g-whips and g-braids

Advanced methods and approaches for solving Sudoku puzzles

Re: g-whips and g-braids

Postby denis_berthier » Mon Sep 12, 2011 12:03 pm

pjb wrote:I saw this puzzle somewhere given as a great example of POM analysis. Following basics, POM first finds two 3's at 1,1 and 4,4, and then no less than 20 9's. After this the puzzle solves easily.


As you may have guessed from my previous posts, what I'm interested in is comparing various ratings of puzzles based on various coherent families of rules.
All the consistent rating systems I've studied, give the same rating to almost all (in the sense of unbiased stats) the puzzles.
In this exceptional example, it appears that the following two ratings both give the same result, 5:
1) the gW rating
2) a tentative rating that would be based only on basic interactions, generalized Subsets (with Finned, Franken and Kraken Fish) and XYZ-Wing .

I'm curious: what's the maximum size of the patterns used in your POM solution?

As for the number of candidates eliminated by a single pattern, this has never impressed me much: often, people like to exhibit artificially big patterns by aggregating smaller ones; as a result, most of, if not all, their eliminations could be done by simpler patterns. Notice that this is the case in the Hodoku solution (but Hodoku is a great solver anyway !)
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Re: g-whips and g-braids

Postby pjb » Tue Oct 25, 2011 10:36 am

denis_berthier wrote:
I'm curious: what's the maximum size of the patterns used in your POM solution?


I gave up identifying patterns as 'a', 'b', etc when I realized there are up to 2000 or more patterns for a number in some puzzles. So I ended up numbering patterns as '0001', '0002' etc, to accommodate that many. As far as I know, the program can generate all possible patterns in any puzzle I've tested it on. If you're interested, the program displays all possible patterns for a number in the output box for the patterns - contradiction method.

Thanks for your interest
pjb
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An extremely rare case of difference in W, B and gB ratings

Postby denis_berthier » Thu Sep 20, 2012 8:12 am



An extremely rare large difference in W, B and gB ratings


Together with his implementation of whips in javascript, Mauricio has proposed a puzzle
http://forum.enjoysudoku.com/whip-solver-in-javascript-t30678.html#p220337
with interesting properties:
- it has a whip of length 31
- its B rating is only 19.

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


The purpose of this post is to show that the rating is largely improved if g-whips are allowed: gW = 12
This puzzle is therefore not only an extremely rare example of an extremely long whip but also one of an extremely large difference in ratings.

In the following g-whip solution, what I now call bivalue-chains are activated. They are what I called previously nrc-chains (i.e. what some would call chains of bivalue/bilocal cells - but as in the supersymmetric view of whips, rc, rn, cn and bn cells are all on the same footing, there is no reason to make any difference between bivalue and bilocal).

Hidden Text: Show
***** SudoRules 15d.1.12 based on CSP-Rules 1.2, config: gW *****
000001002000030040005200100003600010020070008900005700009007000080900004300040080
24 givens, 220 candidates, 1433 csp-links and 1433 links. Initial density = 1.49
g-whip[6]: b2n4{r3c6 r1c4} – c3n4{r1 r456} – r6n4{c2 c3} – b4n8{r6c3 r4c1} – r3n8{c1 c5} – c6n8{r3 .} ==> r3c6 ≠ 6, r3c6 ≠ 9
g-whip[6]: b4n8{r6c3 r4c1} - r3n8{c1 c456} - c6n8{r2 r3} - b2n4{r3c6 r1c4} - c3n4{r1 r5} - r6n4{c3 .} ==> r6c3 <> 6, r6c3 <> 1
whip[11]: r8n1{c1 c5} - r9c4{n1 n5} - c2n5{r9 r4} - b4n7{r4c2 r4c1} - b4n8{r4c1 r6c3} - r6c5{n8 n2} - r4n2{c6 c7} - b6n4{r4c7 r5c7} - b4n4{r5c3 r6c2} - c3n4{r6 r1} - c4n4{r1 .} ==> r7c2 <> 1
whip[12]: r6n4{c2 c4} - r4n4{c6 c7} - b6n2{r4c7 r6c8} - r6n3{c8 c9} - r6n6{c9 c2} - r6n1{c2 c5} - r5c4{n1 n3} - r5c6{n3 n9} - r4n9{c6 c9} - r2n9{c9 c2} - c2n1{r2 r9} - r8n1{c1 .} ==> r5c3 <> 4
g-whip[5]: c3n4{r1 r6} - b4n8{r6c3 r4c1} - r3n8{c1 c456} - c6n8{r2 r3} - b2n4{r3c6 .} ==> r1c2 <> 4
g-whip[5]: c3n4{r1 r6} - b4n8{r6c3 r4c1} - r3n8{c1 c456} - c6n8{r2 r3} - b2n4{r3c6 .} ==> r1c1 <> 4
whip[10]: c3n4{r1 r6} - b4n8{r6c3 r4c1} - b1n8{r1c1 r2c3} - b1n2{r2c3 r2c1} - c1n7{r2 r8} - c2n7{r9 r4} - b4n5{r4c2 r5c1} - c1n1{r5 r7} - b9n1{r7c9 r9c9} - b9n7{r9c9 .} ==> r1c3 <> 7
whip[11]: c3n7{r9 r2} - b1n2{r2c3 r2c1} - b1n1{r2c1 r2c2} - c2n7{r2 r4} - r9n7{c2 c9} - b9n1{r9c9 r7c9} - c1n1{r7 r5} - b4n5{r5c1 r4c1} - r4c9{n5 n9} - r5n9{c8 c6} - r2n9{c6 .} ==> r8c1 <> 7
g-whip[11]: b1n2{r2c1 r2c3} - c3n8{r2 r6} - c3n4{r6 r1} - r3n4{c2 c6} - r3n8{c6 c5} - b8n8{r7c5 r7c4} - b8n3{r7c4 r8c6} - r5c6{n3 n9} - c8n9{r5 r123} - r2n9{c9 c2} - b1n1{r2c2 .} ==> r2c1 <> 8
whip[12]: b9n1{r7c9 r9c9} - r9c4{n1 n5} - c2n5{r9 r4} - r4c9{n5 n9} - b5n9{r4c6 r5c6} - r2n9{c6 c2} - c2n1{r2 r6} - r5c3{n1 n6} - r5c1{n6 n4} - r5n1{c1 c4} - b5n3{r5c4 r6c4} - r6n4{c4 .} ==> r7c9 <> 5
whip[12]: r3c6{n8 n4} - b1n4{r3c2 r1c3} - r6c3{n4 n8} - c4n8{r6 r7} - b8n3{r7c4 r8c6} - r5c6{n3 n9} - r4c5{n9 n2} - r6c5{n2 n1} - b8n1{r8c5 r9c4} - c2n1{r9 r2} - r2n9{c2 c9} - c8n9{r3 .} ==> r3c5 <> 8

