## More Monster Loops, Structure and Symmetry

Advanced methods and approaches for solving Sudoku puzzles
Allan Barker wrote:I took a look at Cola036 and found the following 4 layer loop that eliminates 26 candidates, 29 with a few more cover sets. It is obviously related to the same kind of patterns as in other puzzles. The major difference seems to be the requirment for 3 strong cell sets in the center box. Similar to other monster loops, it has two symmetric layers, a 90 degree rotationally symmetric layer, and a slightly distorted layer.

The "3 strong cell sets in the center box" apparently also need to be in a mini-row or mini-column. This property is useful in at least six of these eighteen puzzles. For cola199 (as morphed by gsf's program), an extra cover set is required in b1. Despite this, the solution is rank 0 due to the extra base set in the center box.

Code: Select all
`cola199 (gsf morph)000000001060009030002030700400805000000006000030900080001000004050008090700000200+--------------------------------------------------------------------------------------+| 3589     89(47)   3-45-789 | 24567    -2-456-78  247    | -45689   56(24)   1        || 58(1)    6        58(47)   | -5(1247) 58(1247) 9        | 58(4)    3        58(2)    || -1589    89(14)   2        | 1456     3        14       | 7        56(4)    5689     |+--------------------------------------------------------------------------------------+| 4        -9(127)  679      | 8        (127)    5        | 1369     -6(127)  23679    || 12589    -8-9(127) 5789    | 3        (1247)   6        | 1459     -5(1247) 2579     || 1256     3        567      | 9        (1247)   1247     | 1456     8        2567     |+--------------------------------------------------------------------------------------+| -23689   89(2)    1        | 2567     -256-79  237      | 3568     56(7)    4        || 36(2)    5        36(4)    | -6(1247) 6(1247)  8        | 36(1)    9        36(7)    || 7        89(4)    3-4689   | 1456     -1-4569  134      | 2        56(1)    3568     |+--------------------------------------------------------------------------------------+cola 60 Nodes, Raw Rank = 0 (linksets - sets)     19 Sets = {1247R2 1247R8 1247C2 1247C8 456N5}     19 Links = {1247c5 45n2 28n4 45n8 1b19 2b37 4b137 7b19}     20 Eliminations, 0 Assignments -->      r1c357<>4, r1c35<>7, r7c15<>2, r9c35<>4, r45c2<>9, r8c4<>6, r1c5<>2,      r2c4<>5, r3c1<>1, r4c8<>6, r5c8<>5, r5c2<>8, r7c5<>7, r9c5<>1`

Allan Barker wrote:
Ronk wrote:Several of the solving methods for these monsters are based on symmetry, and several of us will be tempted to morph the puzzle to visualize this symmetry. Therefore, I've used gsf's solver to obtain a "best symmetry" for all 18 of coloin's puzzles that you've posted. Hopefully then, we can all be posting the same morph.

The morph that makes the logic symetrical (at least for this puzzle) is very different from the morph in the list of gsf's best symmetry morphs, above.

I was hoping the program would take personal bias out of the morphing process, but clearly a number of the morphs are unsatisfactory. Do you think I should manually morph the worst ones? Or maybe just delete the post entirely?
Last edited by ronk on Wed Feb 04, 2009 10:30 pm, edited 1 time in total.
ronk
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RonK wrote:I was hoping the program would take personal bias out of the morphing process, but clearly a number of the morphs are unsatisfactory. Do you think I should manually morph the worst ones? Or maybe just delete the post entirely?

Maybe you said it correctly in your previous post, that there are different goals, or perhaps different properties at work. However, it looks like the symmetry of your cola199 solution is dead on the gsf's minimum distance morph. I would not delete the gsf type morphs because it would be interesting to see which solutions follow the same symmetry and which ones don't.

RonK wrote:The "3 strong cell sets in the center box" apparently also need to be in a mini-row or mini-column. This property is useful in at least six of these eighteen puzzles.

Yes, it looks like cola036, cola259, and cola199 have a number of similarities including the 3 cells in a mini-row. I wonder if I can fit something like your solution into 036?

.
Allan Barker

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Hi allan and ronk,

Impressive work. I compared to the present status of the solver without help of Allan model, nothing comes (no double XWing...) for cola199, and no exocet was found in cola036, so this is really a new pattern.

BTW, I have the feeling that the "hardest puzzle" family using standard rating has, as says Allan "quasi symmetry" properties leading to that very efficient way to fight against.

I thought I would find possibilities to simplify the path on easier puzzles, up to now, I have no clear answer.

One specificity of cola199 is the big number of "mini" ecocets. The path should be shorten when the program will perform well on exocets, but ronk has settled a very high target. Same (may be more) for cola036 where I don't foresee any other way than to apply Allan model or to digest that new pattern.

again, congratulations to both

champagne
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SK + core loops for Cola036 and Cola259

After some search, I was able to find loops that are almost the same as the one for Cola199 found by RonK above. These seem to be made of an SK like outer loop connected to a center box group of 3 strong cell sets, again all in a mini row/column. All these loops are rank 0, like Ron's.

