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by **al_bell** » Mon Nov 24, 2008 6:28 am

I need help with this puzzle - ie what is the next best step and why

The orginal puzzle:

Code: Select all: . . 7 | . . 3 | . . 9 . . . | 1 . 5 | . 6 . 4 . . | . . . | . . . ------+-------+------ . . 5 | . . . | 4 . . 3 . . | . 9 . | . . 7 . . 2 | . . . | 8 . . ------+-------+------ . . . | . . . | . . 3 . 9 . | 2 . 6 | . . . 1 . . | 8 . . | 5 . .

The puzzle with pencil marks:

Code: Select all: 258 1258 7 | 6 248 3 | 12 1458 9 28 238 9 | 1 2478 5 | 237 6 48 4 123568 168 | 9 278 78 | 1237 1578 158 ---------------------+--------------------+------------------- 679 167 5 | 37 1678 178 | 4 39 2 3 148 148 | 45 9 2 | 6 15 7 679 1467 2 | 357 1567 147 | 8 39 15 ---------------------+--------------------+------------------- 25678 25678 68 | 457 157 147 | 9 1278 3 578 9 348 | 2 1357 6 | 17 1478 148 1 247 34 | 8 37 9 | 5 247 6

Thanks for your help.

by **hobiwan** » Mon Nov 24, 2008 10:00 am

al_bell wrote:I need help with this puzzle - ie what is the next best step and why

I don't think there is such a thing like a "best step". The IMO easiest step is an X-Chain (or Multi-Color move or Finned Franken X-Wing):

Code: Select all: .---------------------.----------------.-------------------. | 258 1258 7 | 6 248 3 | 12 1458 9 | | 28 238 9 | 1 2478 5 | 237 6 48 | | 4 123568 *168 | 9 278 78 | 1237 1578 -158 | :---------------------+----------------+-------------------: | 679 167 5 | 37 1678 178 | 4 39 2 | | 3 148 *148 | 45 9 2 | 6 *15 7 | | 679 1467 2 | 357 1567 147 | 8 39 *15 | :---------------------+----------------+-------------------: | 25678 25678 68 | 457 157 147 | 9 1278 3 | | 578 9 348 | 2 1357 6 | 17 1478 148 | | 1 247 34 | 8 37 9 | 5 247 6 | '---------------------'----------------'-------------------' X-Chain: 1 r3c3 =1= r5c3 -1- r5c8 =1= r6c9 => r3c9<>1

It unlocks the following AIC:

Code: Select all: .--------------------.------------------.---------------------. | 258 1258 7 | 6 248 3 | 12 14-58 9 | | 28 238 9 | 1 *2478 5 | *237 6 48 | | 4 123568 168 | 9 278 *78 | 1237 1-578 *58 | :--------------------+------------------+---------------------: | 679 167 5 | 37 1678 178 | 4 39 2 | | 3 148 148 | 45 9 2 | 6 *15 7 | | 679 1467 2 | 357 1567 147 | 8 39 *1-5 | :--------------------+------------------+---------------------: | 25678 25678 68 | 457 157 147 | 9 1278 3 | | 578 9 348 | 2 1357 6 | *17 1478 *148 | | 1 247 34 | 8 37 9 | 5 247 6 | '--------------------'------------------'---------------------' AIC 5- r5c8 -1- r6c9 =1= r8c9 -1- r8c7 -7- r2c7 =7= r2c5 -7- r3c6 -8- r3c9 -5 => r13c8,r6c9<>5

which in turn leed's to a few singles. But you will still need some advanced moves to solve the puzzle.

[edit: incorrect notation fo the AIC, see daj95376's comment below]

by **Luke** » Mon Nov 24, 2008 1:03 pm

Proceeding from hobiwan's PM and after two wee XY chains:

Code: Select all: *-----------------------------------------------------------* | 25 15 7 | 6 4 3 | 12 8 9 | | 28 38 9 | 1 7 5 | 23 6 4 | | 4 *136 *16 | 9 2 8 | 137 17 5 | |-------------------+-------------------+-------------------| | 67 67 5 | 3 8 1 | 4 9 2 | | 3 *18 *18 | 4 9 2 | 6 5 7 | | 9 4 2 | 5 6 7 | 8 3 1 | |-------------------+-------------------+-------------------| | 568 2568 68 | 7 15 4 | 9 12 3 | | 57 9 3 | 2 15 6 | 17 4 8 | | 1 27 4 | 8 3 9 | 5 27 6 | *-----------------------------------------------------------*

As always, there's different ways to go, but this looks like an example of the supposedly rare "Strong Wing" discussed in udosuk's thread "What is this technique called?"

