The New Sudoku Players' Forum

by **Mauriès Robert** » Fri Sep 18, 2020 7:49 am

Hi all,
Here is a fairly easy puzzle.
Good resolution.
Robert

....3.9.....8.1.6.34......8.....943.2.......9.192.....8......75.3.4.7.....5.6....

Puzzle: Show

Resolution: Show

by **SpAce** » Fri Sep 18, 2020 7:38 pm

Step 1. AIC (with two subchains)

Code: Select all: .----------------------.------------------.--------------------. | 1567 c25678 c12678 | 567 3 d456 | 9 c1245 147 | | 579 2579 7-2 | 8 e247 1 | 2357 6 347 | | 3 4 167-2 | 5679 e279 56 | 1257 f125 8 | :----------------------+------------------+--------------------: | 567 5678 678 | 1567 178 9 | 4 3 2 | | 2 5678 3 | 567 478 456 | 178 f18 9 | | 4 1 9 | 2 78 3 | 5678 f58 67 | :----------------------+------------------+--------------------: | 8 69 4 | 139 19 2 | 136 7 5 | | 169 3 a126 | 4 5 7 | b1268 b29-18 16 | | 179 279 5 | 139 6 8 | 123 b249-1 134 | '----------------------'------------------'--------------------' a(2)r8c3 = b(2-894)b9p458 = (42)r1c832a - 4r1c6 = (42)r23c5 - (2=518)r356c8b => -2 r23c3 (a), -18 r89c8 (b)

Step 2. Kraken Cell (259)r2c2

Code: Select all: .-------------------.------------------.----------------------. | b156 2568 1268 | 567 3 456 | 9 1245 14-7 | | b59 *259 7 | 8 a24 1 | 235 6 34 | | 3 4 16 | a5679 a279 56 | a125(7) 125 8 | :-------------------+------------------+----------------------: | b567 5678 68 | 1567 178 9 | 4 3 2 | | 2 5678 3 | 567 478 456 | 1-7 18 9 | | 4 1 9 | 2 78 3 | 56-7 58 bc6(7) | :-------------------+------------------+----------------------: | 8 c69 4 | 139 19 2 | c136 7 5 | | b169 3 b126 | 4 5 7 | 8 29 bc16 | | b179 279 5 | 139 6 8 | 123 249 134 | '-------------------'------------------'----------------------' a:(2)r2c2 - r2c5 = (29-7)r3c54 = (7)r3c7 || b:(5)r2c2 - r12c1 = (57-1)r49c1 = r8c13 - (1=67)r86c9 || c:(9)r2c2 - (9=6)r7c2 - r7c7 = (67)r86c9 => -7 r1c9,r56c7; stte

by **Mauriès Robert** » Sat Sep 19, 2020 9:49 am

Hi Space,
Interesting resolution from you in two steps, but I have difficulty with the AIC and its two sub-chains. Could you split into two AICs?
I think we can make it simpler with a resolution using only two AICs (See my resolution).
Robert

by **SpAce** » Sat Sep 19, 2020 5:54 pm

Hi Robert,

Mauriès Robert wrote:Interesting resolution from you in two steps, but I have difficulty with the AIC and its two sub-chains. Could you split into two AICs?

Sure. Here you are:

SpAce wrote:a(2)r8c3 = b(2-894)b9p458 = (42)r1c832a - 4r1c6 = (42)r23c5 - (2=518)r356c8b => -2 r23c3 (a), -18 r89c8 (b)

As two AICs:

Code: Select all: a: (2)r8c3 = (289-4)b9p458 = (4-2)r1c8 = (2)r1c23 => -2 r23c3 b: (94)r89c8 = 4r1c8 - r1c6 = (4-2)r2c5 = r3c5 - (2=518)r356c8 => -18 r89c8

Subchain b (and the original) could also be written with a different last node to make it more symmetrical:

Code: Select all: b: (94)r89c8 = 4r1c8 - r1c6 = (4-2)r2c5 = r3c5 - r3c8 = (249)r198c8 => -18 r89c8

Note that the original b is practically equivalent to your first anti-track reversed:

Robert wrote:P'(8r56c8) : (-8r56c8) => (1r5c8 and 5r6c8)->2r3c8->2r2c5->4r2c9->4r9c8->9r8c8 => -8r8c8 => r8c7=8

As an AIC:

Code: Select all: (8=15)r56c8 - (1|5=2)r3c8 - r3c5 = (2-4)r2c5 = r2c9 - r9c9 = (4-9)r9c8 = (9)r8c8 => -8 r8c8

The difference is that the way I wrote it allowed to eliminate also 1r89c8, though that's not really necessary.

