## New Solving Technique (I think)

Everything about Sudoku that doesn't fit in one of the other sections

### Re: New Solving Technique (I think)

As ever i was not completly clear in what I was saying !

I appreciate that if a "puzzle" has > 1 sol then template POM wont reliably solve the "puzzle". Actually no technique will !!!!

I really dont want to go down that road much - apart from to say that I was perhaps meaning to find a rating method or perhaps more relevantly finding a logical way to map an initial solving path in any of our hardest puzzles.

Say we look at which level at which there is only 1 combination of templates which fit in the board. This could well be level 3 in our hard puzzles. This would mean say in Easter Monster there is only one way to put in the templates of 5-7-9 clues. If this was the case it sure would go a long way to solve the puzzle. There might be harder puzzles where 4 levels are required - maybe 4 levels or more are required in EM...... However if you find that only one combination of templates fit - then that template combination is the only valid one - regardless whether the "puzzle" has > 1 sol.

Maybe someone could elucidate the occurance of the {5-7-9} 3-rookeries /3-templates in EM ?
Thinking about it ..... if there is only one combination i would be amazed

C
coloin

Posts: 1735
Joined: 05 May 2005

### Re: New Solving Technique (I think)

[quote="coloin"]Maybe someone could elucidate the occurance of the {5-7-9} 3-rookeries / 3-templates in EM ?
Thinking about it ..... if there is only one combination i would be amazed[/quote]

You don't need to worry about being amazed.

300 non-conflicting <579>-templates here: Show
Code: Select all
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daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Re: New Solving Technique (I think)

coloin wrote:Maybe someone could elucidate the occurance of the {5-7-9} 3-rookeries /3-templates in EM ?

daj95375, if your templates implementation allows, is there any 3-template that yields an exclusion for the Easter Monster ("EM")? Secondly, how many 4-templates for digits <1267> yield an exclusion? You'll recognize the latter as requirements for the sk-loop.

[edit: 3-template and 4-templates were 3-rookery and 4-rookeries, respectively]
Last edited by ronk on Sun Jan 29, 2012 11:08 am, edited 1 time in total.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Re: New Solving Technique (I think)

ronk wrote:
coloin wrote:Maybe someone could elucidate the occurance of the {5-7-9} 3-rookeries /3-templates in EM ?

daj95376, if your templates implementation allows, is there any 3-rookery that yields an exclusion for the Easter Monster ("EM")? Secondly, how many 4-rookeries for digits <1267> yield an exclusion? You'll recognize the latter as requirements for the sk-loop.

I created a working copy of my template() routine, and then altered it to specifically create the list coloin requested. The alterations simply took the template results for {5,7,9} and checked to see which cross-products didn't contain a conflict. Out of 2,592 possibilities, 593 didn't contain a conflict. Of the resulting 593 entries, I didn't check to see which one actually appears in the solution.

I'm not sure what you mean by "yields an exclusion", so I'm not sure what modifications would be needed to meet your request.

BTW, I'm not familiar with any of champagne's patterns.
daj95376
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### Re: New Solving Technique (I think)

Thank you very much for looking at this.
I did laugh !!
I was actually amazed that there were so many - but i suppose i hadnt done the 8*18*18 sum

I was expecting the 3-templates to interfere a little more with each other. Maybe it happens in easier puzzles.
The n-template count will be 1 when we get to n=8.
The 4,5,6,7 levels are unchartered.

Of the 593 puzzle plus 3-template combinations - one will be valid and 592 will be demonstratably invalid.
It would be possible to "rate" each of these invalid combinations. But this is approaching systematic T&E.

Applying the SK loop will reduce the 1-template counts - and hence the n-templates generally.
So i think ronk is asking how many 4-templates for digits <1267> are there before and after applying SK loop eliminations.

C
coloin

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### Re: New Solving Technique (I think)

Here is the <1267>-rookery. Of 524,472 combinations, only 23 survived. I couldn't apply an sk-loop for comparison.