While this is not a uniqueness situation, my interest was piqued by the combination of strong inferences on 6 and 8 and the strong link on 1 in c3. The pattern consist of strong links in the "roof" and "floor" and a third connecting the two. Usually, in such a rectangle the value not associated with any of the strong inferences ends up getting edged out.

In this case, after a quick look at the grid it's obvious r3c2<>3, and it's singles night.

by **eleven** » Tue Nov 25, 2008 2:03 am

Still having these almost SdC's in mind, i found this one with the same effect as hobiwans AIC, which is not easy to spot.

Code: Select all: .--------------------.------------------.---------------------. | 258 1258 7 | 6 248 3 | 12 *1458 9 | | 28 238 9 | 1 2478 5 | 237 6 #48 | | 4 123568 168 | 9 278 78 | 1237 *1578 5-8 | :--------------------+------------------+---------------------: | 679 167 5 | 37 1678 178 | 4 39 2 | | 3 148 148 | 45 9 2 | 6 #15 7 | | 679 1467 2 | 357 1567 147 | 8 39 15 | :--------------------+------------------+---------------------: | 25678 25678 68 | 457 157 147 | 9 1278 3 | | 578 9 348 | 2 1357 6 | 17 1478 148 | | 1 247 34 | 8 37 9 | 5 247 6 | '--------------------'------------------'---------------------'

Either r3c8=7 -> r3c6=8 -> r3c9=5
Or Sue de Coq 15|1458|48 -> r3c9<>8 -> r3c9=5

(there is another almost SdC 37|1357|15 removing 5 in r7c4, which does not help much)

by **ronk** » Tue Nov 25, 2008 2:51 am

Luke451 wrote:As always, there's different ways to go, but this looks like an example of the supposedly rare "Strong Wing" discussed in udosuk's thread "What is this technique called?"

A strong wing has four strong inferences rather than three. Your deduction seems to use r2c2, as in ...

Code: Select all: r3c2 -3- r2c2 -8- r5c2 =8= r5c3 =1= r3c3 =6= r3c2 ==> r3c2<>3

by **Luke** » Tue Nov 25, 2008 8:12 am

Thanks, ronk, I see the difference now. Not a strong wing, and "roof and floor" has no bearing. In terms of pattern recognition, one might say a strong wing has a "strong elbow" with two separate strong links meeting on its diagonal; the value in the elbow must be true. From the other thread:

Code: Select all: ... |... |... ..*-1--* |... ..| |..| |... --2-+--3-+--- ..| |..| |... ..*-3--* |... ... |... |... ----+----+--- ... |... |... ... |... |... ... |... |... r5c6=3

Still, it's not a bad idea to examine any rectangle with multiple strong links for obvious contradictions.

by **Jasper32** » Wed Nov 26, 2008 5:33 am

The notation that Hobiwan used in his AIC posted above confuses me.
In no way is this written to be critical of Hobiwan. He is far brighter than I am. I just want to know how he arrived at his notation as shown below.

AIC 5= r5c8 -1- r6c9 =1= r8c9 -1- r8c7 -7- r2c7 =7= r2c5 -7- r3c6 -8- r3c9 =5 => r13c8,r6c9<>5

I have always thought that the notation should alternate, between "=" and "-" in AIC's as written in Andrew C. Stuart’s book, “The Logic of Sudoku”. Below is the text and the notation of the example shown on pages 88 and 89 of his book.

The chief characteristics of all chains is that they start on a candidate which has a strong inference to another candidate, which then has a weak inference to the next candidate, and so on, alternating until the start point is reached.