I think we can make it simpler with a resolution using only two AICs (See my resolution).

Yours is an interesting solution, but I can't agree that it's simpler in any way. For one, I can't write your second step as a reasonable AIC at all. It's clearly a net, and if properly analyzed, it's actually way more complex than my second step (which I can easily write as an AIC, but I think it's simpler and clearer as a kraken). The least horrible chain representation I could come up with for yours is a memory chain (which is not a valid AIC):

Robert wrote:P'(6r7c2) : (-6r7c2) => 9r7c2->9r3c5->(2r2c5->4r2c9->4r9c8->9r7c8->2r8c3)->(7r2c3 and 2r1c2->8r1c3)->6r4c3 => -6r1c2, -6r45c2 => r7c2=6 stte.

As a memory chain:

Code: Select all: (6=9)r7c2 - r7c5 = (9-2)r3c5 = (2*-4)r2c5 = r2c9 - r9c9 = (4-9)r9c8 = (9-2)r8c8 = r8c3 - b1p369*5 = (2^87)b1p236 - (8|7=6)r4c3 - r45^1c2 = (6)r7c2 => +6 r7c2

The complexity is also easily seen in the matrix form:

12x12 TM: Show

It has three complex strong inference sets (SIS): 2b1, r4c3, 6c2, and two memories: 2r2c5, 2r1c2. My kraken only has one 3-SIS: r2c2, and one memory: 6r6c9:

10x10 TM: Show

It can also be written as a fairly simple and valid AIC, either with a nested chain (a) or with split-nodes (b):

Code: Select all: a: (7)r3c7 = (79-2)r3c78 = r2c5 - 2r2c2 = [(76=1)r68c9 - r8c13 = (17-5)r94c1 = r12c1 - (5=96)r27c2 - r7c7 = (67)r86c9] => -7 r1c1,r56c7 b: (7=6)r6c9 - r8c9 = (1,6)b9p61 - 1r8c13|6r7c2 = (17,5)r9421,(9)r7c2 - (5|9=2)r2c2 - r2c5 = (29-7)r3c45 = (7)r3c7 => -7 r1c1,r56c7

...but I chose to present it as a kraken because it's much easier to read that way. Yours is difficult to write even as a multi-kraken:

Code: Select all: (8)r4c3 - r1c3 = (8)r1c2 -. || ) (7)r4c3 - (7=2)r2c3 -. / || ) / (6)r45c2 - (6)r4c3 .--------'- ' || ( (6)r1c2 -------------- (2)r1c2 || || || (2)r123c3 - r8c3 = (2-9)r8c8 = (9-4)r9c8 = r9c9 - r2c9 = (4-2)r2c5 = (2-9)r3c5 = r7c5 - (9=6)r7c2 * || || / || (2)r2c2 --------------------------------------------------' || (6)r7c2 * => +6 r7c2

Would you still say that's a simpler approach?

--
Added. FWIW, I think both second steps (yours and mine) would translate to braids instead of whips in Denis' system, so no difference there.

by **Cenoman** » Sat Sep 19, 2020 9:03 pm

One step:

Code: Select all: +-------------------------+---------------------+----------------------+ | 1567 25678 12678 | 567 3 456 | 9 1245 147 | | 579 2579 27 | 8 247 1 | 2357 6 347 | | 3 4 1267 | 5679 279 56 | 1257 125 8 | +-------------------------+---------------------+----------------------+ | 567 5678 678 | 1567 178 9 | 4 3 2 | | 2 5678 3 | 567 478 456 | 178 18 9 | | 4 1 9 | 2 78 3 | 5678 58 67 | +-------------------------+---------------------+----------------------+ | 8 69 4 | 139 19 2 | 136 7 5 | | 169 3 126 | 4 5 7 | 1268 1289 16 | | 179 279 5 | 139 6 8 | 123 1249 134 | +-------------------------+---------------------+----------------------+