Code: Select all
`17.6....22...17..6..62..71...1.6.27..2..71.6..67..21..71....62.6..12...7..27.6..117.6....22...17..6..62..71...7.2.16..1..76.2.62...1..77..1.26....176.2...62....7117.6....22...17..6..62..71.....2.167.1..76.2.627..1...7..1.26....176.2...62....7117...6..22...17..6..62..71...7.2.16..1.67..2.62...1..77..1.26....176.2...62....7117...6..22...17..6..62..71.....2.167.1.67..2.627..1...7..1.26....176.2...62....711..7.6..2..7.12..6.26...71.2...6.17..1.27..6.67...12..7..1..62...162...7.62..7..117.6....22...17..6..6..271.6...2.17..2.17..6..17..62..7....162...126...7.627....117.6....22...17..6..6..271.6...2.17..2..71.6..17..62..7..1..62...126...7.627....11..6.7..2..7.21..6.26...71.6...1.27..1..72.6.27...61..7..1..62...126...7.627....11..6.7..2..7.21..6.26...71.6...1..27.1.27..6.27...61..7..1.26....176.2...62....7117.....622...671....61.27....1.2..76.2.67..1..67..12..71....62.6..21...7..27.6..11..7...62..7.261...261..7..2.1.6..7..6..72.1..7...12.671....62.6..21...7..26.7..11..7...62..7.261...261..7....1.6.27..6..72.1.27...1..671....62.6..21...7..26.7..11....7.62..7.261...261..7....1.6..27.6.27..1.27...1..671...26..6..71.2....26...711....7.62..7.261...261..7....1.6..27.6..72.1.27...1..671.2..6..6..71.2....26...7117.....622...671....61.27....7.2..16.6..71.2..21..6..771.2..6..6..71.2....26...711....7.62..7.261...261..7..2...6..17.6..71.2..71..2..671.2..6..6..71.2....26...7117.....622...671....62.17..6...1.27..2..76.1..17..2..67..1..62...162...7.627....117.....622...671....62.17......1.276.2..76.1.617..2...7..1..62...162...7.627....11..7...62..7.621...26..17..2...1..76.6.27..1..71..62..71....62.6..12...7..26.7..11..7...62..7.621...26..17......1.276.6.27..1.271..6...71....62.6..12...7..26.7..117.....622...671....62.17....7.2..16.6.17..2..21..6..771...26..6..71.2....26...711....7.62..7.261...26..17..2...6..17.6.17..2..71..2..671.2..6..6..71.2....26...71`
daj95376
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### Re: New Solving Technique (I think)

In the table below, the entries represent the number of templates that don't conflict with each of the acceptable <1267>-rookery combinations.

Code: Select all
`    1    2    3    4    5    6    7    8    9   Value ==== ==== ==== ==== ==== ==== ==== ==== ====  n/a  n/a    8    0    2  n/a  n/a   16    5  n/a  n/a    3    0    2  n/a  n/a   17    5  n/a  n/a    5    0    2  n/a  n/a   11    4  n/a  n/a    5    1    2  n/a  n/a   12    5  n/a  n/a    7    4    2  n/a  n/a    7    4  n/a  n/a    7    2    3  n/a  n/a    8    3  n/a  n/a   15    1    1  n/a  n/a   10    5  n/a  n/a    7    0    3  n/a  n/a   12    5  n/a  n/a    6    0    3  n/a  n/a   15    3  n/a  n/a   13    2    2  n/a  n/a    6    5  n/a  n/a   10    2    2  n/a  n/a   13    5  n/a  n/a    5    0    2  n/a  n/a   11    4  n/a  n/a    7    0    2  n/a  n/a    5    3  n/a  n/a    8    1    3  n/a  n/a    7    5  n/a  n/a    6    0    3  n/a  n/a   15    3  n/a  n/a    7    0    3  n/a  n/a   12    5  n/a  n/a    3    0    3  n/a  n/a   14    3  n/a  n/a    6    0    1  n/a  n/a    5    5  n/a  n/a    4    0    1  n/a  n/a   10    4  n/a  n/a    5    1    2  n/a  n/a   12    5  n/a  n/a    7    4    2  n/a  n/a    7    4  n/a  n/a    7    2    3  n/a  n/a    8    3  n/a  n/a    5    1    4  n/a  n/a    9    3`

I'm not sure if this has any significance.

