The New Sudoku Players' Forum

by **Mauriès Robert** » Sat Sep 26, 2020 8:32 am

Hi all,
I propose you this puzzle. What is its level of difficulty ?
..9..2..6..5.1.9.2..2..9.7..2...14..6.......7..42...8..3.8..7..2.7.5.6..5..9..3..

puzzle: Show

solution: Show

Cordialy,
Robert

by **SpAce** » Sat Sep 26, 2020 11:01 am

Step 1. Double Kraken cell

Code: Select all: .----------------------.------------------.--------------. | 3478 478 9 | 3457 3478 2 | 58 1 6 | | *3478 4678 5 | 3467 1 d478 | 9 a34 2 | | b1348 b1468 2 | 3456 348 9 | 58 7 34 | :----------------------+------------------+--------------: | c789 2 38 | 37 3789 1 | 4 6 5 | | 6 abcd589-1 138 | 34 3489 d458 | 2 39 7 | | c79 abc579 4 | 2 3679 567 | 1 8 39 | :----------------------+------------------+--------------: | b149 3 16 | 8 2 46 | 7 5 149 | | 2 ab489 7 | 1 5 3 | 6 a49 489 | | 5 148 168 | 9 467 467 | 3 2 148 | '----------------------'------------------'--------------' a:(3)r2c1 - (3=4)r2c8 - (4=9)r8c2 - r8c2 = (9-5)r6c2 = (5)r5c2 || || (1)r7c1 - r3c1 = (1)r3c2 || || b:(4)r2c1 - (4)r7c1 || || || (9)r7c1 - r8c2 = (9-5)r6c2 = (5)r5c2 || c:(7)r2c1 - r46c1 = (7-5)r6c2 = (5)r5c2 || d:(8)r2c1 - r2c6 = (8-5)r5c6 = (5)r5c2 => -1 r5c2

10x10 TM: Show

g-braid(10): Show

Step 2. AIC (with nesting)

Code: Select all: .----------------.------------------.-------------. | 3478 g78 9 | 345 F3478 2 | g58 1 6 | | 3478 678 5 | 346 1 E78 | 9 34 2 | | i1348 i168 2 | i3456 348 9 | h58 7 34 | :----------------+------------------+-------------: | 89 2 3 | 7 89 1 | 4 6 5 | | 6 589 1 | 34 3489 E58 | 2 39 7 | | 79 cD579 4 | 2 369 D56 | 1 8 39 | :----------------+------------------+-------------: | a9-1 3 6 | 8 2 4 | 7 5 19 | | 2 b49 7 | 1 5 3 | 6 49 8 | | 5 14 8 | 9 67 67 | 3 2 14 | '----------------'------------------'-------------' (9)r7c1 = r8c2 - r6c2 = [(75)r6c26 = (58-7)r52c6 = (7)r1c5] - (7=85)r1c27 - r3c7 = (561)r3c421 => -1 r7c1; stte

as a Kraken cell: Show

10x10 TM: Show

t-whip(10): Show

--
Edit. Cosmetic changes to the first kraken and matrix. Added the t-whip and g-braid notations.

by **Cenoman** » Sat Sep 26, 2020 3:40 pm

Five steps: four AIc's and one kraken:

Code: Select all: +----------------------+----------------------+------------------+ | 3478 478 9 | 3457 3478 2 | 58 1 6 | |Ww347-8 4678 5 | 3467 1 c478 | 9 x34 2 | | 1348 1468 2 | 3456 348 9 | 58 7 34 | +----------------------+----------------------+------------------+ |Xa789 2 Yb38 | 37 3789 1 | 4 6 5 | | 6 b1589 Yb138 | 34 3489 c458 | 2 A39 7 | |XB79 B579 4 | 2 3679 B567 | 1 8 B39 | +----------------------+----------------------+------------------+ | z19-4 3 Y16 | 8 2 ZC46 | 7 5 y19-4 | | 2 489 7 | 1 5 3 | 6 xA49 489 | | 5 148 168 | 9 467 467 | 3 2 148 | +----------------------+----------------------+------------------+

