r4c9 - {n5 n8}r5c9 - {n8 n3}r5c3 - {n3 n6}r7c3 ==> r7c9 <> 6 nrczt-whip-rn[5] n5{r7c5 r7c6} - n3{r7c6 r7c3} - n6{r7c3 r7c8} - n6{r8c9 r8c3} - {n4r8c3 .} ==> r7c5 <> 4 nrct-chain[5] {n8 n2}r2c6 - n2{r3c5 r9c5} - n4{r9c5 r6c5} - {n4 n5}r6c1 - {n5 n8}r6c4 ==> r5c6 <> 8 nrct-chain[5] {n8 n2}r2c6 - n2{r3c5 r9c5} - n4{r9c5 r6c5} - {n4 n5}r6c1 - {n5 n8}r6c4 ==> r4c6 <> 8 interaction column c6 with block b2 for number 8 ==> r3c4 <> 8 swordfish-in-columns n8{r5 r2 r1}{c3 c6 c9} ==> r5c7 <> 8 hidden-single-in-a-column ==> r9c7 = 8 nrct-chain[2] n1{r5c7 r8c7} - n1{r8c6 r7c6} ==> r5c6 <> 1 nrc-chain[3] n1{r5c7 r8c7} - {n1 n2}r8c6 - n2{r8c4 r5c4} ==> r5c4 <> 1 interaction row r5 with block b6 for number 1 ==> r4c8 <> 1 nrc-chain[5] {n7 n8}r1c6 - {n8 n2}r2c6 - {n2 n1}r8c6 - n1{r8c7 r5c7} - {n1 n7}r5c8 ==> r5c6 <> 7 hidden-singles ==> r5c8 = 7, r5c7 = 1 interaction row r8 with block b8 for number 1 ==> r7c6 <> 1 naked-pairs-in-a-block {n3 n6}{r8c7 r8c9} ==> r7c8 <> 6 interaction block b9 with row r8 for number 6 ==> r8c3 <> 6 nrc-chain[4] n3{r3c1 r3c2} - n8{r3c2 r3c8} - {n8 n9}r6c8 - n9{r6c7 r3c7} ==> r3c1 <> 9 naked-single ==> r3c1 = 3 hidden-pairs-in-a-row {n2 n3}r9{c2 c5} ==> r9c5 <> 4 naked and hidden singles 914378526 658912734 327564981 289137465 463295178 571846392 846753219 792481653 135629847 5.3) Yet another example of nrczt-braids As I mentioned above, #187 (SER = 9.4) is one of the two grids in the top1465 collection that can't be solved by nrczt-whips but can by nrczt-braids (as can be verified with a simple T&E procedure). First tentative solution with whips: ***** SudoRules version 13.7w-bis ***** 6.....3...5..9..8...2..6..98.....7...7..5..4......1..51..3..5...4..2..6...8..7..2 hidden-singles ==> r4c3 = 5, r7c2 = 2 interaction row r4 with block b5 for number 4 ==> r6c5 <> 4, r6c4 <> 4 nrczt-whip-cn[11] {n4 n1}r3c7 - {n1 n9}r9c7 - {n9 n8}r8c7 - n8{r8c9 r5c9} - n1{r5c9 r5c3} - n1{r4c2 r1c2} - n9{r1c2 r1c3} - n9{r8c3 r8c1} - {n9 n5}r8c6 - n1{r9c5 r8c4} - {n1r9c5 .} ==> r2c7 <> 4 ;;; end common part nrczt-whip[21] n8{r8c9 r7c9} - n4{r7c9 r9c7} - {n4 n1}r3c7 - n1{r8c7 r8c9} - n7{r8c9 r7c8} - {n7 n5}r3c8 - {n5 n2}r1c8 - n1{r1c8 r4c8} - n1{r4c2 r1c2} - n9{r1c2 r1c3} - {n9 n6}r7c3 - {n6 n4}r7c5 - {n4 n9}r7c6 - {n9 n5}r8c6 - n5{r9c4 r1c4} - n1{r1c4 r2c4} - n2{r2c4 r2c6} - n2{r4c6 r4c4} - n9{r4c4 r4c2} - {n9 n3}r9c2 - {n3r9c8 .} ==> r8c4 <> 8 GRID 187 NOT SOLVED. 55 VALUES MISSING. Here again, we notice a very long whip (length 21), an indication that this puzzle has few chains. Let us now solve it with braids: ***** SudoRules version 13.7wbis-B2 ***** 6.....3...5..9..8...2..6..98.....7...7..5..4......1..51..3..5...4..2..6...8..7..2 ;;; Same resolution path down to "end common part" ;;; We now get braids much shorter than the whip obtained in the previous path: nrczt-braid-rc[11] n3{r8c9 r9c8} - {n7 n9}r7c8 - {n3 n2}r6c8 - n2{r1c8 r2c7} - n6{r2c7 r2c9} - {n2 n1}r4c8 - n1{r2c9 r1c9} - n1{r1c2 r3c2} - n1{r1c5 r9c5} - n4{r9c7 r7c9} - {n4r9c7 .} ==> r8c9 <> 7 interaction row r8 with block b7 for number 7 ==> r7c3 <> 7 nrczt-braid-cn[4] {n6 n9}r7c3 - n4{r6c3 r6c1} - n9{r9c1 r5c1} - {n2r6c1 .} ==> r6c3 <> 6 nrczt-braid-cn[11] n7{r6c4 r6c5} - n8{r6c5 r6c7} - n6{r6c7 r6c2} - n2{r4c6 r4c8} - n2{r6c7 r2c7} - n6{r2c7 r5c7} - n9{r6c7 r6c8} - n3{r6c8 r9c8} - n2{r6c1 r5c1} - n9{r9c1 r9c2} - {n9r9c1 .} ==> r6c4 <> 2 nrczt-braid-rc[12] n1{r9c4 r9c5} - {n1 n4}r3c7 - n4{r9c5 r9c4} - n6{r9c4 r9c2} - {n6 n9}r7c3 - n9{r1c3 r1c2} - n1{r3c2 r4c2} - n1{r9c8 r1c8} - {n4 n7}r1c9 - {n9 n3}r6c2 - {n1 n4}r1c3 - {n4r6c3 .} ==> r3c4 <> 1 ;;; Even with braids activated, the next eliminations use only whips nrczt-whip-cn[13] {n1 n4}r3c7 - {n4 n9}r9c7 - {n9 n8}r8c7 - n8{r8c9 r5c9} - n1{r5c9 r5c3} - n6{r5c3 r7c3} - n9{r7c3 r7c6} - n8{r7c6 r1c6} - n8{r3c5 r3c2} - n1{r3c2 r3c5} - n3{r3c5 r3c1} - n3{r5c1 r5c6} - {n3r6c5 .} ==> r2c7 <> 1 nrczt-whip-cn[8] n1{r5c9 r5c3} - n1{r2c3 r2c4} - n1{r9c4 r9c5} - n1{r9c8 r8c7} - {n1 n4}r3c7 - n4{r9c7 r9c4} - n6{r9c4 r9c2} - {n6r7c3 .} ==> r4c9 <> 1 nrczt-whip-rc[6] n9{r1c2 r1c3} - {n9 n6}r7c3 - n6{r9c2 r4c2} - {n6 n3}r4c9 - n3{r8c9 r9c8} - {n3r9c2 .} ==> r6c2 <> 9 nrczt-whip-cn[8] n1{r4c8 r4c2} - n1{r3c2 r3c5} - {n1 n4}r3c7 - {n4 n7}r1c9 - {n7 n6}r2c9 - {n6 n3}r4c9 - n3{r4c5 r6c5} - {n7r6c5 .} ==> r1c8 <> 1 nrczt-whip-rn[11] n6{r9c5 r9c4} - n6{r9c2 r4c2} - {n6 n3}r4c9 - n3{r4c6 r5c6} - {n3 n4}r4c5 - n4{r9c5 r7c6} - {n4 n2}r2c6 - n2{r2c7 r1c8} - {n2 n9}r6c8 - {n6 n9}r7c3 - {n6r7c3 .} ==> r6c5 <> 6 ;;;; (I checked that a solution with whips only can be obtained from here, which shows that this puzzle needs only a few eliminations to be solvable by whips). nrczt-braid-bn[11] n6{r9c2 r7c3} - n6{r7c5 r9c5} - {n6 n3}r4c9 - {n6 n3}r6c2 - n3{r6c5 r3c5} - n1{r9c5 r1c5} - n1{r4c2 r3c2} - n3{r5c9 r5c6} - n7{r3c5 r6c5} - n8{r6c4 r3c4} - {n8r6c4 .} ==> r4c2 <> 6 nrczt-braid-rc[8] n6{r5c3 r6c2} - {n6 n3}r4c9 - n3{r8c9 r9c8} - {n6 n9}r9c2 - {n9 n1}r4c2 - n1{r5c9 r5c7} - {n9 n4}r9c7 - {n4r3c7 .