whip[4]: c3n8{r1 r6} - c3n4{r6 r1} - r3n4{c2 c6} - r3n8{c6 .} ==> r1c1 <> 8
whip[10]: r3n8{c1 c6} - b2n4{r3c6 r1c4} - b1n4{r1c3 r3c2} - b1n3{r3c2 r1c2} - r1n7{c2 c8} - r2n7{c9 c4} - b2n5{r2c4 r1c5} - r1n9{c5 c7} - b9n9{r9c7 r9c9} - b9n7{r9c9 .} ==> r3c1 <> 7
whip[11]: r3n7{c9 c2} - b1n3{r3c2 r1c2} - b1n9{r1c2 r2c2} - r2n7{c2 c4} - c9n7{r2 r9} - b9n9{r9c9 r9c7} - r1n9{c7 c5} - b2n5{r1c5 r1c4} - r9c4{n5 n1} - b5n1{r5c4 r6c5} - c2n1{r6 .} ==> r1c8 <> 7
whip[12]: b9n9{r9c7 r9c9} - r4c9{n9 n5} - r2n5{c9 c4} - r9c4{n5 n1} - c5n1{r8 r6} - c2n1{r6 r2} - r2n9{c2 c6} - r3c5{n9 n6} - r1c5{n6 n8} - c5n9{r1 r4} - b5n2{r4c5 r4c6} - c6n8{r4 .} ==> r9c7 <> 5
whip[12]: r6c3{n4 n8} - c1n8{r4 r3} - r3c6{n8 n4} - r4n4{c6 c7} - b6n2{r4c7 r6c8} - r6c5{n2 n1} - r5c4{n1 n3} - r5c6{n3 n9} - b6n9{r5c8 r4c9} - r2n9{c9 c2} - c2n1{r2 r9} - r8n1{c1 .} ==> r5c1 <> 4