[Edit: forgot one entry.]
Last edited by daj95376 on Wed Jan 18, 2012 8:14 pm, edited 1 time in total.
daj95376
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### Re: New Solving Technique (I think)

coloin wrote:Applying the SK loop will reduce the 1-template counts - and hence the n-templates generally.
So i think ronk is asking how many 4-templates for digits <1267> are there before and after applying SK loop eliminations.

My question was poorly phrased, likely because of fuzzy thinking. What I meant was, adopting daj95376's notation ... are there exclusions based on the <1267>-templates over and above exclusions based on the sk-loop, aka hidden-pair-loop?

daj95376 wrote:Here is the <1267>-rookery. Of 524,472 combinations, only 23 survived. I couldn't apply an sk-loop for comparison.

Code: Select all
`.............VV...................V..V.....V..V...................VV.............17.6....22...17..6..62..71...1.6.27..2..71.6..67..21..71....62.6..12...7..27.6..117.6....22...17..6..62..71...7.2.16..1..76.2.62...1..77..1.26....176.2...62....7117.6....22...17..6..62..71.....2.167.1..76.2.627..1...7..1.26....176.2...62....7117...6..22...17..6..62..71...7.2.16..1.67..2.62...1..77..1.26....176.2...62....7117...6..22...17..6..62..71.....2.167.1.67..2.627..1...7..1.26....176.2...62....711..7.6..2..7.12..6.26...71.2...6.17..1.27..6.67...12..7..1..62...162...7.62..7..117.6....22...17..6..6..271.6...2.17..2.17..6..17..62..7....162...126...7.627....117.6....22...17..6..6..271.6...2.17..2..71.6..17..62..7..1..62...126...7.627....11..6.7..2..7.21..6.26...71.6...1.27..1..72.6.27...61..7..1..62...126...7.627....11..6.7..2..7.21..6.26...71.6...1..27.1.27..6.27...61..7..1.26....176.2...62....7117.....622...671....61.27....1.2..76.2.67..1..67..12..71....62.6..21...7..27.6..11..7...62..7.261...261..7..2.1.6..7..6..72.1..7...12.671....62.6..21...7..26.7..11..7...62..7.261...261..7....1.6.27..6..72.1.27...1..671....62.6..21...7..26.7..11....7.62..7.261...261..7....1.6..27.6.27..1.27...1..671...26..6..71.2....26...711....7.62..7.261...261..7....1.6..27.6..72.1.27...1..671.2..6..6..71.2....26...7117.....622...671....61.27....7.2..16.6..71.2..21..6..771.2..6..6..71.2....26...711....7.62..7.261...261..7..2...6..17.6..71.2..71..2..671.2..6..6..71.2....26...7117.....622...671....62.17..6...1.27..2..76.1..17..2..67..1..62...162...7.627....117.....622...671....62.17......1.276.2..76.1.617..2...7..1..62...162...7.627....11..7...62..7.621...26..17..2...1..76.6.27..1..71..62..71....62.6..12...7..26.7..11..7...62..7.621...26..17......1.276.6.27..1.271..6...71....62.6..12...7..26.7..117.....622...671....62.17....7.2..16.6.17..2..21..6..771...26..6..71.2....26...711....7.62..7.261...26..17..2...6..17.6.17..2..71..2..671.2..6..6..71.2....26...71`

I added a "V" header line which marks the 8 cells limited to "sk-loop-digits", aka <1267>-template digits. In grid format, this looks like ...

Code: Select all
` . . . | . . . | . . . . . . | . V V | . . . . . . | . . . | . . .-------+-------+------- . . . | . . . | . V . . V . | . . . | . V . . V . | . . . | . . .-------+-------+------- . . . | . . . | . . . . . . | V V . | . . . . . . | . . . | . . .`

Therefore, we have r28c5,r5c28=126 and r2c6,r4c8,r6c2,r8c2=1267 with the consequential exclusion of other digits in those cells.