1. (8)r4c1 = (8-135)b4p356 = (58)r25c6 => -8 r2c1
2. (4=93)r58c8 - (3=5796)r6c1269 - (6=4)r7c6 => -4 r7c9

3. Kraken cell (347)r2c1
(3)r2c1 - (3=49)r28c8 - r7c9 = (9)r7c1
(4)r2c1
(7)r2c1 - (7=98)r46c1 - (8=136)r457c3 - (6=4)r7c6
---------------------
=> -4 r7c1; 2 placements and basics

Code: Select all: +----------------------+----------------------+------------------+ | 3478 78 9 | 3457 3478 2 | 58 1 6 | | 347 678 5 | 3467 1 78 | 9 34 2 | | 1348 168 2 | 3456 348 9 | 58 7 34 | +----------------------+----------------------+------------------+ | b789 2 b38 | 37 3789 1 | 4 6 5 | | 6 1589 b138 | 34 3489 58 | 2 39 7 | | b79 579 4 | 2 3679 567 | 1 8 39 | +----------------------+----------------------+------------------+ | a19 3 6 | 8 2 4 | 7 5 19 | | 2 489 7 | 1 5 3 | 6 49 489 | | 5 148 8-1 | 9 67 67 | 3 2 148 | +----------------------+----------------------+------------------+

4. (1=9)r7c1 - (9=3781)b4p1367 => -1 r9c3; 5 placements and basics

Code: Select all: +--------------------+---------------------+-----------------+ | 3478 c78 9 | 345 3478 2 | c58 1 6 | | d347 678 5 | d346 1 78 | 9 d34 2 | | 1348 168 2 | a345-6 348 9 | b58 7 34 | +--------------------+---------------------+-----------------+ | 89 2 3 | 7 89 1 | 4 6 5 | | 6 589 1 | 34 3489 58 | 2 39 7 | | 79 579 4 | 2 369 56 | 1 8 39 | +--------------------+---------------------+-----------------+ | 19 3 6 | 8 2 4 | 7 5 19 | | 2 49 7 | 1 5 3 | 6 49 8 | | 5 14 8 | 9 67 67 | 3 2 14 | +--------------------+---------------------+-----------------+

5. (5)r3c4 = r3c7 - (5=87)r1c27 - (7=346)r2c148 => -6 r3c4; ste

Website · by **denis_berthier** » Sat Sep 26, 2020 4:24 pm

Two solutions starting the same way:

- with whips and other chains, all of length ≤ 5:

Code: Select all: (solve "..9..2..6..5.1.9.2..2..9.7..2...14..6.......7..42...8..3.8..7..2.7.5.6..5..9..3..") *********************************************************************************************** *** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = gB+SFin *** Using CLIPS 6.32-r770 *********************************************************************************************** hidden-single-in-a-column ==> r8c4 = 1 hidden-single-in-a-block ==> r8c6 = 3 hidden-single-in-a-column ==> r5c7 = 2 hidden-single-in-a-column ==> r4c8 = 6 175 candidates, 942 csp-links and 942 links. Density = 6.19% whip[1]: c3n6{r9 .} ==> r9c2 ≠ 6 whip[1]: c6n5{r6 .} ==> r5c4 ≠ 5, r4c4 ≠ 5 hidden-single-in-a-row ==> r4c9 = 5 naked-single ==> r6c7 = 1 hidden-single-in-a-block ==> r7c8 = 5 hidden-single-in-a-block ==> r9c8 = 2 hidden-single-in-a-block ==> r7c5 = 2 hidden-single-in-a-column ==> r1c8 = 1 whip[1]: c3n3{r5 .} ==> r6c1 ≠ 3, r4c1 ≠ 3 whip[1]: c7n8{r3 .} ==> r3c9 ≠ 8 whip[1]: c4n6{r3 .} ==> r3c5 ≠ 6, r2c6 ≠ 6 biv-chain[3]: r3c9{n4 n3} - b6n3{r6c9 r5c8} - r5c4{n3 n4} ==> r3c4 ≠ 4 biv-chain[3]: r4n9{c5 c1} - b7n9{r7c1 r8c2} - c8n9{r8 r5} ==> r5c5 ≠ 9 biv-chain[4]: r3c9{n4 n3} - r6n3{c9 c5} - b5n6{r6c5 r6c6} - r7c6{n6 n4} ==> r7c9 ≠ 4 biv-chain[4]: r3c9{n4 n3} - r6c9{n3 n9} - r7c9{n9 n1} - c1n1{r7 r3} ==> r3c1 ≠ 4 biv-chain[4]: r7c6{n4 n6} - r7c3{n6 n1} - b4n1{r5c3 r5c2} - r5n5{c2 c6} ==> r5c6 ≠ 4 z-chain[4]: c1n4{r2 r7} - b7n9{r7c1 r8c2} - r8c8{n9 n4} - b3n4{r2c8 .} ==> r3c2 ≠ 4 ;;; RS1 t-whip[4]: b7n9{r8c2 r7c1} - r6c1{n9 n7} - r4c1{n7 n8} - c3n8{r5 .} ==> r8c2 ≠ 8 hidden-single-in-a-row ==> r8c9 = 8 x-wing-in-rows: n9{r5 r8}{c2 c8} ==> r6c2 ≠ 9 biv-chain[4]: r5n1{c3 c2} - r5n9{c2 c8} - c9n9{r6 r7} - b9n1{r7c9 r9c9} ==> r9c3 ≠ 1 biv-chain[4]: c5n9{r4 r6} - c5n6{r6 r9} - r9c3{n6 n8} - r4c3{n8 n3} ==> r4c5 ≠ 3 biv-chain[4]: r4c4{n7 n3} - r4c3{n3 n8} - r9c3{n8 n6} - c5n6{r9 r6} ==> r6c5 ≠ 7 z-chain[4]: r9c3{n8 n6} - c5n6{r9 r6} - c5n9{r6 r4} - r4n8{c5 .} ==> r5c3 ≠ 8 whip[4]: r2c8{n3 n4} - r8n4{c8 c2} - b1n4{r1c2 r1c1} - r1n3{c1 .} ==> r2c4 ≠ 3 biv-chain[5]: r7c6{n4 n6} - r7c3{n6 n1} - b4n1{r5c3 r5c2} - r5n5{c2 c6} - c6n8{r5 r2} ==> r2c6 ≠ 4 whip[1]: c6n4{r9 .} ==> r9c5 ≠ 4 biv-chain-rc[4]: r9c5{n7 n6} - r9c3{n6 n8} - r4c3{n8 n3} - r4c4{n3 n7} ==> r4c5 ≠ 7 z-chain[4]: r2c6{n7 n8} - r5c6{n8 n5} - c2n5{r5 r6} - c2n7{r6 .} ==> r2c1 ≠ 7 z-chain[5]: c6n8{r2 r5} - r4c5{n8 n9} - r4c1{n9 n7} - b5n7{r4c4 r6c6} - r2c6{n7 .} ==> r2c1 ≠ 8 naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c4 ≠ 4, r2c2 ≠ 4 finned-x-wing-in-rows: n4{r2 r8}{c8 c1} ==> r7c1 ≠ 4 stte

with only reversible, but longer, chains:

Code: Select all: ;;; same path upto RS1 biv-chain[5]: r7c6{n4 n6} - r7c3{n6 n1} - b4n1{r5c3 r5c2} - r5n5{c2 c6} - c6n8{r5 r2} ==> r2c6 ≠ 4 whip[1]: c6n4{r9 .} ==> r9c5 ≠ 4 z-chain[4]: r2c6{n7 n8} - r5c6{n8 n5} - c2n5{r5 r6} - c2n7{r6 .} ==> r2c1 ≠ 7 z-chain[5]: r5c4{n3 n4} - r5c5{n4 n8} - r3c5{n8 n4} - r3c9{n4 n3} - r6n3{c9 .} ==> r4c5 ≠ 3 z-chain[6]: r2n6{c4 c2} - r2n7{c2 c6} - r9n7{c6 c5} - c5n6{r9 r6} - c5n3{r6 r5} - c8n3{r5 .} ==> r2c4 ≠ 3 z-chain[8]: r2n6{c2 c4} - r2n7{c4 c6} - c6n8{r2 r5} - r5n5{c6 c2} - r5n1{c2 c3} - r7c3{n1 n6} - r7c6{n6 n4} - c1n4{r7 .} ==> r2c2 ≠ 4 z-chain[6]: c6n8{r2 r5} - r4n8{c5 c3} - c3n3{r4 r5} - r5c4{n3 n4} - r2n4{c4 c8} - r2n3{c8 .} ==> r2c1 ≠ 8 naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c4 ≠ 4 biv-chain[3]: r2n4{c1 c8} - r8c8{n4 n9} - b7n9{r8c2 r7c1} ==> r7c1 ≠ 4 hidden-single-in-a-row ==> r7c6 = 4 hidden-single-in-a-row ==> r7c3 = 6 whip[1]: b7n4{r9c2 .} ==> r1c2 ≠ 4 biv-chain[4]: r6n5{c2 c6} - r5c6{n5 n8} - r2n8{c6 c2} - r1c2{n8 n7} ==> r6c2 ≠ 7 whip[1]: c2n7{r2 .} ==> r1c1 ≠ 7 z-chain[4]: r4n8{c3 c5} - r4n9{c5 c1} - r7c1{n9 n1} - r9c3{n1 .} ==> r5c3 ≠ 8 z-chain[4]: c2n5{r5 r6} - c2n9{r6 r8} - r7c1{n9 n1} - r3n1{c1 .} ==> r5c2 ≠ 1 stte

by **SpAce** » Sat Sep 26, 2020 10:49 pm

I don't expect an answer, and it doesn't matter a whole lot anyway, but something I've been wondering about for a while...

denis_berthier wrote:
Code: Select all
biv-chain[4]: r7c6{n4 n6} - r7c3{n6 n1} - b4n1{r5c3 r5c2} - r5n5{c2 c6} ==> r5c6 ≠ 4 z-chain[4]: c1n4{r2 r7} - b7n9{r7c1 r8c2} - r8c8{n9 n4} - b3n4{r2c8 .} ==> r3c2 ≠ 4 t-whip[4]: b7n9{r8c2 r7c1} - r6c1{n9 n7} - r4c1{n7 n8} - c3n8{r5 .} ==> r8c2 ≠ 8

Why is it named z-chain instead of z-whip? Since z-chains are written as contradiction chains they resemble whips much more than biv-chains. It's obvious just by looking at the three examples above. Is there a logical reason (beyond "inventor's prerogative") for such an unintuitive naming (or writing)?

That said, I'd rather keep the name and change the notation of z-chains to make them biv-chains with the z-extension. That's what the name implies. There's no need for the ugly contradiction anyway. (Then again, there's no need for that with whips or braids either. They could be written with actual end points, too.)

The biv-chain family might make more sense for z-chains because, unlike whips, they're confluent (as far as I understand the term). Deleting any of its candidates would result in a simpler z-chain or a biv-chain (or eliminating the target), though possibly with a different starting point -- but no braid (which is the fallback pattern for maimed whips). Like biv-chains, but unlike whips, they're also reversible even by your definition. Writing them as (special cases of) biv-chains would make that more obvious, too.

Either way, the current name and notation seem mismatched.