whip[4]: b4n5{r5c1 r4c2} - b4n7{r4c2 r4c1} - c1n8{r4 r3} - c1n4{r3 .} ==> r7c1 <> 5
g-whip[8]: r4c9{n5 n9} - b5n9{r4c6 r5c6} - r5n4{c6 c4} - b5n3{r5c4 r6c4} - b5n1{r6c4 r6c5} - r8n1{c5 c123} - c2n1{r9 r2} - r2n9{c2 .} ==> r5c7 <> 5
whip[12]: r5n5{c1 c8} - r4c9{n5 n9} - r5n9{c8 c6} - r2n9{c6 c2} - c2n1{r2 r9} - r8n1{c1 c5} - c4n1{r9 r6} - b5n3{r6c4 r5c4} - c4n4{r5 r1} - c3n4{r1 r6} - r6c2{n4 n6} - r5c3{n6 .} ==> r5c1 <> 1
biv-chain[3]: r5c1{n5 n6} - r1c1{n6 n7} - r4n7{c1 c2} ==> r4c2 <> 5
whip[1]: c2n5{r9 .} ==> r8c1 <> 5
whip[4]: b7n4{r7c1 r7c2} - b7n5{r7c2 r9c2} - r9c4{n5 n1} - b9n1{r9c9 .} ==> r7c1 <> 1
whip[6]: r6n2{c8 c5} - r7n2{c5 c1} - b7n4{r7c1 r7c2} - b7n5{r7c2 r9c2} - r9c4{n5 n1} - b5n1{r5c4 .} ==> r8c8 <> 2
whip[7]: c1n1{r8 r2} - c1n2{r2 r7} - b7n4{r7c1 r7c2} - b7n5{r7c2 r9c2} - c2n1{r9 r6} - b5n1{r6c5 r5c4} - r9c4{n1 .} ==> r8c1 <> 6
whip[9]: b7n5{r7c2 r9c2} - r9c4{n5 n1} - r5n1{c4 c3} - b4n6{r5c3 r5c1} - b4n5{r5c1 r4c1} - r4c9{n5 n9} - r5n9{c8 c6} - r2n9{c6 c2} - c2n1{r2 .} ==> r7c2 <> 6
g-whip[11]: b7n4{r7c1 r7c2} - b7n5{r7c2 r9c2} - r9c4{n5 n1} - b5n1{r5c4 r6c5} - r6c2{n1 n6} - r6n4{c2 c4} - r5c4{n4 n3} - r5c6{n3 n9} - c8n9{r5 r123} - r2n9{c9 c2} - c2n1{r2 .} ==> r4c1 <> 4
whip[5]: b2n7{r2c4 r1c4} - r1n4{c4 c3} - c1n4{r3 r7} - c1n2{r7 r8} - c1n1{r8 .} ==> r2c1 <> 7
whip[5]: r7c2{n5 n4} - c1n4{r7 r3} - b2n4{r3c6 r1c4} - b2n7{r1c4 r2c4} - b2n5{r2c4 .} ==> r7c5 <> 5
whip[6]: r7c2{n5 n4} - b4n4{r6c2 r6c3} - r1n4{c3 c4} - b2n7{r1c4 r2c4} - c4n5{r2 r9} - r8n5{c5 .} ==> r7c8 <> 5
whip[6]: b2n5{r1c5 r2c4} - r7n5{c4 c2} - r9n5{c2 c9} - r4n5{c9 c1} - c1n7{r4 r1} - b2n7{r1c4 .} ==> r1c7 <> 5
whip[6]: r7c2{n5 n4} - b4n4{r6c2 r6c3} - r1n4{c3 c4} - b2n7{r1c4 r2c4} - c4n5{r2 r9} - r8n5{c5 .} ==> r7c7 <> 5
whip[5]: b9n7{r9c9 r8c8} - b9n5{r8c8 r8c7} - c5n5{r8 r1} - r2n5{c4 c9} - r4c9{n5 .} ==> r9c9 <> 9
hidden-single-in-a-block ==> r9c7 = 9
whip[6]: c6n3{r8 r5} - r5n9{c6 c8} - r4c9{n9 n5} - c8n5{r5 r1} - b9n5{r8c8 r8c7} - c5n5{r8 .} ==> r8c8 <> 3
whip[6]: c1n1{r2 r8} - c1n2{r8 r7} - b9n2{r7c7 r8c7} - r8n3{c7 c6} - c6n6{r8 r9} - b8n2{r9c6 .} ==> r2c1 <> 6
whip[2]: r8c1{n2 n1} - r2c1{n1 .} ==> r7c1 <> 2
whip[7]: c5n5{r1 r8} - r9c4{n5 n1} - r5n1{c4 c3} - c2n1{r6 r2} - r2n9{c2 c9} - r4c9{n9 n5} - b9n5{r9c9 .} ==> r1c5 <> 9
biv-chain[2]: c5n9{r3 r4} - r5n9{c6 c8} ==> r3c8 <> 9
biv-chain[3]: c5n9{r3 r4} - b6n9{r4c9 r5c8} - r1n9{c8 c2} ==> r3c2 <> 9
whip[4]: c5n5{r1 r8} - c7n5{r8 r4} - r4c9{n5 n9} - c8n9{r5 .} ==> r1c8 <> 5
whip[1]: b3n5{r2c9 .} ==> r2c4 <> 5
whip[4]: r8n7{c3 c8} - c8n5{r8 r5} - r4n5{c7 c1} - b4n7{r4c1 .} ==> r9c2 <> 7
whip[1]: b7n7{r9c3 .} ==> r2c3 <> 7
whip[4]: r8n7{c3 c8} - c8n5{r8 r5} - r5c1{n5 n6} - r5c3{n6 .} ==> r8c3 <> 1
whip[4]: c8n7{r3 r8} - c8n5{r8 r5} - r4n5{c7 c1} - b4n7{r4c1 .} ==> r3c2 <> 7
whip[1]: r3n7{c9 .} ==> r2c9 <> 7
whip[4]: c5n5{r1 r8} - c8n5{r8 r5} - c7n5{r4 r2} - b3n8{r2c7 .} ==> r1c5 <> 8
whip[6]: r9c4{n5 n1} - r5n1{c4 c3} - c2n1{r6 r2} - b1n9{r2c2 r1c2} - c8n9{r1 r5} - r4c9{n9 .} ==> r9c9 <> 5
whip[1]: b9n5{r8c7 .} ==> r8c5 <> 5
hidden-single-in-a-column ==> r1c5 = 5
biv-chain[2]: b2n9{r2c6 r3c5} - b2n6{r3c5 r2c6} ==> r2c6 <> 8
whip[2]: c1n8{r4 r3} - c6n8{r3 .} ==> r4c5 <> 8
whip[3]: r9c6{n2 n6} - r8c5{n6 n1} - r8c1{n1 .} ==> r8c6 <> 2
whip[4]: b9n2{r7c7 r8c7} - r8c1{n2 n1} - c5n1{r8 r6} - c5n8{r6 .} ==> r7c5 <> 2
whip[1]: r7n2{c8 .} ==> r8c7 <> 2
whip[4]: r2n5{c7 c9} - r4c9{n5 n9} - r5n9{c8 c6} - r2c6{n9 .} ==> r2c7 <> 6
whip[4]: r2c6{n6 n9} - r2c9{n9 n5} - r4c9{n5 n9} - r5n9{c8 .} ==> r2c2 <> 6, r2c3 <> 6
biv-chain[5]: r6c9{n3 n6} - r2n6{c9 c6} - r3c5{n6 n9} - r4c5{n9 n2} - r6n2{c5 c8} ==> r6c8 <> 3
whip[5]: r1n9{c8 c2} - b1n3{r1c2 r3c2} - b1n6{r3c2 r3c1} - r3n8{c1 c6} - r3n4{c6 .} ==> r1c8 <> 6
biv-chain[6]: c9n1{r7 r9} - b9n7{r9c9 r8c8} - c8n5{r8 r5} - c1n5{r5 r4} - r4n8{c1 c6} - c5n8{r6 r7} ==> r7c5 <> 1
biv-chain[3]: r5n1{c3 c4} - c5n1{r6 r8} - c1n1{r8 r2} ==> r2c3 <> 1
whip[4]: r8n7{c3 c8} - r8n5{c8 c7} - r2c7{n5 n8} - r2c3{n8 .} ==> r8c3 <> 2
whip[2]: r8n1{c5 c1} - r8n2{c1 .} ==> r8c5 <> 6
whip[2]: r2n6{c9 c6} - b8n6{r9c6 .} ==> r7c9 <> 6
whip[3]: r9n7{c9 c3} - b7n1{r9c3 r8c1} - b7n2{r8c1 .} ==> r9c9 <> 1
hidden-single-in-a-block ==> r7c9 = 1
biv-chain[5]: b3n8{r1c7 r2c7} - r2n5{c7 c9} - r2n6{c9 c6} - c5n6{r3 r7} - r7n8{c5 c4} ==> r1c4 <> 8
whip[3]: r4c2{n7 n4} - c3n4{r6 r1} - r1c4{n4 .} ==> r1c2 <> 7
whip[4]: b6n4{r5c7 r4c7} - r4c2{n4 n7} - r2n7{c2 c4} - r1c4{n7 .} ==> r5c4 <> 4
whip[2]: c3n4{r6 r1} - c4n4{r1 .} ==> r6c2 <> 4
whip[2]: r5c3{n6 n1} - r6c2{n1 .} ==> r5c1 <> 6
singles ==> r5c1 = 5, r8c8 = 5, r9c9 = 7, r3c8 = 7, r8c3 = 7
whip[2]: r2n6{c9 c6} - r8n6{c6 .} ==> r1c7 <> 6
whip[1]: b3n6{r3c9 .} ==> r6c9 <> 6
singles ==> r6c9 = 3, r3c2 = 3
whip[1]: r1n6{c1 .} ==> r3c1 <> 6
biv-chain[3]: b4n6{r5c3 r6c2} - r1c2{n6 n9} - c8n9{r1 r5} ==> r5c8 <> 6
singles ==> r5c8 = 9, r4c9 = 5, r1c8 = 3, r1c7 = 8, r2c7 = 5, r1c2 = 9
whip[2]: c2n6{r9 r6} - c8n6{r6 .} ==> r7c1 <> 6
singles to the end
GRID SOLVED. rating-type = gW, MOST COMPLEX RULE = Whip[12]
694751832
172839546
835264179
743698215
526173498
918425763
459387621
287916354
361542987
denis_berthier
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Re: g-whips and g-braids