Multi-krakens: cell (136)r7c7, column (8)r568c8, cell (169)r8c1, cell (1567)r1c1

Code: Select all: (1)r7c7 - [(1)r5c7 = (1-8)r5c8 = r6c8 - (8=7)r6c5 - (7=61)r68c9] = (8)r8c8 - (8=1263)b9p1467 || (3)r7c7 || (6)r7c7 - [(6=9)r7c2 - (9=1)r8c1 - (1=6)r8c9] = (6)r8c1 - (6)r1c1 || (1)r1c1 - r13c3 = (1-2)r8c3 = r9c2 - (2=163)b9p167 || (5)r1c1-r2c12=(5)r2c7 || (7)r1c1-(7=2)r2c3-r8c3=r9c2-(2=163)b9p167 ---------------- => -3 r2c7; ste

Added: matrices of the net above.

First matrix, 12x12 TM with two condensed nodes (ALS b9p1476,Y-Wing r7c2,r8c19)

Hidden Text: Show

Matrix 2, almost everything unfolded, notably the two embedded AIC's, 17x17 BTM

Hidden Text: Show

Matrix 3, everything unfolded, 18x18 (BTM)

Hidden Text: Show

Note the repetion of truth (136)r7c7 in rows 1 & 10, to be considered for the BTM property proof.

Code: Select all: 3r7c7 1r7c7 6r7c7 1r8c9 6r8c9 3r9c7 1r9c7 2r9c7 1r8c7 6r8c7 2r8c7 8r8c7 2r9c2 2r8c3 2r2c3 7r2c3 1r8c3 1r13c3 5r2c7 5r2c12 7r1c1 1r1c1 5r1c1 6r1c1 3r7c7 6r7c7 1r7c7 6r8c9 1r8c9 6r7c2 9r7c2 6r8c1 1r8c1 9r8c1 1r5c7 1r5c8 8r8c8 8r5c8 8r6c8 8r6c5 7r6c5 7r6c9 6r6c9 1r8c9 6r8c9 ---------------- => -3 r2c7; ste

For this matrix, hereafter triangular sub-matrices TMa and TMb for top entries 1r7c7 and 6r7c7 resp, proving matrix 3 to be a valid BTM.

Code: Select all: TMa 10x10 3r7c7 1r7c7 1r8c9 6r8c9 3r9c7 1r9c7 2r9c7 1r8c7 6r8c7 2r8c7 8r8c7 3r7c7 1r7c7 1r5c7 1r5c8 8r8c8 8r5c8 8r6c8 8r6c5 7r6c5 7r6c9 6r6c9 1r8c9 6r8c9 ---------------- => -3 r2c7 TMb 12x12 3r7c7 6r7c7 6r8c9 1r8c9 3r9c7 1r9c7 2r9c7 2r9c2 2r8c3 2r2c3 7r2c3 1r8c3 1r13c3 5r2c7 5r2c12 7r1c1 1r1c1 5r1c1 6r1c1 3r7c7 6r7c7 6r8c9 1r8c9 6r7c2 9r7c2 6r8c1 1r8c1 9r8c1 ---------------- => -3 r2c7

Added: sub matrices TMa and TMb checking BTM validity of matrix 3.

by **SpAce** » Sun Sep 20, 2020 12:00 am

Hi Cenoman,

Cenoman wrote:Multi-krakens: cell (136)r7c7, column (8)r568c8, cell (169)r8c1, cell (1567)r1c1

Very nice! I love the way you used those almost-AICs. It gave me an idea to write Robert's net as a relatively clean AIC:

(6)r7c2 = [(6)r1c2 = r45c2 - (6=8)r4c3 - r1c3 = (8)r1c2]|(72)r42c3 - 2r1c2 = [(2)r2c2 = r123c3 - r8c3 = (2-9)r8c8 = (9-4)r9c8 = r9c9 - r2c9 = (4)r2c5] - 2r2c5 = (2-9)r3c5 = (9)r7c5 => -9 r7c2

Obviously it's horribly long and takes some deciphering, but at least it's a valid AIC. This would be a slightly shorter compromise with a kludge node:

(9)r7c5 = (9-2)r3c5 = 2r2c5 - [(4)r2c5 = r2c9 - r9c9 = (4-9)r9c8 = (9-2)r8c8 = r8c3 - r123c3 = (2)r2c2] = (2,87,6)b1p2,r124c3 - 6r145c2 = (6)r7c2 => -9 r7c2

Btw, how would (or probably did) you write yours a matrix? I'm still incapable of writing valid BTMs (and a bit unwilling too), despite your best effort to help me (which I greatly appreciate), so this is what I came up with:

15x15 ?M: Show

It has the same top entry problem as before (marked with the first set of '|'; the other is just for symmetry), but personally I don't mind. The way I see it is that it's readable both ways using the same exact rules. That way it actually makes both nested AICs quite visible.