} ==> r5c9 <> 6 ;;; There's nothing new in the sequel (whips of length <= 12). Details can be seen on the top1465 solutions page. 5.4) Additional examples of nrczt-braids Examples taken from the suexg-cb collection. (solve "..5..7......5621.........9.24....9..5.6...7.4.8.45..1..3..4..8...962.........1...") ***** SudoRules version 13.7wbis-B2 ***** ..5..7......5621.........9.24....9..5.6...7.4.8.45..1..3..4..8...962.........1... singles ==> r3c6 = 4, r5c4 = 2, r5c8 = 3, r4c9 = 8, r4c8 = 5, r4c6 = 6 interaction row r6 with block b4 for number 7 ==> r4c3 <> 7 interaction row r2 with block b1 for number 8 ==> r3c3 <> 8, r3c1 <> 8, r1c1 <> 8 interaction row r2 with block b1 for number 9 ==> r1c2 <> 9, r1c1 <> 9 naked-pairs-in-a-column {n4 n7}{r2 r8}c8 ==> r9c8 <> 7, r9c8 <> 4, r1c8 <> 4 hidden-pairs-in-a-column {n4 n8}{r2 r9}c3 ==> r9c3 <> 7, r9c3 <> 2, r2c3 <> 7, r2c3 <> 3 nrc-chain[2] n4{r1c1 r1c7} - n4{r2c8 r8c8} ==> r8c1 <> 4 interaction row r8 with block b9 for number 4 ==> r9c7 <> 4 nrct-chain[2] n6{r1c8 r9c8} - n6{r9c2 r7c1} ==> r1c1 <> 6 nrc-chain[3] n7{r6c3 r6c1} - n9{r6c1 r2c1} - {n9 n7}r2c2 ==> r3c3 <> 7 nrczt-whip-rc[4] n9{r9c9 r7c9} - {n9 n5}r7c6 - {n5 n2}r7c7 - {n2r9c8 .} ==> r9c9 <> 6 nrczt-whip-rn[4] n3{r9c5 r8c6} - {n3 n9}r6c6 - n9{r6c1 r2c1} - {n3r2c1 .} ==> r9c9 <> 3 nrczt-whip-rc[4] n9{r9c9 r7c9} - {n9 n5}r7c6 - {n5 n6}r7c7 - {n6r9c8 .} ==> r9c9 <> 2 nrczt-whip-rn[4] n2{r7c3 r9c2} - {n2 n6}r9c8 - {n6 n5}r7c7 - {n5r9c9 .} ==> r7c9 <> 2 nrct-chain[3] {n6 n2}r1c8 - n2{r1c9 r6c9} - n6{r6c9 r6c7} ==> r3c7 <> 6, r1c7 <> 6 nrczt-whip-cn[5] {n2 n6}r6c7 - {n6 n5}r7c7 - {n5 n3}r9c7 - n3{r9c5 r8c6} - {n5r8c6 .} ==> r1c7 <> 2, r3c7 <> 2 nrct-chain[3] {n6 n2}r9c8 - n2{r9c7 r6c7} - {n2 n6}r6c9 ==> r7c9 <> 6 nrczt-braid-rc[5] n9{r7c9 r9c9} - n5{r9c9 r9c2} - n1{r7c9 r8c9} - n7{r7c9 r8c8} - {n7r8c2 .} ==> r7c9 <> 5 <------- nrczt-braid-rn[5] n6{r7c7 r7c1} - n2{r7c7 r7c3} - n5{r9c9 r3c9} - n6{r3c1 r3c2} - {n2r3c9 .} ==> r7c7 <> 5 <------- hidden-single-in-a-row ==> r7c6 = 5 interaction column c6 with block b5 for number 9 ==> r5c5 <> 9 naked-pairs-in-a-block {n2 n6}{r7c7 r9c8} ==> r9c7 <> 6, r9c7 <> 2 nrct-chain[6] {n8 n1}r5c5 - {n1 n9}r5c2 - {n9 n7}r2c2 - n7{r3c2 r3c9} - n5{r3c9 r3c7} - n8{r3c7 r1c7} ==> r1c5 <> 8 nrct-chain[6] n5{r8c2 r9c2} - {n5 n3}r9c7 - n3{r9c5 r8c6} - n8{r8c6 r5c6} - n9{r5c6 r5c2} - {n9 n7}r2c2 ==> r8c2 <> 7 nrc-chain[5] {n5 n1}r8c2 - {n1 n9}r5c2 - {n9 n7}r2c2 - {n7 n4}r2c8 - n4{r8c8 r8c7} ==> r8c7 <> 5 nrct-chain[6] n5{r3c9 r3c7} - {n5 n3}r9c7 - n3{r9c5 r8c6} - {n3 n9}r6c6 - n9{r6c1 r2c1} - n3{r2c1 r2c9} ==> r3c9 <> 3 nrczt-whip-bn[6] n7{r2c9 r3c9} - n7{r3c2 r9c2} - n2{r9c2 r7c3} - n2{r3c3 r3c2} - n6{r3c2 r3c1} - {n6r9c1 .} ==> r2c1 <> 7 nrczt-whip-rc[6] n4{r1c1 r1c7} - n8{r1c7 r1c4} - n9{r1c4 r1c5} - n3{r1c5 r1c9} - {n3 n7}r2c9 - {n7r2c8 .} ==> r1c1 <> 1 nrczt-whip-bn[2] n1{r5c5 r5c2} - {n1r1c2 .} ==> r3c5 <> 1 nrc-chain[3] {n3 n8}r3c5 - n8{r5c5 r5c6} - {n8 n3}r8c6 ==> r9c5 <> 3 nrct-chain[3] {n3 n8}r3c5 - n8{r3c4 r9c4} - n3{r9c4 r9c7} ==> r3c7 <> 3 nrct-chain[6] n8{r1c7 r1c4} - n9{r1c4 r1c5} - n1{r1c5 r1c2} - {n1 n9}r5c2 - {n9 n7}r2c2 - {n7 n3}r2c9 ==> r1c7 <> 3 interaction column c7 with block b9 for number 3 ==> r8c9 <> 3 nrc-chain[5] {n3 n4}r1c1 - n4{r1c7 r8c7} - n3{r8c7 r8c6} - {n3 n9}r6c6 - n9{r6c1 r2c1} ==> r2c1 <> 3 hidden-single-in-a-row ==> r2c9 = 3 naked-pairs-in-a-block {n2 n6}{r1c8 r1c9} ==> r3c9 <> 6 interaction row r3 with block b1 for number 6 ==> r1c2 <> 6 naked-pairs-in-a-block {n2 n6}{r1c8 r1c9} ==> r3c9 <> 2 interaction row r3 with block b1 for number 2 ==> r1c2 <> 2 naked-singles GRID 0 SOLVED. LEVEL = B-NRCZT6, MOST COMPLEX RULE = NRCZT6 415937862 978562143 623184597 241376958 596218734 387459216 132745689 859623471 764891325 (solve ".9.5..8.7....2......3.49.2.1....67.......4..38..1526....5....7...8.9.2..3.....96.") ***** SudoRules version 13.7wbis-B2 ***** .9.5..8.7....2......3.49.2.1....67.......4..38..1526....5....7...8.9.2..3.....96. singles ==> r6c2 = 3, r6c3 = 7, r7c1 = 9, r4c9 = 2 interaction row r6 with block b6 for number 9 ==> r5c8 <> 9, r4c8 <> 9 interaction row r6 with block b6 for number 4 ==> r4c8 <> 4 naked-pairs-in-a-column {n1 n5}{r3 r5}c7 ==> r7c7 <> 1, r2c7 <> 5, r2c7 <> 1 nrc-chain[2] n5{r3c7 r5c7} - n5{r4c8 r4c2} ==> r3c2 <> 5 nrczt-whip-rc[4] n5{r9c6 r9c9} - n8{r9c9 r7c9} - {n8 n3}r7c6 - {n3r1c6 .} ==> r9c6 <> 1 nrczt-whip-rn[5] n6{r7c5 r1c5} - n1{r1c5 r9c5} - {n1 n3}r7c6 - n3{r8c6 r8c8} - {n3r1c8 .} ==> r7c5 <> 8 nrczt-whip-rn[6] n4{r7c7 r2c7} - n4{r2c1 r1c1} - n2{r1c1 r1c3} - n6{r1c3 r1c5} - n6{r7c5 r7c4} - {n2r7c4 .} ==> r7c2 <> 4 nrczt-braid-cn[6] {n1 n3}r1c6 - n1{r1c5 r2c6} - n1{r1c3 r9c3} - {n1 n8}r7c6 - {n1 n7}r9c5 - {n7r9c6 .} ==> r1c8 <> 1 <------- naked-pairs-in-a-block {n3 n4}{r1c8 r2c7} ==> r2c9 <> 4, r2c8 <> 4, r2c8 <> 3 nrct-chain[5] {n4 n3}r7c7 - n3{r8c8 r1c8} - {n3 n1}r1c6 - {n1 n8}r7c6 - n8{r9c6 r9c9} ==> r9c9 <> 4 nrct-chain[6] n6{r3c9 r2c9} - n9{r2c9 r2c8} - {n9 n4}r6c8 - {n4 n3}r1c8 - {n3 n1}r1c6 - {n1 n6}r1c5 ==> r3c4 <> 6 nrczt-whip-rn[6] {n4 n3}r7c7 - n3{r8c8 r1c8} - {n3 n1}r1c6 - {n1 n6}r1c5 - n6{r7c5 r7c2} - {n2r7c2 .