Postby Mauricio » Thu Sep 20, 2012 10:16 pm

Complementing the previous post, I decided to make my gBraid solver output the solution in humand readable form, the gBraid rating of the puzzle is 11:

Hidden Text: Show
Code: Select all
000001002000030040005200100003600010020070008900005700009007000080900004300040080
Minirow and minicolumn assignments
 61) r3c6<>6,  gBraid[6]  n4b2{r3c6 r1c4} - n4r6{c4 c123} - n4{r5 r6}c3 - n8b4{r6c3 r4c1} - n8r3{c1 c5} - n8{r2 .}c6
 62) r3c6<>9,  gBraid[6]  n4b2{r3c6 r1c4} - n4r6{c4 c123} - n4{r5 r6}c3 - n8b4{r6c3 r4c1} - n8r3{c1 c5} - n8{r2 .}c6
 63) r6c3<>1,  gBraid[6]  n8b4{r6c3 r4c1} - n8r3{c1 c456} - n8{r2 r3}c6 - n4b2{r3c6 r1c4} - n4r6{c4 c2} - n4{r5 .}c3
 64) r6c3<>6,  gBraid[6]  n8b4{r6c3 r4c1} - n8r3{c1 c456} - n8{r2 r3}c6 - n4b2{r3c6 r1c4} - n4r6{c4 c2} - n4{r5 .}c3
 65) r9c9<>5,  gBraid[8]  {n5 n9}r4c9 - {n5 n1}r9c4 - n9r5{c8 c6} - n1{r8 r6}c5 - n1r8{c5 c123} - n1{r7 r2}c2 - n9r2{c2 c7} - n9r9{c7 .}
 66) r1c5<>8, gBraid[10]  {n8 n4}r3c6 - n8r7{c5 c4} - n3b8{r7c4 r8c6} - {n3 n9}r5c6 - {n9 n2}r4c5 - {n2 n1}r6c5 - n1r8{c5 c123} - n1{r9 r2}c2 - n9r2{c2 c789} - n9{r3 .}c8
 67) r1c7<>9, gBraid[10]  n8{r1 r2}c7 - n9{r3 r5}c8 - {n9 n5}r4c9 - n5r2{c9 c4} - n7{r2 r1}c4 - n4b2{r1c4 r3c6} - {n4 n3}r5c6 - n3{r6 r7}c4 - n8r7{c4 c5} - n8b2{r3c5 .}
 68) r3c5<>8, gBraid[10]  {n8 n4}r3c6 - n8r7{c5 c4} - n3b8{r7c4 r8c6} - {n3 n9}r5c6 - {n9 n2}r4c5 - {n2 n1}r6c5 - n1r8{c5 c123} - n1{r9 r2}c2 - n9r2{c2 c789} - n9{r3 .}c8
 69) r1c1<>8,  gBraid[5]  n8r3{c1 c6} - n8{r2 r6}c3 - n4b2{r3c6 r1c4} - n4r6{c4 c2} - n4{r5 .}c3
 70) r1c7<>5,  gBraid[5]  n5r2{c9 c4} - n7{r2 r1}c4 - n8r1{c4 c3} - n8r3{c1 c6} - n4b2{r3c6 .}
 71) r2c1<>8,  gBraid[5]  n8r3{c1 c6} - n8{r1 r6}c3 - n4b2{r3c6 r1c4} - n4r6{c4 c2} - n4{r5 .}c3
 72) r3c1<>7,  gBraid[6]  n8r3{c1 c6} - n4r3{c6 c2} - {n4 n6}r1c1 - {n6 n8}r1c3 - {n8 n3}r1c7 - n3r3{c9 .}
 73) r1c8<>7,  gBraid[9]  n7r3{c9 c2} - n3{r3 r1}c2 - n9r1{c2 c5} - n5r1{c5 c4} - {n5 n1}r9c4 - n1{r8 r6}c5 - n1{r9 r7}c9 - n1{r7 r2}c2 - n9{r2 .}c2
 74) r7c5<>5,  gBraid[9]  {n5 n1}r9c4 - n1{r8 r6}c5 - n8{r6 r4}c5 - n2b5{r4c5 r4c6} - n9b5{r4c6 r5c6} - n9{r5 r123}c8 - n9r2{c9 c2} - n1{r2 r7}c2 - n1r8{c3 .}
 75) r8c8<>3,  gBraid[9]  n3{r8 r5}c6 - n3{r5 r1}c7 - n9r5{c6 c789} - n8{r1 r2}c7 - {n9 n5}r4c9 - n5r2{c9 c4} - n7{r2 r1}c4 - n4b2{r1c4 r3c6} - n8b2{r3c6 .}
 76) r2c7<>9, gBraid[10]  n9{r3 r5}c8 - {n9 n5}r4c9 - n5r2{c9 c4} - n7{r2 r1}c4 - n4b2{r1c4 r3c6} - {n4 n3}r5c6 - n8r3{c6 c1} - n8{r2 r6}c3 - n8{r6 r7}c4 - n3{r7 .}c4
 77) r5c1<>4,  gBraid[9]  n4r6{c3 c4} - n5r5{c1 c789} - {n5 n9}r4c9 - n9r5{c8 c6} - n9r2{c6 c2} - n3b5{r5c6 r5c4} - n1b5{r5c4 r6c5} - n1r8{c5 c123} - n1{r9 .}c2
 78) r5c3<>4, gBraid[10]  {n4 n8}r6c3 - n4{r5 r4}c7 - n2b6{r4c7 r6c8} - {n2 n1}r6c5 - {n1 n3}r5c4 - {n3 n9}r5c6 - n9r4{c5 c9} - n9r2{c9 c2} - n1{r2 r789}c2 - n1r8{c3 .}
 79) r1c1<>4,  gBraid[4]  n4r3{c2 c6} - n8r3{c6 c1} - n8{r2 r6}c3 - n4{r6 .}c3
 80) r1c2<>4,  gBraid[4]  n4r3{c1 c6} - n8r3{c6 c1} - n8{r2 r6}c3 - n4{r6 .}c3
 81) r7c1<>5,  gBraid[4]  n5{r9 r4}c2 - n7r4{c2 c1} - n4{r4 r3}c1 - n8{r3 .}c1
 82) r7c1<>1,  gBraid[7]  n4r7{c1 c2} - n1r8{c3 c5} - {n1 n5}r9c4 - n5{r9 r4}c2 - {n5 n6}r5c1 - {n6 n7}r1c1 - n7r4{c1 .}
 83) r1c3<>7,  gBraid[6]  {n7 n6}r1c1 - n4{r1 r6}c3 - n8{r6 r2}c3 - {n8 n4}r3c1 - {n4 n2}r7c1 - n2r2{c1 .}
 84) r5c7<>5,  gBraid[8]  {n5 n9}r4c9 - n9r5{c8 c6} - n9r2{c6 c2} - n4r5{c6 c4} - n3b5{r5c4 r6c4} - n1b5{r6c4 r6c5} - n1r8{c5 c123} - n1{r9 .}c2
 85) r8c1<>7,  gBraid[8]  n7{r9 r2}c3 - n2r2{c3 c1} - n1r2{c1 c2} - n1{r2 r5}c1 - n5r5{c1 c8} - {n5 n9}r4c9 - n9r2{c9 c6} - n9r5{c6 .}
 86) r8c8<>2,  gBraid[9]  n2r6{c8 c5} - n2r7{c5 c1} - n1{r6 r789}c5 - n4r7{c1 c2} - {n1 n5}r9c4 - n5{r9 r4}c2 - n7r4{c2 c1} - n4{r4 r3}c1 - n8{r3 .}c1
 87) r7c2<>1, gBraid[10]  n1r8{c3 c5} - {n1 n5}r9c4 - n5{r9 r4}c2 - n7r4{c2 c1} - n8{r4 r3}c1 - {n8 n4}r3c6 - n4r4{c6 c7} - n2b6{r4c7 r6c8} - {n2 n8}r6c5 - n8r4{c6 .}