Website · by **denis_berthier** » Sun Sep 27, 2020 4:02 am

It is clearly stated in PBCS that z-chains are the z-extension of bivalue-chains and that they are reversible. They could be called z-biv-chains, but then some systematic opponents would whine that "biv" is incorrect because they are not really bivalue.

In chains, the target(s) is (are) linked to both ends. Not in whips or braids. I think the names are perfectly suggestive of this difference.

Already in HLS, I had defined nrczt-chains; I still called them chains because they also had this property. I moved the emphasis to whips when I found that raising this condition led to a slightly more general pattern (and that it was essential for the T&E vs braids theorem).

Definitions are not arbitrary: if they allow no proof of anything, they are useless.

by **SpAce** » Sun Sep 27, 2020 5:55 pm

denis_berthier wrote:It is clearly stated in PBCS that z-chains are the z-extension of bivalue-chains and that they are reversible.

So why are they written like whips? That's my question, and I don't think you answered it at all. If the above is the intended interpretation then z-chains shouldn't be written as contradiction chains -- they should be written with actual end-points like biv-chains (as the old nrcz-chains apparently were).

They could be called z-biv-chains, but then some systematic opponents would whine that "biv" is incorrect because they are not really bivalue.

One doesn't have to be a "systematic opponent" to make that judgment call. 'z-biv-chain' would be a poor naming option regardless of how the chain is written, but especially in the current form. As currently written, 'z-whip' would be best. If written as a non-contradiction chain (as I would recommend), then the current name 'z-chain' would do.

In chains, the target(s) is (are) linked to both ends. Not in whips or braids. I think the names are perfectly suggestive of this difference.

I would like to agree, hence my whole question!!! The target of a z-chain is NOT linked to both ends, because one end is a contradiction -- just like in whips and braids. Thus not a chain by that definition. What am I missing?

Already in HLS, I had defined nrczt-chains; I still called them chains because they also had this property.

They indeed seem to be 'chains' by having actual nodes at both ends, at least based on these examples:

denis_berthier wrote:nrczt6-chain {n9 n2}r6c1 - {n2 n6}r4c1 - {n6 n8}r5c1 - n9{r5c1 r4c3} - {n9 n7}r1c3 - {n7 n9}r2c1 ==> r8c1 <> n9

No dot at the end -- thus a 'chain'. Since we seem to agree on that, I don't understand why you muddy the definition by keeping the 'chain' in z-chains while writing them like whips. You have not answered that question at all. You've instead helped argue my point!

slightly off-topic: Show

PS. I made a mistake in the previous post:

SpAce wrote:Writing them [z-chains] as (special cases of) biv-chains would make that more obvious, too.

Obviously z-chains would not be special cases of biv-chains even in that case. The other way around would be true, since a biv-chain would be a restricted subtype of the more generic z-chain. As currently written they're not related, as z-chains look like special cases of whips.

--
Added 1. The difference between a 'whip' and a 'chain' as I see it, using my second move as an example (t-whip[10] vs "t-chain"[10]):

r3n1{c1 c2} - r3n6{c2 c4} - r3n5{c4 c7} - r1c7{n5 n8} - r1c2{n8 n7} - b2n7{r1c5 r2c6} - c6n8{r2 r5} - c6n5{r5 r6} - r6c2{n5 n9} - b7n9{r8c2 .} => -1 r7c1
r3n1{c1 c2} - r3n6{c2 c4} - r3n5{c4 c7} - r1c7{n5 n8} - r1c2{n8 n7} - b2n7{r1c5 r2c6} - c6n8{r2 r5} - c6n5{r5 r6} - r6c2{n5 n9} - b7n9{r8c2 r7c1} => -1 r7c1

(The mandatory chain prefixes omitted to save space and to highlight the difference.)

Obviously the only difference is that the whip-form has a contradiction at the end. My main question repeated: why does a z-chain have the whip-form despite being called a 'chain'? My secondary question: why is the whip-form used for almost everything in your system, including z-chains, even though the cleaner and more intuitive chain-form could replace it in most cases?

a bonus braid question: Show

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