} ==> r7c4 <> 4 interaction row r7 with block b9 for number 4 ==> r8c9 <> 4 interaction row r7 with block b9 for number 4 ==> r8c8 <> 4 nrczt-whip-cn[6] n3{r8c8 r1c8} - {n3 n1}r1c6 - {n1 n6}r1c5 - n6{r7c5 r7c4} - n2{r7c4 r9c4} - {n4r9c4 .} ==> r8c4 <> 3 nrc-chain[2] n3{r2c7 r7c7} - n3{r8c8 r8c6} ==> r2c6 <> 3 nrczt-braid-cn[6] n4{r4c2 r4c3} - n9{r4c3 r4c4} - {n4 n3}r2c7 - n3{r4c4 r7c4} - n4{r9c3 r9c4} - {n2r9c4 .} ==> r2c2 <> 4 <------- nrczt-braid-rc[6] n6{r3c9 r2c9} - n6{r2c4 r1c5} - n5{r3c7 r5c7} - n6{r2c3 r5c3} - {n6 n2}r5c2 - {n6r5c1 .} ==> r3c9 <> 5 <------- nrczt-whip-rn[7] {n7 n8}r3c4 - {n8 n1}r2c6 - {n1 n3}r1c6 - n3{r8c6 r8c8} - n1{r8c8 r5c8} - n8{r5c8 r5c5} - {n7r5c5 .} ==> r2c4 <> 7 nrczt-braid-rc[6] n6{r3c9 r2c9} - n6{r2c3 r5c3} - n9{r5c3 r5c4} - n8{r3c2 r3c4} - {n8 n3}r4c4 - {n3r2c4 .} ==> r3c2 <> 6 <------- nrczt-braid-cn[6] {n7 n8}r3c4 - n7{r9c6 r9c2} - {n7 n1}r3c2 - n4{r8c4 r9c4} - n2{r9c4 r9c3} - {n1r9c3 .} ==> r8c4 <> 7 <------- nrczt-whip-rn[7] n7{r2c6 r3c4} - n8{r3c4 r2c4} - n3{r2c4 r2c7} - n3{r7c7 r8c8} - n1{r8c8 r5c8} - n8{r5c8 r5c5} - {n7r5c5 .} ==> r2c6 <> 1 interaction block b2 with row r1 for number 1 ==> r1c3 <> 1 naked-pairs-in-a-block {n7 n8}{r2c6 r3c4} ==> r2c4 <> 8 nrc-chain[3] n6{r7c5 r1c5} - {n6 n3}r2c4 - n3{r4c4 r4c5} ==> r7c5 <> 3 nrczt-whip-cn[4] {n4 n3}r2c7 - {n3 n6}r2c4 - {n6 n4}r8c4 - {n4r8c1 .} ==> r2c3 <> 4 nrc-chain[5] {n1 n6}r2c3 - {n6 n3}r2c4 - n3{r2c7 r1c8} - n4{r1c8 r6c8} - n9{r6c8 r2c8} ==> r2c8 <> 1 hxy-cn-chain[4] {r4 r5}c8n8 - {r5 r8}c8n1 - {r8 r1}c8n3 - {r1 r4}c5n3 ==> r4c5 <> 8 naked-single ==> r4c5 = 3 naked-pairs-in-a-column {n1 n6}{r1 r7}c5 ==> r9c5 <> 1 x-wing-in-rows n3{r1 r8}{c6 c8} ==> r7c6 <> 3 naked-triplets-in-a-column {n7 n8 n9}{r3 r4 r5}c4 ==> r9c4 <> 8, r9c4 <> 7, r7c4 <> 8 nrczt-whip-rc[3] n1{r9c2 r9c9} - n8{r9c9 r7c9} - {n8r7c6 .} ==> r7c2 <> 1 nrc-chain[4] n6{r2c4 r1c5} - {n6 n1}r7c5 - {n1 n8}r7c6 - n8{r2c6 r2c2} ==> r2c2 <> 6 nrc-chain[4] n4{r2c1 r2c7} - n4{r7c7 r7c9} - n8{r7c9 r7c6} - {n8 n7}r2c6 ==> r2c1 <> 7 hidden-pairs-in-a-row {n7 n8}r2{c2 c6} ==> r2c2 <> 5 interaction column c2 with block b4 for number 5 ==> r5c1 <> 5 hidden-pairs-in-a-row {n7 n8}r2{c2 c6} ==> r2c2 <> 1 nrc-chain[2] n7{r2c6 r2c2} - n7{r3c1 r8c1} ==> r8c6 <> 7 interaction row r8 with block b7 for number 7 ==> r9c2 <> 7 naked-triplets-in-a-row {n2 n1 n4}r9{c2 c3 c4} ==> r9c9 <> 1 interaction row r9 with block b7 for number 1 ==> r8c2 <> 1 nrc-chain[4] n1{r5c8 r8c8} - {n1 n5}r8c9 - {n5 n8}r9c9 - n8{r9c5 r5c5} ==> r5c8 <> 8 singles ==> r4c8 = 8, r4c4 = 9, r4c3 = 4, r4c2 = 5, r5c3 = 9 interaction column c3 with block b1 for number 6 ==> r3c1 <> 6 hidden-single-in-a-row ==> r3c9 = 6 interaction column c3 with block b1 for number 6 ==> r2c1 <> 6, r1c1 <> 6 interaction column c2 with block b7 for number 4 ==> r8c1 <> 4 naked-pairs-in-a-column {n2 n6}{r5 r7}c2 ==> r9c2 <> 2, r8c2 <> 6 nrc-chain[3] n6{r1c5 r1c3} - n2{r1c3 r9c3} - {n2 n6}r7c2 ==> r7c5 <> 6 naked-singles GRID 0 SOLVED. LEVEL = B-NRCZT7, MOST COMPLEX RULE = NRCZT7 492561837 586327491 713849526 154936782 269784153 837152649 925618374 678493215 341275968 (solve "..53.81.2........982..9.3..9..1.5.2..429..5...1..7...82..7...1..5........3.4....6") ***** SudoRules version 13.7wbis-B2 ***** ..53.81.2........982..9.3..9..1.5.2..429..5...1..7...82..7...1..5........3.4....6 singles ==> r1c2 = 9, r5c5 = 8, r8c4 = 8, r5c9 = 1, r6c1 = 5 interaction column c4 with block b2 for number 5 ==> r2c5 <> 5 x-wing-in-rows n7{r1 r5}{c1 c8} ==> r9c8 <> 7, r9c1 <> 7 naked-single ==> r9c1 = 1 x-wing-in-rows n7{r1 r5}{c1 c8} ==> r8c8 <> 7, r8c1 <> 7 interaction block b7 with column c3 for number 7 ==> r4c3 <> 7, r3c3 <> 7, r2c3 <> 7 x-wing-in-rows n7{r1 r5}{c1 c8} ==> r3c8 <> 7, r2c8 <> 7, r2c1 <> 7 xyz-chain[3] {n6 n7}r2c2 - {n7 n4}r1c1 - {n4 n6}r8c1 ==> r2c1 <> 6 nrc-chain[3] n1{r3c3 r3c6} - n7{r3c6 r2c6} - {n7 n6}r2c2 ==> r3c3 <> 6 nrct-chain[3] n4{r6c6 r4c5} - {n4 n6}r1c5 - n6{r3c4 r6c4} ==> r6c6 <> 6 nrczt-whip-rn[3] n6{r3c4 r3c8} - n6{r5c8 r5c1} - {n6r1c1 .} ==> r2c6 <> 6 nrct-chain[4] {n4 n6}r1c5 - n6{r3c4 r6c4} - {n6 n3}r5c6 - {n3 n4}r4c5 ==> r2c5 <> 4 nrct-chain[4] n4{r6c6 r4c5} - {n4 n6}r1c5 - n6{r3c4 r6c4} - n2{r6c4 r6c6} ==> r6c6 <> 3 nrczt-whip-cn[4] n7{r3c6 r2c6} - n4{r2c6 r6c6} - n2{r6c6 r6c4} - {n6r6c4 .} ==> r3c6 <> 6 nrczt-whip-rc[4] n6{r3c4 r6c4} - {n6 n3}r5c6 - {n3 n4}r4c5 - {n4r1c5 .} ==> r2c5 <> 6 nrczt-whip-rn[5] n1{r8c6 r8c5} - {n1 n2}r2c5 - {n2 n5}r9c5 - n5{r9c8 r7c9} - {n3r7c9 .} ==> r8c6 <> 3 nrczt-braid-rn[5] {n4 n6}r1c5 - n6{r3c4 r6c4} - n7{r1c1 r5c1} - n6{r5c1 r5c8} - {n6r3c8 .} ==> r1c1 <> 4 <------- naked-pairs-in-a-block {n6 n7}{r1c1 r2c2} ==> r2c3 <> 6 nrc-chain[2] n4{r1c8 r1c5} - n4{r4c5 r6c6} ==> r6c8 <> 4 nrczt-whip-cn[2] n4{r1c8 r8c8} - {n4r8c1 .} ==> r2c7 <> 4 nrczt-whip-bn[3] n7{r2c6 r3c6} - n1{r3c6 r3c3} - {n4r3c3 .} ==> r2c6 <> 4 nrct-chain[5] n4{r8c1 r2c1} - n3{r2c1 r2c3} - n3{r6c3 r6c8} - n3{r4c9 r4c5} - n3{r8c5 r8c9} ==> r8c9 <> 4 nrct-chain[4] n7{r3c6 r3c9} - n5{r3c9 r7c9} - n4{r7c9 r4c9} - n4{r6c7 r6c6} ==> r3c6 <> 4 singles ==> r1c5 = 4, r6c6 = 4, r6c4 = 2 nrct-chain[5] {n5 n6}r2c4 - n6{r3c4 r3c8} - n6{r1c8 r1c1} - {n6 n4}r8c1 - n4{r8c8 r2c8} ==> r2c8 <> 5 singles ==> r2c4 = 5; r3c4 = 6 nrczt-whip-cn[5] n4{r8c1 r2c1} - n4{r2c8 r3c8} - n5{r3c8 r9c8} - {n5 n2}r9c5 - {n2r9c7 .