 88) r7c9<>5,  gBraid[9]  {n5 n9}r4c9 - n1{r7 r9}c9 - n9r5{c8 c6} - {n1 n5}r9c4 - n9r2{c6 c2} - n5r8{c5 c1} - n1{r2 r6}c2 - {n1 n6}r5c1 - {n6 .}r5c3
 89) r9c7<>5,  gBraid[9]  {n5 n1}r9c4 - n9r9{c7 c9} - n1{r8 r6}c5 - {n9 n5}r4c9 - n1{r6 r2}c2 - n5r2{c9 c4} - n9r2{c2 c6} - {n9 n6}r1c5 - {n6 .}r3c5
 90) r1c4<>8, gBraid[11]  n8r7{c4 c5} - n7{r1 r2}c4 - n8{r1 r2}c7 - n5r2{c7 c9} - {n5 n9}r4c9 - {n9 n2}r4c5 - {n2 n1}r6c5 - n1r8{c5 c123} - n1{r9 r2}c2 - n9r2{c2 c6} - n9r5{c6 .}
 91) r1c2<>7,  gBraid[9]  n7r4{c2 c1} - n8{r4 r3}c1 - {n8 n4}r3c6 - {n4 n5}r1c4 - {n5 n1}r9c4 - n1{r8 r6}c5 - n1{r6 r2}c2 - n9{r2 r3}c2 - n3{r3 .}c2
 92) r4c2<>5,  gBraid[8]  {n5 n9}r4c9 - n7r4{c2 c1} - n5{r4 r2}c9 - n7r1{c1 c4} - {n7 n8}r2c4 - {n8 n6}r2c7 - {n6 n9}r2c6 - n9r5{c6 .}
 93) r789c2==5
 94) r456c1==5
 95) r9c2<>7,  gBraid[6]  n7r4{c2 c1} - n7r8{c3 c8} - n5r9{c2 c4} - n5r8{c5 c7} - n5r2{c7 c9} - n5r4{c9 .}
 96) r789c3==7
 97) r8c7<>5,  gBraid[9]  n5{r8 r1}c5 - n5r2{c4 c9} - {n5 n9}r4c9 - n5r4{c9 c1} - n8{r4 r3}c1 - n8r1{c3 c7} - {n8 n6}r2c7 - n6b2{r2c6 r3c5} - n9{r3 .}c5
 98) r9c9<>1,  gBraid[3]  {n1 n5}r9c4 - n5r8{c5 c8} - n7b9{r8c8 .}
 99) r7c9==1,  n1{. r7}c9
100) r7c789==1
101) r3c2<>7,  gBraid[7]  {n7 n4}r4c2 - n7{r3 r8}c8 - n5r8{c8 c5} - n5r9{c4 c2} - {n5 n6}r7c2 - {n6 n1}r6c2 - n1{r6 .}c5
102) r3c789==7
103) r5c1<>1,  gBraid[7]  n5r5{c1 c8} - {n5 n9}r4c9 - n9r5{c7 c6} - n9r2{c6 c2} - n1r2{c2 c3} - n1r8{c3 c5} - n5r8{c5 .}
104) r8c3<>1,  gBraid[2]  n1r5{c3 c4} - n1r9{c4 .}
105) r2c3<>1,  gBraid[3]  n1r5{c3 c4} - n1r9{c4 c2} - n1r6{c2 .}
106) r5c8<>5,  gBraid[6]  {n5 n9}r4c9 - n9r5{c7 c6} - n9r2{c6 c2} - n1r2{c2 c1} - n1r8{c1 c5} - n5r8{c5 .}
107) r5c1==5,  n5r5{. c1}
108) r4c789==5
109) r5c123==5
110) r6c2<>4,  gBraid[2]  n1b4{r6c2 r5c3} - n6b4{r5c3 .}
111) r5c4<>4,  gBraid[2]  n4r1{c4 c3} - n4r6{c3 .}
112) r1c2<>6,  gBraid[4]  {n6 n1}r6c2 - {n1 n5}r9c2 - {n5 n1}r9c4 - n1r5{c4 .}
113) r2c2<>6,  gBraid[4]  {n6 n1}r6c2 - {n1 n5}r9c2 - {n5 n1}r9c4 - n1r5{c4 .}
114) r3c2<>6,  gBraid[4]  {n6 n1}r6c2 - {n1 n5}r9c2 - {n5 n1}r9c4 - n1r5{c4 .}
115) r7c2<>6,  gBraid[4]  {n6 n1}r6c2 - n1r5{c3 c4} - {n1 n5}r9c4 - {n5 .}r9c2
116) r8c1<>6,  gBraid[5]  n1{r8 r2}c1 - n2r2{c1 c3} - n6b1{r2c3 r1c3} - n4{r1 r6}c3 - n8{r6 .}c3
117) r8c6<>2,  gBraid[4]  {n2 n1}r8c1 - n1r9{c3 c4} - {n1 n3}r5c4 - n3{r5 .}c6
118) r3c8<>9,  gBraid[7]  {n9 n6}r3c5 - n7r3{c8 c9} - n3r3{c9 c2} - {n3 n9}r1c2 - {n9 n5}r1c5 - n5r8{c5 c8} - n7{r8 .}c8
119) r1c5<>9,  gBraid[7]  n5{r1 r8}c5 - {n5 n1}r9c4 - {n1 n3}r5c4 - {n3 n8}r7c4 - {n8 n4}r6c4 - {n4 n9}r5c6 - n9{r5 .}c8
120) r2c4<>5,  gBraid[3]  {n5 n6}r1c5 - {n6 n7}r1c1 - n7r2{c2 .}
121) r2c789==5
122) r1c456==5
123) r789c8==5
124) r1c7<>3,  gBraid[5]  {n3 n9}r1c2 - n8{r1 r2}c7 - n5r2{c7 c9} - {n5 n9}r4c9 - n9{r5 .}c8
125) r1c8<>6,  gBraid[2]  n3r1{c8 c2} - n9r1{c2 .}
126) r6c8<>3,  gBraid[3]  {n3 n9}r1c8 - {n9 n6}r5c8 - {n6 .}r6c9
127) r5c7<>3,  gBraid[4]  {n3 n6}r6c9 - {n6 n9}r5c8 - {n9 n3}r1c8 - n3{r3 .}c9
128) r789c7==3
129) r7c7<>2,  gBraid[5]  n3r7{c7 c4} - {n3 n1}r5c4 - n8r7{c4 c5} - {n8 n2}r6c5 - n2r4{c6 .}
130) r2c7<>6,  gBraid[5]  {n6 n3}r7c7 - n3r8{c7 c6} - n6{r8 r9}c6 - n6{r9 r6}c2 - n6{r6 .}c9
131) r8c5<>2,  gBraid[5]  n2r6{c5 c8} - n5r8{c5 c8} - {n5 n6}r7c8 - {n6 n3}r7c7 - {n3 .}r8c7
132) r1c3<>6,  gBraid[6]  {n6 n5}r1c5 - n6{r3 r7}c1 - n5r8{c5 c8} - {n5 n2}r7c8 - {n2 n6}r6c8 - n6r5{c7 .}
133) r2c3<>8,  gBraid[2]  {n8 n4}r1c3 - {n4 .}r6c3
134) r4c7<>4,  gBraid[4]  n4r5{c7 c6} - {n4 n8}r3c6 - n8r2{c4 c7} - n5{r2 .}c7
135) r5c7==4,  n4{. r5}c7
136) r5c789==4
137) r3c2<>9,  gBraid[3]  n9r1{c2 c8} - n9r5{c8 c6} - n9r2{c6 .}
138) r3c8<>6,  gBraid[3]  n7r3{c8 c9} - n3{r3 r6}c9 - n6b6{r6c9 .}
139) r8c8<>6,  gBraid[3]  n5r8{c8 c5} - {n5 n6}r1c5 - n6{r1 .}c7
140) r2c2<>1,  gBraid[4]  {n1 n6}r6c2 - n6r5{c3 c8} - n9{r5 r1}c8 - n9{r1 .}c2
singles
Most difficult rule: gBraid[11]
Mauricio
 