--
Btw, if one wants to go overboard with nested AICs, I guess it could combine two branches in the 4-kraken:

Code: Select all: - (6)r1c1 || (5)r1c1 - r2c12 = (5)r2c7 || [(1)r8c3 = r13c3 - (1=7)r1c1 - (7=2)r2c3] - 2r8c3 = r9c2 - (2=163)b9p167

(Of course I'm not saying it should.)

by **Mauriès Robert** » Sun Sep 20, 2020 6:54 am

Hi Space,

SpAce wrote:Yours is an interesting solution, but I can't agree that it's simpler in any way. For one, I can't write your second step as a reasonable AIC at all. It's clearly a net, and if properly analyzed, it's actually way more complex than my second step (which I can easily write as an AIC, but I think it's simpler and clearer as a kraken). The least horrible chain representation I could come up with for yours is a memory chain (which is not a valid AIC):

The notion of simplicity is relative to the experience one has of the techniques one applies. For me, that of AICs is not familiar so I thought that my second step could be written as an AIC as easily as the track P(9r7c2) quickly leads to a sequence of 2 quite visible as soon as you have placed the 8r8c7, hence the importance of having previously studied the 8s.
As for multi-kraken, they are for me bifurcations (or) therefore more complex than direct sequences (and).
Thank you for all your explanations.
Cordially
Robert

by **SpAce** » Sun Sep 20, 2020 2:04 pm

Mauriès Robert wrote:The notion of simplicity is relative to the experience one has of the techniques one applies.

True, but it doesn't mean there aren't objective measures of complexity as well. Those can only be seen and analyzed with a proper notation. Both your and Denis' notations lack the details necessary for such analysis, which makes moves expressed with them appear simpler than they are. Matrices and AICs/Krakens/Nets expressed with Eureka don't lack those details, so they represent the reality much more accurately.

I thought that my second step could be written as an AIC as easily

Like I just said -- when details are omitted, many things can appear simpler than they are.

As for multi-kraken, they are for me bifurcations (or) therefore more complex than direct sequences (and).

It's just a different point of view of the same exact logic, so it's not inherently any more or less complex. If you read a kraken or a multi-kraken backwards you have "AND-logic". Just try it with the one I wrote for your logic -- you'll see that it has your logic precisely when read from right to left. (If seen as pure Boolean logic there's no directionality at all, because ANDs, ORs, and NANDs are all commutative operations.)

That's the beauty of bidirectional (or non-directional) notations -- they're always reversible in that sense. (Not by Denis' definition of reversibility, but I don't use that.) The only time when you don't get to choose between AND/OR points of view is when both are needed going either way, but those are examples of more complex logic. (Ironically, Denis' definition counts such mixed cases as reversible, but not the simpler ones.)

Anything that can be written as a whip or a braid (AND-logic) is a kraken or a multi-kraken (OR-logic) going backwards, and vice versa. Denis' notation just doesn't allow seeing that duality (TDP even less) because it lacks the details necessary for reading them backwards (and forwards, as far as I'm concerned). He does acknowledge the duality itself, though -- just not that they're equivalent points of view.

Why do we normally prefer to write our net moves as krakens/multi-krakens instead of the admittedly more sequential AND-logic? One advantage is that it allows writing and following each branch linearly, though possibly introducing repetition if written that way. That's often easier to read than complex nets requiring join points.

Another benefit is that it produces natural verities. When more than two end points (z-candidates in Denis' parlance) are needed, the only way to write pure AND-logic is to start with the elimination and end up with a contradiction (as whips and braids do). As you know, we don't like contradictions, so we rather flip it around and use OR-logic to produce a verity with multiple end points. The end result is the same.

by **Cenoman** » Sun Sep 20, 2020 2:16 pm

SpAce wrote:Btw, how would (or probably did) you write yours a matrix?