There are other exclusions however, some part of the sk-loop technique, and some not. As an example of the former, either r1c2=7 or r2c3=7. Templating should eliminate any 7s candidate that sees both of these. As an example of the latter, r4c5=126 for exclusion r4c5<>4, which AFAIK is beyond the scope of the sk-loop.
ronk
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### Re: New Solving Technique (I think)

Well the sk loop eliminates the 7@r1c3 and the 2@r3c1 - neither of which survives in any template.
The reduction from 524,472 combinations to 23 might be because of the sk loop. It seems a big reduction compared to the best clues at the 3-level.
daj95376 wrote:In the table below, the entries represent the number of templates that don't conflict with each of the acceptable <1267>-rookery combinations.......
I'm not sure if this has any significance.

Well it does mean that if there is 0 template which doesnt conflict - therefore that means that all templates conflict - therefore this must rule out 11 of the templates. But not all sadly.

C
coloin

Posts: 1735
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### Re: New Solving Technique (I think)

!!! Okay, I understand how ronk derived some of his eliminations. This is it using templates.

Code: Select all
` <1267>-rookery with at least one value assigned to a cell by Templates +-----------------------+ | . . . | . . . | . . . | | . . . | . T T | . . . | | . . . | . . . | . . . | |-------+-------+-------| | . . . | . T . | . T . | | . T . | . . . | . T . | | . T . | . . T | . . . | |-------+-------+-------| | . . . | . . . | . . . | | . . . | T T . | . . . | | . . . | . . . | . . . | +-----------------------++--------------------------------------------------------------------------------------+|  1        478      34578   |  3567     3689     5678    |  3489     369      2       ||  238      9        378     |  4       T126-38  T1267-8  |  138      5        368     ||  23458    248      6       |  1235     12389    1258    |  7        139      3489    ||----------------------------+----------------------------+----------------------------||  2468     5        1478    |  9       T126-4    3       |  128     T1267     678     ||  234689  T126-48   13489   |  126      7        1246    |  123589  T126-39   35689   ||  2369    T1267     1379    |  8        5       T126     |  1239     4        3679    ||----------------------------+----------------------------+----------------------------||  7        148      14589   |  1235     12348    12458   |  6        239      3459    ||  456      3        145     | T1267-5  T126-4    9       |  245      8        457     ||  45689    468      2       |  3567     3468     45678   |  3459     379      1       |+--------------------------------------------------------------------------------------+`
daj95376
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### Re: New Solving Technique (I think)

coloin wrote:Well the sk loop eliminates the 7@r1c3 and the 2@r3c1 - neither of which survives in any template.

But of course, just look for cells without a digit in all of the <1267>-templates. I was making it much more complicated than that.
ronk
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### Re: New Solving Technique (I think)

When I first saw the list of 23 (1267)template combinations I guessed it could be reduced by removing those that contained uncovered <Unavoidable Sets>, but on inspection I can't find any! In this case I think it's because (7)r5c5 forces every digit pair in the SK loop to contain a single truth (one proof of that is <here>).

I guess that checking for UAs as every template is added would consume more time than it would save in running a backtracking algorithm, but it might be worthwhile for say when the second and third digits are brought into the mix.
Last edited by David P Bird on Thu Jan 19, 2012 11:29 pm, edited 1 time in total.
David P Bird
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### Re: New Solving Technique (I think)

ronk wrote:
coloin wrote:Well the sk loop eliminates the 7@r1c3 and the 2@r3c1 - neither of which survives in any template.

But of course, just look for cells without a digit in all of the <1267>-templates.

!!! Okay, are there any eliminations that I'm missed?