Posts: 1174
Joined: 22 March 2006

Re: g-whips and g-braids

Postby denis_berthier » Fri Sep 21, 2012 5:25 am

Mauricio wrote:I have updated the solver, now it can also solve using gBraids, I propose as a test case the following puzzle, gB=6 and not even a single vanilla Braid can be applied.
Code: Select all
001002003000030040200500100004006001070000080100900700002007006050080000300200900


OK, but g-braids are not necessary, g-whips are enough.
g-bivalue chains appearing in the following path are g-whips without distant z- or t-candidates.

Hidden Text: Show
***** SudoRules 16.2 based on CSP-Rules 1.2, config: gW *****
001002003000030040200500100004006001070000080100900700002007006050080000300200900
24 givens, 213 candidates, 1289 csp-links and 1289 links. Initial density = 1.43
g-biv-chain[4]: r5n2{c5 c789} - c8n2{r4 r8} - r8n1{c8 c456} - c5n1{r9 .} ==> r5c5 <> 4, r5c5 <> 5
g-biv-chain[4]: c8n2{r8 r456} - r5n2{c7 c5} - c5n1{r5 r789} - r8n1{c6 .} ==> r8c8 <> 3, r8c8 <> 7
g-whip[4]: c8n2{r4 r8} - r8n1{c8 c456} - c5n1{r7 r5} - r5n2{c5 .} ==> r4c7 <> 2
g-whip[4]: c8n2{r6 r8} - r8n1{c8 c456} - c5n1{r7 r5} - r5n2{c5 .} ==> r6c9 <> 2
g-whip[4]: c5n1{r7 r5} - r5n2{c5 c789} - c8n2{r4 r8} - r8n1{c8 .} ==> r7c4 <> 1
g-whip[4]: c5n1{r9 r5} - r5n2{c5 c789} - c8n2{r4 r8} - r8n1{c8 .} ==> r9c6 <> 1
whip[5]: r4c7{n5 n3} - b9n3{r8c7 r7c8} - r7n5{c8 c5} - r9c6{n5 n4} - r7c4{n4 .} ==> r5c7 <> 5, r1c7 <> 5, r2c7 <> 5
biv-chain[3]: r6c9{n4 n5} - c7n5{r4 r7} - b9n8{r7c7 r9c9} ==> r9c9 <> 4
whip[5]: r7c4{n4 n3} - r8n3{c6 c7} - b9n4{r8c7 r8c9} - r6c9{n4 n5} - r4c7{n5 .} ==> r7c5 <> 4, r7c1 <> 4
whip[4]: r1n5{c8 c1} - c1n4{r1 r8} - b9n4{r8c9 r7c7} - c7n5{r7 .} ==> r4c8 <> 5, r6c8 <> 5
whip[4]: b7n1{r9c2 r7c2} - b7n4{r7c2 r8c1} - b9n4{r8c9 r7c7} - b9n8{r7c7 .} ==> r9c2 <> 8
whip[5]: c1n4{r1 r8} - b9n4{r8c9 r7c7} - c7n5{r7 r4} - r4c1{n5 n9} - r7c1{n9 .} ==> r1c1 <> 8
whip[5]: c1n4{r1 r8} - b9n4{r8c9 r7c7} - c7n5{r7 r4} - r4c1{n5 n8} - r7c1{n8 .} ==> r1c1 <> 9
whip[5]: r1n5{c8 c1} - c1n4{r1 r8} - c1n7{r8 r2} - c1n6{r2 r5} - r6n6{c3 .} ==> r1c8 <> 6
whip[5]: r7c4{n4 n3} - r8n3{c6 c7} - b9n4{r8c7 r8c9} - r6c9{n4 n5} - r4c7{n5 .} ==> r7c2 <> 4
whip[3]: b9n8{r9c9 r7c7} - r7n4{c7 c4} - r9c6{n4 .} ==> r9c9 <> 5
whip[3]: c7n8{r2 r7} - c7n5{r7 r4} - c9n5{r5 .} ==> r2c9 <> 8
whip[4]: c1n4{r8 r1} - r1n5{c1 c8} - b9n5{r7c8 r7c7} - b9n4{r7c7 .} ==> r8c4 <> 4, r8c6 <> 4
whip[4]: r7n4{c7 c4} - r7n3{c4 c8} - b9n5{r7c8 r9c8} - r9c6{n5 .} ==> r7c7 <> 8
hidden-single-in-a-block ==> r9c9 = 8
whip[4]: r4c7{n3 n5} - r7c7{n5 n4} - r7c4{n4 n3} - r8n3{c6 .} ==> r5c7 <> 3
g-biv-chain[4]: b9n4{r7c7 r8c789} - c1n4{r8 r1} - r1n5{c1 c8} - b9n5{r7c8 .} ==> r7c7 <> 3
biv-chain[2]: c7n3{r4 r8} - r7n3{c8 c4} ==> r4c4 <> 3