Hi SpAce,
I missed time to check the validity of my matrices. Now completed. I have added three of them (from compacted to unfolded) in my initial post.
Still short of time, I will add later the sub-matrices proving matrix 3 being actually a BTM (Edit: carried out)

PS my preference is matrix 2.

by **SpAce** » Mon Sep 21, 2020 1:06 am

Hi Cenoman,

Thank you very much for the matrix demonstration, once again! I agree that 2 and 3 are valid BTMs (and the first one a valid TM, of course). You proved that it can be done here without any special tricks. Nice job!

Personally I'm still inclined to break the rules in this case, because for me it produces a much more natural logic flow. I think there's room for both styles, as long as they have different names to avoid confusion.

Since matrices are more like personal analysis tools than actual presentations, I think it's best to use whatever style works for oneself -- as long as one doesn't claim it to be something that it isn't (like in my case a valid BTM). I can still consider it a fun challenge to try to write valid BTMs, but I don't think I'm going to obsess about it the same way I obsess about valid AICs

PS. One suggestion. I don't know about you and others, but I find such large matrices really difficult to read without any line helpers (like the dots I use). The SISs get spread out so widely that it's almost impossible to see which nodes are in the same rows. I don't know if the dots are the best solution, but I think something is needed.

by **Cenoman** » Mon Sep 21, 2020 12:19 pm

SpAce wrote:Thank you very much for the matrix demonstration, once again!

I'm glad you found it helpful

PS. One suggestion. I don't know about you and others, but I find such large matrices really difficult to read without any line helpers (like the dots I use). The SISs get spread out so widely that it's almost impossible to see which nodes are in the same rows. I don't know if the dots are the best solution, but I think something is needed.

Something like that ?

Hidden Text: Show

I have kept a density of one dot out of three blanks (easy to do with "replace all") and made some manual re-alignments (maybe dispensable)

by **SpAce** » Mon Sep 21, 2020 11:47 pm

Cenoman wrote:I'm glad you found it helpful

Always!

Something like that ?
...
I have kept a density of one dot out of three blanks (easy to do with "replace all") and made some manual re-alignments (maybe dispensable)

That's a great idea! Works like magic and looks good too. Thanks for the tip! It actually provides column helpers as well, though they're not so crucial.
--

Added. If I flip the upper part of my matrix upside down, wouldn't it result in a valid BTM?

15x15 BTM: Show

Code: Select all: 3r79c7 8216b9p4716 3r7c7 ........... 1r7c7 ....................... 6r7c7 1r5c7 1r5c8 . 1r8c9 ..... 6r8c9 . 6r6c9 7r6c9 . 7r6c5 8r6c5 . 8r8c8 ........... 8r5c8 ........... 8r6c8 . 6r8c9 1r8c9 6r7c2 ..... 9r7c2 1r8c1 9r8c1 6r8c1 2r9c2 ........................................................... 2r8c3 1r8c3 1r13c3 2r2c3 ...... 7r2c3 6r1c1 ..... 1r1c1 7r1c1 5r1c1 5r2c7 ..................................................................................... 5r2c12 =================================================================================================== -3r2c7

7x7 TM:

Code: Select all: 3r79c7 8216b9p4716 3r7c7 ........... 1r7c7 1r5c7 1r5c8 1r8c9 ..... 6r8c9 6r6c9 7r6c9 7r6c5 8r6c5 8r8c8 ........... 8r5c8 ........... 8r6c8 ================================================= -3r2c7

10x10 TM

Code: Select all: 3r79c7 8216b9p4716 3r7c7 ........... 6r7c7 6r8c9 1r8c9 6r7c2 ..... 9r7c2 1r8c1 9r8c1 6r8c1 2r9c2 ............................. 2r8c3 1r8c3 1r13c3 2r2c3 ...... 7r2c3 6r1c1 ..... 1r1c1 7r1c1 5r1c1 5r2c7 ....................................................... 5r2c12 ===================================================================== -3r2c7

The New Sudoku Players' Forum

Robert's puzzles 2020-09-18

Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18

Re: Robert's puzzles 2020-09-18