1[a9]=3[a9]-3[a7]=3[a7]-3[j3]=1[j3]-1[j9]

While I followed Hobiwan’s notation, I do not understand it at all. Is there a reference that explains Hobiwan's technique or can someone explain it to me. I think I am missing something! Thank you.

by **Luke** » Wed Nov 26, 2008 11:38 am

I think this is simply shorthand for the notation you already understand. Notice which of the inferences are omitted:

Code: Select all: AIC 5= r5c8 -1- r6c9 =1= r8c9 -1- r8c7 -7- r2c7 =7= r2c5 -7- r3c6 -8- r3c9 =5

5[r5c8]=5[r6c9]-1[r6c9]=1[r8c9]-1[r8c7]=7[r8c7]-7[r2c7]=7[r2c5]-7[r3c6]=8[r3c6]-8[r3c9]=5[r3c9]

All in the bivalue cells, right? I think it's just a way to save the time and not make the chain appear so cumbersome.

by **hobiwan** » Wed Nov 26, 2008 12:18 pm

Jasper32 wrote:The notation that Hobiwan used in his AIC posted above confuses me. ... Is there a reference that explains Hobiwan's technique or can someone explain it to me. I think I am missing something! Thank you.

I don't have a link, I'm afraid. I use Nice Loop Notation, and what's confusing (at first) in Nice Loop Notation is, that links within cells are not spelled out, you see only links between cells.

In short: The notation follows the Nice loop propagation rules (see http://www.paulspages.co.uk/sudokuxp/howtosolve/niceloops.htm for a tutorial).

In more detail:

1. If a cell is reached via a strong link and left via a strong link on a different digit, a weak link between those two digits is implied within the cell. I don't have an example of this in my chain, but assume the following:

=x= r1c1 =y= would mean
=x= r1c1 -x- r1c1 =y=

2. If a cell is reached via a weak link and left via a weak link on a different digit, a strong link between those two digits is implied within the cell.

As Luke451 already wrote:

-1- r8c7 -7- means
-1- r8c7 =7= r8c7 -7-

Notation aside, inferences in any type of chain have to alternate between weak and strong, you are of course right about that.

by **daj95376** » Wed Nov 26, 2008 4:22 pm

hobiwan wrote:I use Nice Loop Notation, and what's confusing (at first) in Nice Loop Notation is, that links within cells are not spelled out, you see only links between cells.

I don't agree that links within cells aren't spelled out. Here's your AIC w/corrections and decomposed into lines of strong inference followed by lines of weak inference -- ending in a line of strong inference. The links within cells are marked with an (@).

Code: Select all: 5-[r5c8]-1-[r6c9]=1=[r8c9]-1-[r8c7]-7-[r2c7]=7=[r2c5]-7-[r3c6]-8-[r3c9]-5 => r13c8,r6c9<>5 @ 5-[r5c8]-1 [r5c8]-1-[r6c9] [r6c9]=1=[r8c9] [r8c9]-1-[r8c7] @ __________________________ 1-[r8c7]-7 [r8c7]-7-[r2c7] [r2c7]=7=[r2c5] [r2c5]-7-[r3c6] @ _____________________________________________________ 7-[r3c6]-8 [r3c6]-8-[r3c9] @ ______________________________________________________________ 8-[r3c9]-5 _____________________________________________________________________________________________

by **Luke** » Wed Nov 26, 2008 4:31 pm

Jasper, here's a seminal thread on Nice Loops and Nice Loop Notation. It has graphics to check out as one follows the notation. Also, from Sudopedia, check out Nice Loop.

Daj, I'm trying to understand why your notation starts with 5-[r5c8] and the other with 5=r5c8. Doesn't this chain have to begin and end on a strong link to allow the eliminations? Does 5-[r5c8] imply a strong link somehow? You end with [r3c9]-5, the other with r3c9=5.

Seems as different folk have different ways of notating the same thing.

by **DonM** » Wed Nov 26, 2008 6:10 pm

I think Jasper is justifiably confused. In order to not cause just the above confusion, (IMO

) if a chain is labelled as an AIC it would be preferable to use the AIC notation which shows the start and finish on strong links, the alternating inferences and the links within cells.

Thus, the chain above as AICs:

(5)r5c8 = (5-1)r6c9 = (1)r8c9 - (1=7)r8c7 - (7)r2c7 = (7)r2c5 - (7=8)r3c6 - (8=5)r3c9 => r13c8<> 5

(1)r6c9 = (1)r8c9 - (1=7)r8c7 - (7)r2c7 = (7)r2c5 - (7=8)r3c6 - (8=5)r3c9 => r6c9<>5

(For Jasper's benefit: the 2nd chain is basically the same as the first, with the first part removed to show more clearly the basis for the other elimination. Hobiwan's chain is one of those clever '2 for 1' specials and would be considered an advanced chain as these things go...)