} ==> r8c7 <> 4 nrczt-whip-cn[5] n3{r8c9 r4c9} - n4{r4c9 r4c7} - n4{r7c7 r7c9} - n5{r7c9 r7c5} - {n3r7c5 .} ==> r8c8 <> 3 interaction column c8 with block b6 for number 3 ==> r4c9 <> 3 nrc-chain[3] n7{r5c1 r5c8} - n3{r5c8 r6c8} - {n3 n6}r6c3 ==> r5c1 <> 6 xyt-chain[4] {n9 n4}r8c8 - {n4 n6}r8c1 - {n6 n8}r7c2 - {n8 n9}r7c7 ==> r9c8 <> 9 hxy-cn-chain[4] {r8 r6}c8n9 - {r6 r5}c8n3 - {r5 r2}c1n3 - {r2 r8}c1n4 ==> r8c8 <> 4 singles ==> r8c8 = 9, r6c7 = 9 interaction column c8 with block b3 for number 4 ==> r3c9 <> 4 interaction row r8 with block b7 for number 4 ==> r7c3 <> 4 naked-triplets-in-a-column {n6 n7 n3}{r1 r5 r6}c8 ==> r2c8 <> 6 nrct-chain[3] n6{r5c6 r4c5} - n6{r4c7 r2c7} - n6{r2c2 r7c2} ==> r7c6 <> 6 nrc-chain[3] {n3 n6}r4c5 - n6{r5c6 r8c6} - n1{r8c6 r8c5} ==> r8c5 <> 3 hidden-single-in-a-row ==> r8c9 = 3 interaction block b9 with column c7 for number 7 ==> r4c7 <> 7, r2c7 <> 7 hidden-pairs-in-a-column {n2 n7}{r8 r9}c7 ==> r9c7 <> 8 xy-chain[3] {n6 n4}r4c7 - {n4 n8}r7c7 - {n8 n6}r7c2 ==> r4c2 <> 6 interaction block b4 with column c3 for number 6 ==> r8c3 <> 6, r7c3 <> 6 nrc-chain[3] n8{r7c7 r2c7} - n6{r2c7 r2c2} - {n6 n8}r7c2 ==> r7c3 <> 8 singles ==> r7c3 = 9, r7c6 = 3, r5c6 = 6, r4c5 = 3, r9c6 = 9 xy-chain[4] {n1 n2}r8c6 - {n2 n7}r8c7 - {n7 n4}r8c3 - {n4 n1}r3c3 ==> r3c6 <> 1 naked-singles GRID 0 SOLVED. LEVEL = B-NRCZT5, MOST COMPLEX RULE = B-NRCZT5 695348172 374512689 821697345 986135427 742986531 513274968 269753814 457861293 138429756 (solve "6.9.3......1.456..4.26..83....72...4.....81......5..7.9...7...3.14.8......3..92..") ***** SudoRules version 13.7wbis-B2 ***** 6.9.3......1.456..4.26..83....72...4.....81......5..7.9...7...3.14.8......3..92.. hidden-single-in-a-column ==> r5c3 = 7 interaction block b1 with column c2 for number 5 ==> r9c2 <> 5, r7c2 <> 5, r5c2 <> 5, r4c2 <> 5 nrczt-whip-bn[3] n6{r7c3 r9c2} - {n6 n1}r9c5 - {n1r9c9 .} ==> r7c8 <> 6 nrczt-whip-bn[4] n9{r3c9 r3c5} - {n9 n6}r5c5 - n6{r5c9 r4c8} - {n8r4c8 .} ==> r6c9 <> 9 nrczt-whip-cn[5] {n6 n8}r6c3 - n8{r6c9 r9c9} - {n8 n7}r9c2 - n7{r9c1 r2c1} - {n8r2c1 .} ==> r4c2 <> 6 nrczt-whip-rn[5] {n6 n8}r6c3 - n8{r6c9 r4c8} - n6{r4c8 r4c3} - n6{r7c3 r7c2} - {n8r7c2 .} ==> r6c6 <> 6 nrczt-whip-cn[5] n4{r5c2 r6c2} - n2{r6c2 r7c2} - n6{r7c2 r9c2} - n6{r7c3 r7c6} - {n4r7c6 .} ==> r5c2 <> 9, r5c2 <> 3 nrczt-whip-cn[5] {n6 n8}r6c3 - n8{r6c9 r9c9} - n8{r9c2 r7c2} - n2{r7c2 r6c2} - {n4r6c2 .} ==> r5c2 <> 6 nrczt-whip-cn[5] {n6 n8}r6c3 - n8{r6c9 r9c9} - n8{r9c2 r7c2} - n2{r7c2 r5c2} - {n4r5c2 .} ==> r6c2 <> 6 interaction column c2 with block b7 for number 6 ==> r7c3 <> 6 nrczt-whip-cn[6] n6{r9c5 r5c5} - n6{r5c9 r6c9} - {n6 n8}r6c3 - n8{r6c9 r4c8} - n8{r7c8 r7c2} - {n6r7c2 .} ==> r9c8 <> 6 nrczt-whip-rc[6] n1{r9c5 r3c5} - {n1 n7}r3c6 - {n7 n2}r1c6 - n2{r7c6 r7c2} - n6{r7c2 r9c2} - {n6r9c5 .} ==> r7c4 <> 1 nrct-chain[6] n4{r6c6 r7c6} - {n4 n5}r7c7 - {n5 n2}r7c4 - n2{r8c6 r8c1} - n5{r8c1 r8c4} - n3{r8c4 r8c6} ==> r6c6 <> 3 nrczt-braid-rn[6] n2{r8c1 r7c2} - n2{r6c2 r6c9} - n8{r6c9 r4c8} - n6{r6c9 r6c3} - n6{r4c3 r4c6} - {n6r7c6 .} ==> r5c1 <> 2 <------- nrczt-whip-rc[4] n4{r6c2 r5c2} - n2{r5c2 r6c1} - {n2 n6}r6c9 - {n6r6c3 .} ==> r6c2 <> 8 nrczt-whip-bn[6] n2{r8c1 r7c2} - n6{r7c2 r7c6} - {n6 n3}r8c6 - {n3 n1}r4c6 - n1{r6c6 r6c1} - {n2r6c1 .} ==> r8c4 <> 2 nrct-chain[5] n9{r4c2 r6c2} - n4{r6c2 r5c2} - n2{r5c2 r6c1} - n2{r8c1 r8c6} - n3{r8c6 r4c6} ==> r4c2 <> 3 xyzt-chain[5] {n9 n8}r4c2 - {n8 n6}r6c3 - {n6 n5}r4c3 - {n5 n3}r4c7 - {n3 n9}r6c7 ==> r4c8 <> 9 nrczt-whip-cn[4] {n6 n9}r5c5 - n9{r6c4 r2c4} - n9{r2c8 r8c8} - {n6r8c8 .} ==> r5c9 <> 6 nrczt-whip-rn[5] n4{r6c2 r5c2} - n2{r5c2 r6c1} - n2{r8c1 r8c6} - n3{r8c6 r8c4} - {n3r5c4 .} ==> r6c2 <> 3 hidden-single-in-a-column ==> r2c2 = 3 nrc-chain[2] n7{r8c7 r1c7} - n7{r2c9 r2c1} ==> r8c1 <> 7 interaction row r8 with block b9 for number 7 ==> r9c9 <> 7 nrczt-whip-rn[5] n8{r9c9 r7c8} - n1{r7c8 r7c6} - n1{r9c5 r3c5} - n9{r3c5 r2c4} - {n8r2c4 .} ==> r9c1 <> 8 nrczt-whip-rc[5] n5{r5c9 r5c1} - {n5 n7}r9c1 - n7{r2c1 r2c9} - {n7 n4}r1c7 - {n4r7c7 .} ==> r4c7 <> 5 naked-pairs-in-a-block {n3 n9}{r4c7 r6c7} ==> r5c9 <> 9, r5c8 <> 9 interaction row r5 with block b5 for number 9 ==> r6c4 <> 9 interaction block b6 with column c7 for number 9 ==> r8c7 <> 9 nrc-chain[4] n1{r4c1 r4c6} - n6{r4c6 r5c5} - n9{r5c5 r5c4} - n3{r5c4 r5c1} ==> r4c1 <> 3 nrct-chain[6] n3{r8c6 r4c6} - n3{r4c7 r6c7} - n9{r6c7 r6c2} - n4{r6c2 r5c2} - n2{r5c2 r6c1} - n2{r8c1 r8c6} ==> r8c6 <> 6 interaction row r8 with block b9 for number 6 ==> r9c9 <> 6 hidden-pairs-in-a-block {n6 n9}{r8c8 r8c9} ==> r8c9 <> 7 hidden-single-in-a-block ==> r8c7 = 7 hidden-pairs-in-a-block {n6 n9}{r8c8 r8c9} ==> r8c9 <> 5, r8c8 <> 5 nrct-chain[4] n6{r7c6 r9c5} - n1{r9c5 r3c5} - {n1 n7}r3c6 - {n7 n2}r1c6 ==> r7c6 <> 2 hxyt-cn-chain[5] {r1 r8}c6n2 - {r8 r6}c1n2 - {r6 r4}c1n1 - {r4 r2}c1n8 - {r2 r1}c4n8 ==> r1c4 <> 2 nrc-chain[4] n7{r2c9 r2c1} - n8{r2c1 