g-whip[4]: r1n5{c1 c8} - b9n5{r7c8 r7c7} - b9n4{r7c7 r8c789} - c1n4{r8 .} ==> r1c1 <> 6
g-whip[4]: r1n5{c1 c8} - b9n5{r7c8 r7c7} - b9n4{r7c7 r8c789} - c1n4{r8 .} ==> r1c1 <> 7
whip[3]: r1n7{c5 c8} - b9n7{r9c8 r8c9} - c1n7{r8 .} ==> r2c4 <> 7
whip[4]: b7n8{r7c1 r7c2} - b7n1{r7c2 r9c2} - b7n4{r9c2 r8c1} - c1n7{r8 .} ==> r2c1 <> 8
whip[5]: b1n4{r1c2 r3c2} - b1n3{r3c2 r3c3} - r3n8{c3 c6} - b5n8{r6c6 r4c4} - c4n7{r4 .} ==> r1c4 <> 4
whip[6]: r4c7{n3 n5} - r7c7{n5 n4} - r8n4{c9 c1} - c1n7{r8 r2} - c1n6{r2 r5} - r6n6{c3 .} ==> r6c8 <> 3
whip[1]: b6n3{r4c7 .} ==> r4c2 <> 3
whip[4]: b3n2{r2c9 r2c7} - b3n8{r2c7 r1c7} - c7n6{r1 r5} - r6c8{n6 .} ==> r5c9 <> 2
biv-chain[3]: b3n5{r1c8 r2c9} - c9n2{r2 r8} - b9n7{r8c9 r9c8} ==> r9c8 <> 5
whip[1]: b9n5{r7c7 .} ==> r7c5 <> 5
biv-chain[3]: b3n5{r1c8 r2c9} - c9n2{r2 r8} - b9n7{r8c9 r9c8} ==> r1c8 <> 7
whip[1]: r1n7{c4 .} ==> r3c5 <> 7
biv-chain[3]: r9c3{n6 n7} - c8n7{r9 r3} - c8n6{r3 r6} ==> r6c3 <> 6
biv-chain[3]: b9n3{r7c8 r8c7} - r4c7{n3 n5} - b9n5{r7c7 r7c8} ==> r7c8 <> 1
whip[4]: c2n3{r3 r6} - r6n6{c2 c8} - r3c8{n6 n7} - r3c9{n7 .} ==> r3c2 <> 9
whip[5]: r9c3{n6 n7} - b9n7{r9c8 r8c9} - r2n7{c9 c1} - r2n5{c1 c9} - c9n2{r2 .} ==> r2c3 <> 6
whip[5]: r1c8{n9 n5} - r1c1{n5 n4} - b7n4{r8c1 r9c2} - b7n1{r9c2 r7c2} - r7c5{n1 .} ==> r1c5 <> 9
whip[4]: r7c1{n9 n8} - r4c1{n8 n5} - r1n5{c1 c8} - r1n9{c8 .} ==> r2c1 <> 9
whip[4]: b3n8{r1c7 r2c7} - b3n2{r2c7 r2c9} - b3n5{r2c9 r1c8} - r1n9{c8 .} ==> r1c2 <> 8
whip[4]: r1n9{c2 c8} - r3c9{n9 n7} - r3c8{n7 n6} - r6n6{c8 .} ==> r1c2 <> 6
biv-chain[3]: r1c2{n4 n9} - r1c8{n9 n5} - r1c1{n5 n4} ==> r3c2 <> 4
whip[1]: r3n4{c6 .} ==> r1c5 <> 4
whip[4]: b2n9{r3c6 r2c6} - r8n9{c6 c1} - b7n4{r8c1 r9c2} - r1c2{n4 .} ==> r3c3 <> 9
biv-chain[5]: r9c3{n6 n7} - b9n7{r9c8 r8c9} - r3c9{n7 n9} - r1n9{c8 c2} - c2n4{r1 r9} ==> r9c2 <> 6
whip[2]: c8n6{r3 r6} - c2n6{r6 .} ==> r3c3 <> 6
whip[4]: b2n4{r3c6 r3c5} - c5n9{r3 r7} - r7n1{c5 c2} - r9c2{n1 .} ==> r9c6 <> 4
naked-single ==> r9c6 = 5
whip[4]: r6c9{n4 n5} - b5n5{r6c5 r4c5} - b5n7{r4c5 r4c4} - b5n8{r4c4 .} ==> r6c6 <> 4
whip[3]: c3n8{r2 r6} - r6c6{n8 n3} - c2n3{r6 .} ==> r3c2 <> 8
biv-chain[3]: b1n3{r3c2 r3c3} - r3n8{c3 c6} - r6c6{n8 n3} ==> r6c2 <> 3
hidden-single-in-a-column ==> r3c2 = 3
whip[1]: b1n6{r2c1 .} ==> r2c7 <> 6, r2c4 <> 6
biv-chain[3]: r9n6{c3 c5} - c4n6{r8 r1} - c7n6{r1 r5} ==> r5c3 <> 6
whip[1]: c3n6{r9 .} ==> r8c1 <> 6
biv-chain[3]: r9n6{c3 c5} - r3n6{c5 c8} - c8n7{r3 r9} ==> r9c3 <> 7
singles ==> r9c3 = 6, r8c4 = 6, r9c8 = 7, r8c8 = 1
whip[1]: b8n1{r9c5 .} ==> r5c5 <> 1
naked-single ==> r5c5 = 2
biv-chain[2]: c4n4{r5 r7} - c4n3{r7 r5} ==> r5c4 <> 1
singles ==> r5c6 = 1, r2c4 = 1, r3c6 = 4, r3c3 = 8, r3c9 = 7
biv-chain[2]: r6n6{c2 c8} - r6n2{c8 c2} ==> r6c2 <> 8
singles to the end