Champage wrote:I compared to the present status of the solver without help of Allan model, nothing comes (no double XWing...) for cola199, and no exocet was found in cola036, so this is really a new pattern.

In order to find the "SK + core" loop for Cola036, I had to remove a given 7. IF you look, the Cola036 loop is slightly damaged on one corner but is still logically equivalent. This might be one reason it is more difficult to find.

Edit:RonK has pointed out in a PM that removing given 7r5c9 is not required, which is true. It was removed while trying to form a typical 1357 SK loop outer ring.

1. Cola036 Morph A - original loop with box structures in box 2468. 26 eliminations. Given 7r7c5
2. Cola036 Morph A Transpose - SK + core without Given 7r7c5(Trans=7r5c9). 20 eliminations.
3. Cola036 Morph A Transpose - SK + core without Given 7r5c9. [extra cover sets] 29 eliminations.

4. Cola259 Morph B. - original layered loops. 21 eliminations.
5. Cola259 Morph B. - SK + core 17 eliminations.

6. Cola259 Morph B. - SK + core 24 eliminations. [extra cover sets]

Cola036.

Cola259.

Cola036 Morph A.
Code: Select all
` (example 1)  +--------------------------------------------------------------------------------------------------------+  | 2469       46(7)      1          | 24(37)     248(-3)    289(37)    | 5          469(3)     -34689     |  | 49(5)      8          479        | 6          -4(135)    -9(1357)   | 149        2          349        |  | 3          456        2469       | 245(1)     24-58(-1)  2589(1)    | -14689     469(1)     7          |  +--------------------------------------------------------------------------------------------------------+  | 124(5)     9          2347       | 8          -12-3-5    6          | 147        -4(1357)   34(5)      |  | -1468(-5)  46(1357)   -346-78    | (1357)     (135)      (1357)     | 2          469(1357)  -3469(-5)  |  | 126(5)     -6(1357)   2367       | 12357      9          4          | 167        8          36(5)      |  +--------------------------------------------------------------------------------------------------------+  | -14689     46(1)      4689       | 45(1)      7          58(1)      | 3          456        2          |  | 1468       2          3468       | 9          -4-6-8(135) -8(135)    | 4678       -4-6(57)   468(5)    |  | 7          46(3)      5          | 24(3)      2468(-3)   28(3)      | 4689       469        1          |  +--------------------------------------------------------------------------------------------------------+001000500080600020300000007090806000000000200000094080000070302020900000705000001`

Cola259 Morph B.
Code: Select all
` (example 6)  +-----------------------------------------------------------------------------------------------+  | 3568      38(4)     7         | 134       -13-4589  1458      | 5689      56(1)     2         |  | 58(2)     9         58(4)     | (1247)    58(1247)  6         | 58(7)     3         58(1)     |  | 1         38(2)     3568      | 237       -235-789  2578      | 4         56(7)     5689      |  +-----------------------------------------------------------------------------------------------+  | 2357      -3(1247)  1345      | 8         (1247)    9         | 2356      -5-6(124) 13456     |  | -2589     -8(124)   -1-4589   | -1-2-46   (124)     3         | -2568     -5-6(124) 7         |  | 2378      6         1348      | 5         (1247)    1247      | 238       9         1348      |  +-----------------------------------------------------------------------------------------------+  | 4         38(17)    -1389     | 1237      6         12578     | -235-79   5(27)     359       |  | 36(7)     5         36(1)     | 9         3(1247)   (1247)    | 36(27)    8         36(4)     |  | 36-789    38(7)     2         | 347       3-45-78   4578      | 1         56(47)    3-4569    |  +-----------------------------------------------------------------------------------------------+007000002090006030100000400000809000000003007060500090400060000050900080002000100`
Last edited by Allan Barker on Thu Feb 05, 2009 5:47 am, edited 1 time in total.
Allan Barker

Posts: 266
Joined: 20 February 2008

Allan Barker wrote:
Champage wrote:I compared to the present status of the solver without help of Allan model, nothing comes (no double XWing...) for cola199, and no exocet was found in cola036, so this is really a new pattern.

In order to find the "SK + core" loop for Cola036, I had to remove a given 7. IF you look, the Cola036 loop is slightly damaged on one corner but is still logically equivalent. This might be one reason it is more difficult to find.

Hi allan,

I register it. I am short in time to benefit of it immediately, but for sure, I will come back to that finding as soon as possible.

All that is very interesting.

champagne
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### cola548

cola548 is not as similar to coly510 as are many of the others, but there is a strong resemblance nonetheless. The morph below is the p-2 (pi radians rotational) symmetry output by gsf's program.