Code: Select all
`......................1...............1...1.............1.1...................... elim..................2...2.............2.....2...............2...................... elim.............3.............................3..................................... elim...............................4.....4.............................4............. elim..................................................................5.............. elim....6...............................6.......6...........................6...6.... elim..7.............................................................................. elim.............88......................8........................................... elim...........................................9..................................... elim Eliminations for <1267>-templates +-----------------------------------------------+ |   .   .  -7   |   .  -6   .   |   .   .   .   | |   .   .   .   |   .  -38 -8   |   .   .   .   | |  -2   .   .   |   .  -12  .   |   .   .   .   | |---------------+---------------+---------------| |   .   .   .   |   .  -4   .   |   .   .   .   | |  -26 -48 -1   |   .   .   .   |  -12 -39 -6   | |   .   .   .   |   .   .   .   |   .   .   .   | |---------------+---------------+---------------| |   .   .  -1   |   .  -12  .   |   .   .   .   | |   .   .   .   |  -5  -4   .   |   .   .   .   | |  -6   .   .   |   .  -6   .   |   .   .   .   | +-----------------------------------------------+ Easter Monster grid after eliminations for <1267>-templates +--------------------------------------------------------------------------------------+ |  1        478      3458-7  |  3567     389-6    5678    |  3489     369      2       | |  238      9        378     |  4        126-38   1267-8  |  138      5        368     | |  3458-2   248      6       |  1235     389-12   1258    |  7        139      3489    | |----------------------------+----------------------------+----------------------------| |  2468     5        1478    |  9        126-4    3       |  128      1267     678     | |  3489-26  126-48   3489-1  |  126      7        1246    |  3589-12  126-39   3589-6  | |  2369     1267     1379    |  8        5        126     |  1239     4        3679    | |----------------------------+----------------------------+----------------------------| |  7        148      4589-1  |  1235     348-12   12458   |  6        239      3459    | |  456      3        145     |  1267-5   126-4    9       |  245      8        457     | |  4589-6   468      2       |  3567     348-6    45678   |  3459     379      1       | +--------------------------------------------------------------------------------------+`

[Edit: added missing eliminations and replace use of "rookery".]
Last edited by daj95376 on Thu Jan 19, 2012 3:27 pm, edited 3 times in total.
daj95376
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### Re: New Solving Technique (I think)

Maybe what we have here is a way to pinpoint solving paths in a [hard] puzzle.
Although we havnt looked at any other template combinations in EM - the <1267> rookery does show a surprisingly reduced number of possible templates - and We know that this will be because the SK loop is there. None of those <6> or <1> eliminations are in the 23 viable <1267>s.

So we could deduce all the eliminations of the SK loop by looking at the possible <1267> templates.

The low numbers of <12467> templates im guessing is because of the eliminations which ronk rightly indicated. But still it doesnt go to 1. Im surmising to solve a puzzle with POM or pjb's technique it has to get down to 1.

I dont know how easy it is for danny to compute these figures.

there are 126 4 and 5 templates in total ......we could see every elimination that there is

Sudoku Explainer - eat your heart out

champagne mentions "floors" level 4 and 5 - i presumed this to mean the number of different <clues> in a chain or loop. It could well be similar to the number of templates.

C
coloin

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Joined: 05 May 2005

### Re: New Solving Technique (I think)

daj95376 wrote:Okay, are there any eliminations that I'm missed?

There is r1c5<>6 and r5c7<>12 also.

here coloin wrote:The low numbers of <12467> templates im guessing is because of the eliminations which ronk rightly indicated. But still it doesnt go to 1. Im surmising to solve a puzzle with POM or pjb's technique it has to get down to 1.

After <1267>-templating, there is a <1246> naked quad in b6 (and r5), so adding the template for digit 4 would add eliminations. I'm curious, what was your reason for adding this 5th digit.

I'd like to see us use rookery and template according to dukuso's earlier definitions.

here dukuso wrote:a k-rookery is a subset of the 81 cells in a 9*9 square which contains exactly k-cells
from each row,column,block

a k-template is a k-rookery whose 9*k cells are filled with numbers from {1,2,..,k},
such that the rookery-cells from each row,column,block contain each digit from {1,..,k}
exactly once

a sudokugrid is a 9-template, there are ~6.67e21 of them

for some k-rookeries there is no k-template, for others there are many
what's the maximum number of ways(k) to fill the cells in a k-rookery
to obtain a k-template ?

(1):1
(2):16? (O..O..... O..O..... ......OO. ......OO. .O..O.... .O..O.... .....O..O ..O..O... ..O.....O)
(8):1.59e16?
(9):6.67e21

Here k-template is normalized to digits 1, 2, 3, ... k, but this isn't convenient for discussing templates in actual puzzles. Also, isn't "{ ... }" commonly used for sets or a collection? Taking these two together, I now think {1267}-templates is better than <1267>-templates or <1267>-rookery or {1267]-rookery. Opinions?
ronk
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