r2c4} - n2{r2c4 r1c6} - n7{r1c6 r3c6} ==> r3c9 <> 7 nrc-chain[5] n7{r9c1 r9c2} - n6{r9c2 r9c5} - n1{r9c5 r3c5} - {n1 n8}r1c4 - n8{r2c4 r2c1} ==> r2c1 <> 7 singles ==> r2c1 = 8, r1c4 = 8, r9c1 = 7, r2c9 = 7 hidden-pairs-in-a-row {n6 n8}r6{c3 c9} ==> r6c9 <> 2 interaction row r6 with block b4 for number 2 ==> r5c2 <> 2 naked-single ==> r5c2 = 4 nrc-chain[3] n1{r6c4 r9c4} - {n1 n6}r9c5 - n6{r7c6 r4c6} ==> r4c6 <> 1 hidden-single-in-a-row ==> r4c1 = 1 hidden-pairs-in-a-row {n1 n4}r6{c4 c6} ==> r6c4 <> 3 nrc-chain[3] n5{r4c3 r5c1} - n3{r5c1 r5c4} - {n3 n6}r4c6 ==> r4c3 <> 6 singles GRID 0 SOLVED. LEVEL = B-NRCZT6, MOST COMPLEX RULE = B-NRCZT6 679832541 831945627 452617839 198726354 547398162 326154978 965271483 214583796 783469215 (solve ".1.5..69.....9...3...3.7.4..5.....8.....3....6.....1..9..........29485...31..6...") ***** SudoRules version 13.7wbis-B2 ***** .1.5..69.....9...3...3.7.4..5.....8.....3....6.....1..9..........29485...31..6... singles ==> r8c1 = 7, r8c2 = 6, r8c9 = 1, r8c8 = 3, r3c5 = 1, r2c8 = 1, r3c9 = 5, r2c4 = 6, r4c5 = 6, r3c3 = 6, r3c2 = 9, r1c5 = 8, r6c3 = 3, r1c1 = 3, r4c7 = 3, r7c6 = 3, r7c4 = 1 interaction column c9 with block b9 for number 8 ==> r9c7 <> 8, r7c7 <> 8 interaction column c6 with block b5 for number 5 ==> r6c5 <> 5 interaction column c4 with block b5 for number 4 ==> r6c6 <> 4, r5c6 <> 4, r4c6 <> 4 interaction block b2 with column c6 for number 2 ==> r6c6 <> 2, r5c6 <> 2, r4c6 <> 2 naked-pairs-in-a-row {n2 n7}r9{c4 c8} ==> r9c9 <> 7, r9c9 <> 2, r9c7 <> 7, r9c7 <> 2, r9c5 <> 7, r9c5 <> 2 singles ==> r9c5 = 5, r7c3 = 5, r2c1 = 5 nrc-chain[2] n2{r6c5 r7c5} - n2{r9c4 r9c8} ==> r6c8 <> 2 nrc-chain[2] n7{r6c5 r7c5} - n7{r9c4 r9c8} ==> r6c8 <> 7 singles ==> r6c8 = 5, r6c6 = 9, r4c6 = 1, r5c6 = 5, r5c1 = 1 nrczt-whip-bn[2] n7{r1c3 r1c9} - {n7r6c9 .} ==> r5c3 <> 7 nrc-chain[3] n2{r2c2 r3c1} - n8{r3c1 r9c1} - n4{r9c1 r7c2} ==> r2c2 <> 4 interaction block b1 with column c3 for number 4 ==> r5c3 <> 4, r4c3 <> 4 nrczt-whip-rn[3] n7{r1c9 r1c3} - n7{r4c3 r4c4} - {n7r9c4 .} ==> r7c9 <> 7 nrct-chain[4] {n7 n2}r1c9 - n2{r3c7 r3c1} - n2{r4c1 r4c4} - {n2 n7}r6c5 ==> r6c9 <> 7 xyzt-chain[4] {n4 n2}r6c9 - {n2 n7}r6c5 - {n7 n2}r4c4 - {n2 n4}r4c1 ==> r4c9 <> 4 nrczt-whip-rn[4] {n7 n2}r1c9 - {n2 n9}r4c9 - {n9 n7}r4c3 - {n7r1c3 .} ==> r5c9 <> 7 nrczt-braid-rn[4] n2{r9c8 r9c4} - n2{r2c7 r1c9} - n2{r3c7 r3c1} - {n2r4c9 .} ==> r7c7 <> 2 <------- nrc-chain[4] n4{r4c4 r4c1} - n4{r9c1 r7c2} - {n4 n7}r7c7 - n7{r9c8 r9c4} ==> r4c4 <> 7 naked-pairs-in-a-row {n2 n4}r4{c1 c4} ==> r4c9 <> 2 x-wing-in-rows n7{r1 r4}{c3 c9} ==> r2c3 <> 7 xy-chain[3] {n2 n8}r3c1 - {n8 n4}r2c3 - {n4 n2}r2c6 ==> r2c2 <> 2 singles GRID 0 SOLVED. LEVEL = B-NRCZT4, MOST COMPLEX RULE = B-NRCZT4 314582697 578694213 296317845 457261389 129835476 683479152 945123768 762948531 831756924 (solve "...1.....68..35...1.9..8..3.......5.9.7....1.4.....927.1..5.6....6341...5.42..7..") ***** SudoRules version 13.7wbis-B2 ***** ...1.....68..35...1.9..8..3.......5.9.7....1.4.....927.1..5.6....6341...5.42..7.. singles ==> r2c3 = 2, r9c9 = 1, r2c7 = 1 interaction row r8 with block b7 for number 7 ==> r7c1 <> 7 interaction column c8 with block b3 for number 6 ==> r1c9 <> 6 xy-chain[3] {n3 n6}r6c6 - {n6 n9}r9c6 - {n9 n3}r9c2 ==> r6c2 <> 3 nrc-chain[3] n3{r9c8 r9c2} - {n3 n8}r7c3 - n8{r7c4 r9c5} ==> r9c8 <> 8 singles ==> r9c5 = 8, r9c6 = 6, r6c6 = 3 interaction block b8 with row r7 for number 9 ==> r7c9 <> 9, r7c8 <> 9 nrc-chain[4] n5{r5c4 r5c2} - n5{r3c2 r3c7} - n2{r3c7 r3c5} - {n2 n6}r5c5 ==> r5c4 <> 6 nrc-chain[4] {n6 n2}r5c5 - n2{r3c5 r3c7} - n5{r3c7 r3c2} - {n5 n6}r6c2 ==> r6c5 <> 6 singles ==> r6c5 = 1, r4c3 = 1 nrczt-whip-bn[3] {n3 n8}r7c3 - n8{r8c1 r4c1} - {n3r4c1 .} ==> r9c2 <> 3 singles ==> r9c2 = 9, r9c8 = 3 nrc-chain[4] n5{r8c9 r1c9} - {n5 n3}r1c3 - n3{r7c3 r7c1} - n2{r7c1 r7c9} ==> r8c9 <> 2 nrc-chain[4] n6{r6c2 r6c4} - {n6 n2}r5c5 - n2{r3c5 r3c7} - n5{r3c7 r3c2} ==> r6c2 <> 5 naked-single ==> r6c2 = 6 nrct-chain[4] n7{r2c8 r2c4} - n7{r3c5 r4c5} - n9{r4c5 r1c5} - n6{r1c5 r1c8} ==> r1c8 <> 7 nrczt-braid-cn[4] n2{r1c9 r7c9} - n5{r1c9 r8c9} - {n2 n8}r8c7 - {n8r8c8 .} ==> r1c9 <> 8 <------- nrc-chain[4] n8{r1c7 r1c8} - n6{r1c8 r1c5} - {n6 n2}r5c5 - n2{r3c5 r3c7} ==> r1c7 <> 2 nrc-chain[3] n2{r1c9 r3c7} - n5{r3c7 r3c2} - n4{r3c2 r1c2} ==> r1c9 <> 4 nrc-chain[3] n5{r3c2 r3c7} - n2{r3c7 r8c7} - {n2 n7}r8c2 ==> r3c2 <> 7 interaction block b1 with row r1 for number 7 ==> r1c6 <> 7, r1c5 <> 7 xy-chain[4] {n4 n5}r3c2 - {n5 n3}r1c3 - {n3 n8}r7c3 - {n8 n4}r7c8 ==> r3c8 <> 4 nrc-chain[3] n6{r4c4 r3c4} - {n6 n7}r3c8 - n7{r3c5 r4c5} ==> r4c5 <> 6, r4c4 <> 7 nrc-chain[3] n9{r1c5 r4c5} - n7{r4c5 r4c6} - {n7 n9}r7c6 ==> r1c6 <> 9 naked-pairs-in-a-column {n2 n4}{r1 r5}c6 ==> r4c6 <> 4, r4c6 <> 2 nrc-chain[3] n6{r1c8 r1c5} - n9{r1c5 r2c4} - {n9 n4}r2c9 ==> r1c8 <> 4 nrc-chain[3] n2{r3c7 r1c9} - {n2 n4}r1c6 - n4{r1c2 r3c2} ==> r3c7 <> 4 nrczt-whip-cn[2] n4{r5c6 r1c6} - {n4r1c7 .