(BTW: Andrew Stuart uses what is basically a variation of the above.)

by **hobiwan** » Wed Nov 26, 2008 9:31 pm

daj95376 wrote:I don't agree that links within cells aren't spelled out. Here's your AIC w/corrections and decomposed into lines of strong inference followed by lines of weak inference -- ending in a line of strong inference. The links within cells are marked with an (@).

Code: Select all
5-[r5c8]-1-[r6c9]=1=[r8c9]-1-[r8c7]-7-[r2c7]=7=[r2c5]-7-[r3c6]-8-[r3c9]-5 => r13c8,r6c9<>5 @ 5-[r5c8]-1 [r5c8]-1-[r6c9] [r6c9]=1=[r8c9] [r8c9]-1-[r8c7] @ __________________________ 1-[r8c7]-7 [r8c7]-7-[r2c7] [r2c7]=7=[r2c5] [r2c5]-7-[r3c6] @ _____________________________________________________ 7-[r3c6]-8 [r3c6]-8-[r3c9] @ ______________________________________________________________ 8-[r3c9]-5 _____________________________________________________________________________________________

Corrections duly noted

(edited my post above).

I see your point but I stick with what I said. If you see "7-r3c6-8" as notation for a strong link you would have a strong link written with a single dash - and that would be really confusing. I see it as direct notation of the propagation rule: r3c6 reached and left via weak links on digits 7 and 8, which implies a strong link between 7 and 8 whithin r3c6 that is not spelled out.

DonM wrote:Thus, the chain above as AICs:

(5)r5c8 = (5-1)r6c9 = (1)r8c9 - (1=7)r8c7 - (7)r2c7 = (7)r2c5 - (7=8)r3c6 - (8=5)r3c9 => r13c8<> 5

That's a nice AIC, but not mine

. "[r5c8]-1-[r6c9]" mean's that the link between r5c8 and r6c9 has to be a weak link on digit 1. Thus:

(5=1)r5c8 - (1)r6c9 = (1)r8c9 - (1=7)r8c7 - (7)r2c7 = (7)r2c5 - (7=8)r3c6 - (8=5)r3c9 => r13c8<> 5

That means, that 5 in r6c9 is not part of the AIC and see's both ends of the AIC. I therefore think that the 2 for 1 comment is not correct.

by **daj95376** » Wed Nov 26, 2008 10:15 pm

Luke451 wrote:Daj, I'm trying to understand why your notation starts with 5-[r5c8] and the other with 5=r5c8. Doesn't this chain have to begin and end on a strong link to allow the eliminations? Does 5-[r5c8] imply a strong link somehow? You end with [r3c9]-5, the other with r3c9=5.

Seems as different folk have different ways of notating the same thing.

If you check, you'll see that hobiwan corrected his chain to match mine. To answer your questions: yes ... and yes.

by **daj95376** » Wed Nov 26, 2008 10:20 pm

hobiwan wrote:I see your point but I stick with what I said. If you see "7-r3c6-8" as notation for a strong link you would have a strong link written with a single dash - and that would be really confusing. I see it as direct notation of the propagation rule: r3c6 reached and left via weak links on digits 7 and 8, which implies a strong link between 7 and 8 whithin r3c6 that is not spelled out.

This is from my notes. I consider it spelled out as a strong link.

Code: Select all: Strong Inference (SI): ~A => B Weak Inference (WI): A => ~B Strong Link (SL): e.g., ( bilocation []=n=[] ) or ( bivalue cell m-[]-n ) Weak Link (WL): e.g., ( peers []-n-[] ) or ( ?-value cell m=[]=n ) bilocation [a]=n=[b]: if [a] is not 'n', then [b] is 'n' bivalue cell m-[c]-n : if [c] is not 'm', then [c] is 'n' peers [d]-n-[e]: if [d] is 'n', then [e] is not 'n' ?-value cell m=[f]=n : if [f] is 'm', then [f] is not 'n'

What's not included in my notes is grouped strong/weak links.

DonM: I agree with hobiwan for the AIC in Eureka notation.

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