I worked on this a long long time, and it was only after paying attention to the two AALSs (r4c12 and r6c89) -- each being one "bi" of a potential "bi bi", I guess -- that I actually made some progress, and then stumbled onto this logic set. In addition to the placement r2c7=5, note the naked single created in r4c7.

Code: Select all
`cola548 (gsf morph)  +--------------------------------------------------------------------------------------+  | 269-37   2689-13  (37-1)-69| 2468-3   (13)-68  5        | (17)-49   246-17   249-17  |  | 269-5    2689-15  69(1-5)  | 7        68(1)    2468-1   | (+5-1)-49 246-1    3       |  | 23567    12356    4        | 236      9        126      | 8         1267     1257    |  +--------------------------------------------------------------------------------------+  | (357)    (135)    2        | 3458     (1357)-8 1478     | 4(-1357)  9        6       |  | 35679    13569    (1357)-69| 345689   2        146789   | (137-5)-4 13478    14578   |  | 8        4        69(-1357)| 369-5    (1357)-6 1679     | 2         (137)    (157)   |  +--------------------------------------------------------------------------------------+  | 2359     2359     8        | 259      4        297      | 6         1237     1297    |  | 1        269      69       | 2689     68(7)    3        | 49(7)     5        2489-7  |  | 2469-35  7        69(35)   | 1        68(5)    2689     | 49(3)     248-3    2489    |  +--------------------------------------------------------------------------------------+000005000000700003004090800002000096000020000840000200008040600100003050070100000cola 54 Nodes, Raw Rank = 17 (linksets - sets)     16 Sets = {1357C3 1357C5 1357C7 4N12 6N89}     33 Links = {1r1246 3r1469 5r2469 7r1468 125n3 146n5 125n7 1357b4 1357b6}     44 Eliminations, 1 Assignments -->      r1c2389<>1, r2c2678<>1, r4c7<>1357, r6c3<>1357, r1c124<>3, r1c189<>7,      r2c123<>5, r125c7<>4, r1c35<>6, r1c37<>9, r9c18<>3, r14c5<>8, r5c3<>69,      r2c7<>9, r5c7<>5, r6c4<>5, r6c5<>6, r8c9<>7, r9c1<>5,      r2c7=5`

Additional combinations of cover sets (Links) can raise the counts to 57 eliminations and 2 assignments.

Normally I wouldn't list the eliminations above, but my indication of eliminations in the pencilmarks is considerably different this time. No matter the number of eliminations per cell, there are never more than two minus signs ('-') per cell, and usually only one. One or more eliminated candidate(s) follows the minus sign, and they are always listed last for that cell. Two minus signs occur only when there is an elimination in the base set as well, as in r2c7 for this example. This has the effect of requiring much less space for the pencilmarks.

[edits: 1) decrease the number of Sets and Links; 2) add images]
ronk
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Allan Barker wrote:Set Logic Diagrams and Recent BB Pattern Solutions

A) Tarx0134, Digits 3458, 18 base sets, 23 eliminations

For tarx0134, there is also a "raw rank 0" logic set with 16 base sets and 22 of those 23 eliminations.

Code: Select all
`+--------------------------+-------------------------------+--------------------------+| 167-58  7(58)     3      | 19(58)    1679-8    67(58)    | 2       19(5)     4      || 1578    9         147    | 2         13478     -7(3458)  | 157     6         137    || 1267-5  27(45)    1267-4 | 19(345)   1679-34   67(345)   | 8       19(35)    179-3  |+--------------------------+-------------------------------+--------------------------+| 12358   -2(3458)  124    | 7         348       9         | 1456    -1(3458)  1368   || 1389    6         149    | (348)     5         2         | 149     7         1389   || 35789   -7(3458)  479    | 6         348       1         | 459     -9(3458)  2      |+--------------------------+-------------------------------+--------------------------+| 2679-3  27(3)     8      | 19(34)    12679-34  67(34)    | 1679-4  19(4)     5      || 3679    1         679    | -9(3458)  346789    -67(3458) | 4679    2         6789   || 4       27        5      | 19(8)     12679-8   67(8)     | 3       19(8)     1679-8 |+--------------------------+-------------------------------+--------------------------+tarx0134 57 Nodes, Raw Rank = 0 (linksets - sets)     16 Sets = {3C2468 4C2468 5C2468 8C2468}     16 Links = {58r1 345r3 34r7 8r9 46n2 58n4 28n6 46n8}     22 Eliminations, 0 Assignments -->      r1c15<>8, r3c59<>3, r3c35<>4, r7c15<>3, r7c57<>4, r9c59<>8, r13c1<>5,      r28c6<>7, r4c8<>1, r4c2<>2, r6c2<>7, r6c8<>9, r8c6<>6, r8c4<>9`

ronk
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