} ==> r5c9 <> 4 nrc-chain[4] {n2 n6}r5c5 - n6{r4c4 r3c4} - {n6 n7}r3c8 - n7{r3c5 r4c5} ==> r4c5 <> 2 interaction row r4 with block b4 for number 2 ==> r5c2 <> 2 naked-pairs-in-a-block {n7 n9}{r4c5 r4c6} ==> r4c4 <> 9 nrc-chain[4] n2{r8c7 r3c7} - n5{r3c7 r3c2} - {n5 n3}r5c2 - {n3 n2}r4c2 ==> r8c2 <> 2 singles ==> r8c2 = 7, r1c1 = 7, r4c2 = 2 hxyzt-cn-chain[4] c2n3{r1 r5} - c7n3{r5 r4} - c7n4{r4 r5} - c6n4{r5 r1} ==> r1c2 <> 4 singles GRID 0 SOLVED. LEVEL = B-NRCZT4, MOST COMPLEX RULE = B-NRCZT4 735194862 682735149 149628573 321879456 957462318 468513927 213957684 876341295 594286731 (solve "...2.....87.9.......5836...3.......5.....36...214....7.......1...431..8.6..5.94..") ***** SudoRules version 13.7wbis-B2 ***** ...2.....87.9.......5836...3.......5.....36...214....7.......1...431..8.6..5.94.. singles ==> r6c5 = 6, r7c4 = 6, r8c9 = 6, r9c2 = 1 interaction column c4 with block b5 for number 7 ==> r5c5 <> 7, r4c6 <> 7, r4c5 <> 7 interaction column c4 with block b5 for number 1 ==> r4c6 <> 1 interaction row r3 with block b3 for number 7 ==> r1c8 <> 7, r1c7 <> 7 interaction column c8 with block b3 for number 5 ==> r2c7 <> 5, r1c7 <> 5 hidden-pairs-in-a-column {n5 n6}{r1 r2}c8 ==> r2c8 <> 4, r2c8 <> 3, r2c8 <> 2, r1c8 <> 9, r1c8 <> 4, r1c8 <> 3 nrc-chain[3] {n9 n5}r6c1 - n5{r6c6 r5c5} - n9{r5c5 r4c5} ==> r4c3 <> 9, r4c2 <> 9 nrczt-whip-rn[5] n9{r8c2 r8c7} - n5{r8c7 r7c7} - n7{r7c7 r9c8} - n3{r9c8 r6c8} - {n9r6c8 .} ==> r7c1 <> 9 nrczt-whip-rn[7] n8{r1c9 r5c9} - n1{r5c9 r4c7} - {n1 n2}r2c7 - n2{r3c9 r3c1} - n2{r8c1 r8c6} - {n2 n8}r4c6 - {n8r6c6 .} ==> r1c9 <> 3 nrczt-braid-rn[7] {n9 n5}r6c1 - n5{r6c6 r5c5} - {n5 n4}r2c5 - {n9 n4}r3c2 - n4{r1c1 r1c9} - n9{r1c1 r1c7} - {n8r1c9 .} ==> r3c1 <> 9 <------- nrczt-braid-rn[7] n9{r6c8 r3c8} - n7{r3c8 r3c7} - {n9 n4}r3c2 - n1{r4c7 r5c9} - n1{r3c7 r3c1} - {n1 n9}r1c1 - {n9r6c7 .} ==> r4c7 <> 9 <------- nrczt-whip-rn[7] n2{r3c9 r3c1} - n1{r3c1 r1c1} - n4{r1c1 r5c1} - n4{r5c9 r4c8} - n9{r4c8 r4c5} - n2{r4c5 r4c6} - {n2r8c6 .} ==> r2c7 <> 2 nrct-chain[3] {n3 n1}r2c7 - n1{r3c9 r3c1} - n2{r3c1 r2c3} ==> r2c3 <> 3 interaction row r2 with block b3 for number 3 ==> r1c7 <> 3 nrczt-whip-bn[4] n1{r5c9 r4c7} - n8{r4c7 r6c7} - {n8 n9}r1c7 - {n9r8c7 .} ==> r5c9 <> 9 nrct-chain[5] n8{r1c9 r5c9} - n1{r5c9 r4c7} - {n1 n3}r2c7 - {n3 n9}r6c7 - n9{r8c7 r7c9} ==> r1c9 <> 9 nrct-chain[5] n9{r4c5 r4c8} - n4{r4c8 r4c2} - {n4 n9}r3c2 - n9{r3c9 r7c9} - n9{r7c3 r5c3} ==> r5c5 <> 9 hidden-single-in-a-block ==> r4c5 = 9 nrct-chain[5] {n4 n2}r4c8 - {n2 n8}r4c6 - n8{r6c6 r6c7} - {n8 n1}r4c7 - {n1 n4}r5c9 ==> r5c8 <> 4 nrct-chain[5] {n4 n9}r3c2 - n9{r3c9 r7c9} - n9{r7c3 r5c3} - {n9 n2}r5c8 - {n2 n4}r4c8 ==> r4c2 <> 4 hidden-single-in-a-row ==> r4c8 = 4 nrct-chain[4] n4{r5c1 r5c2} - {n4 n9}r3c2 - n9{r3c9 r1c7} - n9{r8c7 r8c1} ==> r5c1 <> 9 nrct-chain[5] n4{r5c1 r5c2} - {n4 n9}r3c2 - n9{r3c9 r7c9} - n9{r7c3 r5c3} - {n9 n5}r6c1 ==> r5c1 <> 5 nrct-chain[6] {n2 n9}r5c8 - n9{r6c8 r6c1} - n5{r6c1 r5c2} - n4{r5c2 r5c1} - {n4 n1}r1c1 - {n1 n2}r3c1 ==> r3c8 <> 2 nrct-chain[4] {n3 n2}r9c9 - n2{r3c9 r3c7} - n7{r3c7 r3c8} - {n7 n3}r9c8 ==> r9c3 <> 3 interaction row r9 with block b9 for number 3 ==> r7c9 <> 3, r7c7 <> 3 xyt-chain[4] {n9 n2}r7c9 - {n2 n3}r9c9 - {n3 n7}r9c8 - {n7 n9}r3c8 ==> r3c9 <> 9 hidden-single-in-a-column ==> r7c9 = 9 nrczt-whip-bn[4] {n2 n9}r5c8 - n9{r5c3 r1c3} - n9{r3c2 r3c7} - {n2r3c7 .} ==> r5c9 <> 2 nrct-chain[4] n8{r6c6 r6c7} - {n8 n1}r5c9 - {n1 n2}r4c7 - n2{r5c8 r5c5} ==> r5c5 <> 8 interaction column c5 with block b8 for number 8 ==> r7c6 <> 8 nrct-chain[5] n9{r5c3 r1c3} - {n9 n4}r3c2 - {n4 n1}r1c1 - {n1 n8}r1c7 - n8{r6c7 r5c9} ==> r5c3 <> 8 nrc-chain[3] n5{r5c5 r5c2} - n8{r5c2 r5c9} - n8{r6c7 r6c6} ==> r6c6 <> 5 singles ==> r6c6 = 8, r4c6 = 2, r5c5 = 5, r2c5 = 4, r1c5 = 7, r8c6 = 7, r7c6 = 4, r6c1 = 5, r5c8 = 2 hidden-triplets-in-a-column {n2 n5 n7}{r3 r8 r7}c7 ==> r3c7 <> 9, r3c7 <> 1 nrc-chain[3] {n7 n2}r3c7 - n2{r8c7 r8c1} - {n2 n7}r7c1 ==> r7c7 <> 7 singles GRID 0 SOLVED. LEVEL = B-NRCZT7, MOST COMPLEX RULE = B-NRCZT7 139275864 876941352 245836791 368792145 497153628 521468937 783624519 954317286 612589473 (solve "...6..1..17..38...6......83.4.2.3..8....4.2.........49.5.3...26....2..7...8..1.9.") ***** SudoRules version 13.7wbis-B2 ***** ...6..1..17..38...6......83.4.2.3..8....4.2.........49.5.3...26....2..7...8..1.9. singles ==> r1c8 = 5, r2c8 = 6, r4c8 = 1, r5c8 = 3,> r7c3 = 1, r8c9 = 1 interaction column c3 with block b4 for number 7 ==> r6c1 <> 7, r5c1 <> 7, r4c1 <> 7 interaction column c1 with block b4 for number 5 ==> r6c3 <> 5, r5c3 <> 5, r4c3 <> 5 nrc-chain[3] n9{r2c7 r3c7} - n7{r3c7 r1c9} - {n7 n9}r1c5 ==> r2c4 <> 9 xyzt-chain[5] {n7 n9}r1c5 - {n9 n8}r7c5 - {n8 n4}r7c7 - {n4 n9}r2c7 - {n9 n7}r3c7 ==> r3c5 <> 7 nrczt-whip-rc[7] {n4 n5}r2c4 - n5{r3c6 r3c3} - n4{r3c3 r2c3} - n4{r2c9 r9c9} - n4{r7c7 r7c1} - n7{r7c1 r9c1} - {n7r9c4 .} ==> r1c6 <> 4 nrczt-whip-rc[7] n7{r5c9 r1c9} - n2{r1c9 r2c9} - n4{r2c9 r9c9} - {n4 n8}r7c7 - n8{r8c7 r8c4} - n9{r8c4 r3c4} - {n9r1c5 .} ==> r5c4 <> 7 <------- nrczt-braid-rc[7] {n5 n4}r9c9 - {n4 n2}r2c9 - {n2 n7}r1c9 - {n7 n9}r1c5 - {n4 n8}r7c7 - {n8 n7}r7c5 - {n7r9c4 .} ==> r9c5 <> 5 <------- nrczt-whip-rc[6] {n9 n2}r3c2 - n2{r3c3 r6c3} - n3{r6c3 r8c3} - {n3 n6}r9c2 - {n6 n7}r9c5 - {n7r1c5 .} ==> r1c3 <> 9 nrczt-braid-bn[6] n8{r6c5 r7c5} - {n8 n4}r7c7 - n6{r5c6 r8c6} - n4{r8c6 r3c6} - {n4 n5}r2c4 - {n5r8c4 .} ==> r6c5 <> 6 <------- nrczt-whip-rn[7] n9{r5c6 r4c5} - {n9 n7}r1c5 - n7{r1c9 r5c9} - {n7 n6}r5c3 - n6{r5c6 r6c6} - n7{r6c6 r7c6} - {n9r7c6 .} ==> r5c1 <> 9 nrczt-whip-cn[5] {n8 n5}r5c1 - {n5 n9}r4c1 - n9{r5c3 r5c6} - n9{r7c6 r7c5} - {n8r7c5 .} ==> r5c4 <> 8 interaction row r5 with block b4 for number 8 ==> r6c2 <> 8, r6c1 <> 8 nrct-chain[3] n8{r6c4 r6c5} - n1{r6c5 r3c5} - n5{r3c5 r4c5} ==> r6c4 <> 5 nrczt-whip-cn[6] n1{r3c5 r6c5} - n1{r6c2 r5c2} - n8{r5c2 r1c2} - n9{r1c2 r1c1} - {n9 n5}r4c1 - {n5r4c5 .} ==> r3c5 <> 9 nrct-chain[7] n8{r8c4 r6c4} - n8{r6c5 r7c5} - {n8 n4}r7c7 - {n4 n9}r2c7 - {n9 n7}r3c7 - n7{r3c4 r9c4} - {n7 n9}r7c6 ==> r8c4 <> 9 nrczt-whip-cn[3] n5{r4c5 r3c5} - n1{r3c5 r3c4} - {n9r3c4 .} ==> r5c4 <> 5 nrc-chain[4] n8{r1c2 r5c2} - n1{r5c2 r5c4} - n9{r5c4 r3c4} - {n9 n2}r3c2 ==> r1c2 <> 2 nrczt-whip-rn[4] {n7 n9}r1c5 - n9{r3c4 r5c4} - n1{r5c4 r6c4} - {n8r6c4 .} ==> r6c5 <> 7 nrct-chain[5] n8{r1c2 r1c1} - n8{r5c1 r5c2} - n1{r5c2 r5c4} - n9{r5c4 r3c4} - n9{r1c6 r1c2} ==> r1c2 <> 3 nrczt-whip-cn[6] {n7 n9}r1c5 - {n9 n8}r7c5 - n8{r8c4 r6c4} - n7{r6c4 r3c4} - n1{r3c4 r5c4} - {n9r5c4 .} ==> r9c5 <> 7 naked-single ==> r9c5 = 6 nrczt-braid-rn[4] {n5 n7}r5c9 - {n5 n9}r4c1 - {n7 n6}r5c3 - {n6r4c3 .} ==> r4c7 <> 5 hxyt-cn-chain[4] {r6 r7}c5n8 - {r7 r8}c7n8 - {r8 r9}c7n3 - {r9 r6}c7n5 ==> r6c5 <> 5 nrc-chain[3] n5{r4c5 r3c5} - n1{r3c5 r3c4} - {n1 n9}r5c4 ==> r4c5 <> 9 interaction row r4 with block b4 for number 9 ==> r5c3 <> 9, r5c2 <> 9 xy-chain[3] {n9 n7}r1c5 - {n7 n5}r4c5 - {n5 n9}r4c1 ==> r1c1 <> 9 nrc-chain[4] {n4 n5}r2c4 - {n5 n1}r3c5 - {n1 n8}r6c5 - n8{r7c5 r8c4} ==> r8c4 <> 4 nrct-chain[4] n1{r5c2 r5c4} - n9{r5c4 r3c4} - n9{r1c5 r1c2} - n8{r1c2 r5c2} ==> r5c2 <> 6 nrczt-whip-rc[4] n9{r1c2 r8c2} - n6{r8c2 r6c2} - {n6 n7}r5c3 - {n7r4c3 .} ==> r3c3 <> 9 nrczt-whip-rc[4] n9{r1c2 r8c2} - n6{r8c2 r6c2} - {n6 n7}r5c3 - {n7r4c3 .} ==> r2c3 <> 9 hidden-single-in-a-row ==> r2c7 = 9 interaction block b1 with column c2 for number 9 ==> r8c2 <> 9 nrc-chain[4] n9{r3c2 r1c2} - n8{r1c2 r5c2} - n1{r5c2 r5c4} - n9{r5c4 r5c6} ==> r3c6 <> 9 nrczt-whip-rc[5] {n9 n7}r1c5 - n7{r1c9 r5c9} - {n7 n6}r5c3 - {n6 n5}r5c6 - {n5r4c5 .} ==> r1c6 <> 9 nrczt-whip-bn[5] {n4 n5}r2c4 - n5{r8c4 r8c6} - n4{r8c6 r7c6} - n9{r7c6 r7c5} - {n9r1c5 .} ==> r3c4 <> 4 nrc-chain[3] n2{r1c6 r3c6} - n4{r3c6 r2c4} - {n4 n2}r2c9 ==> r1c9 <> 2 hidden-single-in-a-block ==> r2c9 = 2 nrczt-braid-bn[5] n5{r8c4 r8c6} - {n5 n4}r2c4 - n4{r9c4 r7c6} - n9{r7c6 r7c5} - {n9r1c5 .} ==> r3c4 <> 5 nrct-chain[5] n3{r9c7 r8c7} - n8{r8c7 r8c4} - n5{r8c4 r8c6} - n5{r9c4 r2c4} - n4{r2c4 r9c4} ==> r9c7 <> 4 nrct-chain[5] {n3 n5}r9c7 - {n5 n4}r9c9 - n4{r9c4 r2c4} - n5{r2c4 r8c4} - n8{r8c4 r8c7} ==> r8c7 <> 3 singles ==> r9c7 = 3, r9c2 = 2, r3c2 = 9, r1c2 = 8, r5c2 = 1, r5c4 = 9, r5c1 = 8, r1c5 = 9 hidden-pairs-in-a-row {n1 n8}r6{c4 c5} ==> r6c4 <> 7 nrc-chain[2] n5{r9c4 r9c9} - n5{r5c9 r5c6} ==> r8c6 <> 5 interaction block b8 with column c4 for number 5 ==> r2c4 <> 5 singles ==> r2c4 = 4, r2c3 = 5 hidden-pairs-in-a-column {n4 n9}{r7 r8}c6 ==> r7c6 <> 7 hidden-pairs-in-a-row {n5 n8}r8{c4 c7} ==> r8c7 <> 4 nrc-chain[3] n7{r7c1 r7c5} - {n7 n5}r4c5 - {n5 n9}r4c1 ==> r7c1 <> 9 singles ==> r7c6 = 9, r8c6 = 4 interaction column c3 with block b1 for number 4 ==> r1c1 <> 4 hxy-rn-chain[4] {c1 c5}r4n5 - {c5 c6}r3n5 - {c6 c3}r3n2 - {c3 c1}r6n2 ==> r6c1 <> 5 singles GRID 0 SOLVED. LEVEL = B-NRCZT7, MOST COMPLEX RULE = B-NRCZT7 382697154 175438962 694152783 549273618 816945237 237816549 751389426 963524871 428761395 (solve "15.....6.8.4..95..........42.6.43....1.7....6..5.1.2.7....2...5.4..9.1..3....5...") ***** SudoRules version 13.7wbis-B2 ***** 15.....6.8.4..95..........42.6.43....1.7....6..5.1.2.7....2...5.4..9.1..3....5... singles ==> r4c2 = 7, r5c6 = 2, r8c1 = 5 interaction row r8 with block b8 for number 6 ==> r9c5 <> 6 interaction column c5 with block b2 for number 6 ==> r3c6 <> 6, r3c4 <> 6, r2c4 <> 6 interaction row r8 with block b8 for number 6 ==> r9c4 <> 6, r7c6 <> 6, r7c4 <> 6 interaction column c5 with block b2 for number 3 ==> r3c4 <> 3, r2c4 <> 3, r1c4 <> 3 naked-pairs-in-a-column {n4 n9}{r5 r6}c1 ==> r7c1 <> 9, r3c1 <> 9 interaction column c1 with block b4 for number 9 ==> r6c2 <> 9 interaction column c1 with block b4 for number 9 ==> r5c3 <> 9 xyz-chain[3] {n8 n6}r6c6 - {n6 n7}r8c6 - {n7 n8}r9c5 ==> r7c6 <> 8 nrc-chain[3] {n7 n6}r3c1 - n6{r2c2 r2c5} - n7{r2c5 r2c8} ==> r3c8 <> 7, r3c7 <> 7 nrczt-whip-cn[3] n7{r2c5 r2c8} - n7{r8c8 r8c3} - {n7r7c1 .} ==> r3c6 <> 7 nrct-chain[4] {n8 n7}r9c5 - n7{r1c5 r1c6} - n7{r1c7 r7c7} - n6{r7c7 r9c7} ==> r9c7 <> 8 nrct-chain[5] n6{r9c7 r7c7} - n6{r7c1 r3c1} - n6{r3c5 r2c5} - n7{r2c5 r2c8} - n7{r9c8 r9c7} ==> r9c7 <> 9, r9c7 <> 4 nrct-chain[4] {n8 n7}r9c5 - n7{r1c5 r1c6} - n4{r1c6 r7c6} - n4{r9c4 r9c8} ==> r9c8 <> 8 nrczt-whip-bn[5] {n2 n1}r2c4 - n1{r9c4 r7c6} - {n1 n8}r3c6 - {n8 n4}r1c4 - {n4r9c4 .} ==> r3c4 <> 2 nrczt-whip-rn[3] {n2 n1}r2c4 - n1{r3c6 r3c8} - {n2r3c8 .} ==> r2c2 <> 2 nrczt-braid-rc[5] n1{r9c4 r7c6} - n4{r7c6 r1c6} - {n1 n2}r2c4 - {n2 n8}r1c4 - {n8r3c6 .} ==> r3c4 <> 1 <------- nrczt-braid-rn[5] n6{r2c2 r2c5} - n7{r2c5 r2c8} - {n6 n7}r7c1 - n7{r1c7 r9c7} - {n6r9c7 .} ==> r7c2 <> 6 <------- nrczt-whip-rn[6] {n9 n8}r4c7 - {n8 n3}r3c7 - {n3 n7}r1c7 - n7{r2c8 r2c5} - n6{r2c5 r2c2} - {n3r2c2 .} ==> r7c7 <> 9 nrczt-whip-rn[6] {n8 n9}r4c7 - {n9 n3}r3c7 - {n3 n7}r1c7 - n7{r2c8 r2c5} - n6{r2c5 r2c2} - {n3r2c2 .} ==> r7c7 <> 8 nrczt-whip-cn[6] n5{r4c8 r4c4} - {n5 n8}r3c4 - {n8 n1}r3c6 - n1{r3c8 r2c8} - n7{r2c8 r1c7} - {n8r1c7 .} ==> r4c8 <> 8 nrczt-whip-rc[6] n7{r2c8 r2c5} - n7{r9c5 r9c3} - n1{r9c3 r7c3} - n9{r7c3 r7c2} - n8{r7c2 r7c4} - {n8r9c5 .} ==> r7c8 <> 7 nrczt-whip-cn[6] {n8 n7}r9c5 - n7{r2c5 r2c8} - n7{r9c8 r7c7} - n3{r7c7 r7c8} - n3{r6c8 r5c7} - {n4r5c7 .} ==> r7c4 <> 8 nrczt-whip-rn[6] {n9 n8}r4c7 - {n8 n3}r3c7 - {n3 n7}r1c7 - n7{r2c8 r2c5} - n6{r2c5 r2c2} - {n3r2c2 .} ==> r5c7 <> 9 nrczt-whip-rn[6] {n8 n9}r4c7 - {n9 n3}r3c7 - {n3 n7}r1c7 - n7{r2c8 r2c5} - n6{r2c5 r2c2} - {n3r2c2 .} ==> r5c7 <> 8 nrczt-whip-rn[5] n8{r7c2 r7c8} - n8{r5c8 r5c5} - n5{r5c5 r3c5} - {n5 n8}r3c4 - {n8r9c4 .} ==> r8c3 <> 8 nrczt-whip-rn[6] n3{r6c8 r5c7} - n4{r5c7 r7c7} - n3{r7c7 r7c4} - n4{r7c4 r9c4} - n1{r9c4 r2c4} - {n1r3c6 .} ==> r3c8 <> 3 nrczt-braid-rc[6] {n8 n5}r5c5 - n5{r5c8 r4c8} - {n8 n1}r3c6 - n1{r3c8 r2c8} - n7{r2c8 r2c5} - {n8r9c5 .} ==> r1c5 <> 8 <------- xyt-chain[5] {n8 n7}r9c5 - {n7 n3}r1c5 - {n3 n6}r2c5 - {n6 n3}r2c2 - {n3 n8}r6c2 ==> r9c2 <> 8 nrct-chain[6] n6{r8c4 r6c4} - {n6 n8}r6c6 - {n8 n3}r6c2 - {n3 n6}r2c2 - n6{r3c2 r3c5} - n8{r3c5 r9c5} ==> r8c4 <> 8 nrczt-braid-rn[6] {n8 n7}r9c5 - n7{r2c5 r2c8} - {n8 n1}r3c6 - {n8 n5}r3c4 - n1{r2c8 r4c8} - {n5r4c8 .} ==> r3c5 <> 8 <------- nrc-chain[3] n8{r9c5 r5c5} - n5{r5c5 r3c5} - {n5 n8}r3c4 ==> r9c4 <> 8 nrczt-whip-cn[5] n8{r9c5 r5c5} - n5{r5c5 r4c4} - {n5 n8}r3c4 - n8{r4c4 r4c7} - {n8r6c8 .} ==> r9c9 <> 8 nrczt-whip-rc[3] n9{r9c2 r3c2} - n2{r3c2 r9c2} - {n2r9c9 .} ==> r9c3 <> 9 xyzt-chain[4] {n2 n9}r9c9 - {n9 n6}r9c2 - {n6 n7}r7c1 - {n7 n2}r8c3 ==> r9c3 <> 2 nrct-chain[5] {n9 n2}r9c9 - n2{r9c2 r3c2} - n2{r3c8 r2c8} - {n2 n1}r2c4 - n1{r2c9 r4c9} ==> r4c9 <> 9 nrct-chain[6] {n9 n2}r9c9 - n2{r9c2 r3c2} - n2{r3c8 r2c8} - n7{r2c8 r2c5} - n6{r2c5 r2c2} - {n6 n9}r9c2 ==> r9c8 <> 9 nrct-chain[6] {n9 n2}r9c9 - n2{r9c2 r3c2} - n2{r3c8 r2c8} - n7{r2c8 r1c7} - {n7 n3}r1c5 - {n3 n9}r1c3 ==> r1c9 <> 9 hidden-single-in-a-column ==> r9c9 = 9 naked-triplets-in-a-block {n6 n7 n2}{r7c1 r8c3 r9c2} ==> r9c3 <> 7 nrczt-whip-rn[2] n7{r2c8 r2c5} - {n7r9c5 .} ==> r8c8 <> 7 naked-triplets-in-a-block {n6 n7 n2}{r7c1 r8c3 r9c2} ==> r7c3 <> 7 xyt-chain[4] {n4 n1}r9c4 - {n1 n8}r9c3 - {n8 n7}r9c5 - {n7 n4}r7c6 ==> r7c4 <> 4 nrc-chain[4] {n8 n9}r4c7 - n9{r1c7 r1c3} - n9{r3c2 r7c2} - n8{r7c2 r6c2} ==> r6c8 <> 8 nrc-chain[4] n4{r1c6 r7c6} - {n4 n1}r9c4 - {n1 n8}r9c3 - n8{r9c5 r8c6} ==> r1c6 <> 8 nrczt-whip-rc[4] n8{r3c6 r1c4} - {n8 n5}r3c4 - {n5 n9}r4c4 - {n9r4c7 .} ==> r3c7 <> 8 xyzt-chain[4] {n3 n9}r3c7 - {n9 n8}r4c7 - {n8 n7}r1c7 - {n7 n3}r1c5 ==> r1c9 <> 3 xyzt-chain[4] {n3 n6}r2c2 - {n6 n2}r9c2 - {n2 n9}r3c2 - {n9 n3}r3c7 ==> r3c3 <> 3 nrc-chain[3] n3{r5c3 r1c3} - n9{r1c3 r1c7} - {n9 n3}r3c7 ==> r5c7 <> 3 naked-single ==> r5c7 = 4 naked-single ==> r5c1 = 9 naked-single ==> r6c1 = 4 interaction block b6 with column c8 for number 3 ==> r8c8 <> 3 interaction block b6 with column c8 for number 3 ==> r7c8 <> 3 interaction block b6 with column c8 for number 3 ==> r2c8 <> 3 nrct-chain[4] {n2 n1}r2c4 - {n1 n3}r7c4 - n3{r8c4 r8c9} - {n3 n2}r2c9 ==> r2c8 <> 2 nrct-chain[4] {n8 n9}r4c7 - n9{r4c8 r3c8} - n1{r3c8 r3c6} - n8{r3c6 r3c4} ==> r4c4 <> 8 interaction row r4 with block b6 for number 8 ==> r5c8 <> 8 x-wing-in-rows n8{r5 r9}{c3 c5} ==> r7c3 <> 8 nrc-chain[4] {n7 n3}r1c5 - n3{r1c3 r5c3} - n8{r5c3 r9c3} - {n8 n7}r9c5 ==> r3c5 <> 7 interaction row r3 with block b1 for number 7 ==> r1c3 <> 7 nrc-chain[4] {n7 n3}r1c5 - n3{r1c3 r5c3} - n8{r5c3 r9c3} - {n8 n7}r9c5 ==> r2c5 <> 7 hidden-single-in-a-row ==> r2c8 = 7 naked-pairs-in-a-row {n3 n6}r2{c2 c5} ==> r2c9 <> 3 singles ==> r8c9 = 3, r8c4 = 6, r6c6 = 6, r7c4 = 3 interaction column c9 with block b3 for number 2 ==> r3c8 <> 2 interaction row r3 with block b1 for number 2 ==> r1c3 <> 2 interaction block b9 with column c8 for number 8 ==> r3c8 <> 8 interaction row r3 with block b2 for number 8 ==> r1c4 <> 8 naked-pairs-in-a-block {n7 n8}{r8c6 r9c5} ==> r7c6 <> 7 hidden-pairs-in-a-column {n2 n7}{r3 r8}c3 ==> r3c3 <> 9 xy-chain[4] {n9 n5}r4c4 - {n5 n8}r3c4 - {n8 n1}r3c6 - {n1 n9}r3c8 ==> r4c8 <> 9 nrc-chain[3] {n3 n9}r3c7 - n9{r4c7 r6c8} - n3{r6c8 r6c2} ==> r3c2 <> 3 hxy-cn-chain[4] {r3 r6}c8n9 - {r6 r5}c8n3 - {r5 r1}c3n3 - {r1 r3}c7n3 ==> r3c7 <> 9 naked-single ==> r3c7 = 3 hxy-cn-chain[4] {r7 r8}c8n8 - {r8 r9}c8n2 - {r9 r3}c2n2 - {r3 r7}c2n9 ==> r7c2 <> 8 singles GRID 0 SOLVED. LEVEL = B-NRCZT6, MOST COMPLEX RULE = B-NRCZT6 159437862 834269571 762851394 276543918 913782456 485916237 691324785 547698123 328175649 |