491872653
567139842
238564197
984756321
675321489
123948765
812497536
759683214
346215978


As your previous example, this shows that obstructions to the extension of a partial whip [which are the main resolution pattern in my approach] can be overcome by two main techniques, involving two different kinds of mild branching: braids and g-whips [where branching is localised in a single step]. From all the examples I've studied, the g-whip technique is much more powerful than the braid one (although none of the two subsumes the other).
denis_berthier
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Re: g-whips and g-braids

Postby Mauricio » Thu Oct 04, 2012 2:24 am

denis_berthier wrote:From all the examples I've studied, the g-whip technique is much more powerful than the braid one (although none of the two subsumes the other).
OK. Since I don't have a gWhip solver, may I propose a few examples for you to examine? The first ones are puzzles not solvable by whips that have low B-rating (7,6,6,8, respectively).

Code: Select all
000000001020003040004050300000030100006700080070002005007800900040010020800006000
000001002000020030004500600006050700070600008300008060005200800010005040700080009
000001002010030040005400600003500700040000020100008009002900100030060007600007090
000000001000002030004050600000700080001008002060090700002007009030900040500060800
For the third one, not even a single whip can be found.

This one has B-rating=29 and gB-rating=28, which surprises me, since I would think that the gB-rating would be a lot lower for difficult puzzles that the B-rating.
Code: Select all
001002003000010040500300100006007002010000080700900300007006008090040000300700500

gB solution: Show
Code: Select all
1)r2c7<>9, Braid[12]  {n9 n4}r4c7 - {n4 n2}r7c7 - n4{r6 r9}c9 - n9{r9 r5}c9 - n9r4{c8 c1} - n9r1{c1 c5} - n9{r3 r9}c6 - n1r9{c6 c8} - {n1 n5}r4c8 - n5r1{c8 c4} - {n5 n1}r7c4 - n1r4{c4 .}
2)r5c7<>9, gBraid[12]  {n9 n4}r4c7 - n9r4{c8 c1} - {n4 n2}r7c7 - n4{r6 r9}c9 - n9{r9 r123}c9 - n9r1{c8 c5} - n9{r3 r9}c6 - n1r9{c6 c8} - {n1 n5}r4c8 - n5r1{c8 c4} - {n5 n1}r7c4 - n1r4{c4 .}
3)r2c3<>2, gBraid[15]  n3r2{c3 c2} - n3{r2 r5}c3 - n7r2{c2 c789} - n9{r5 r3}c3 - n7{r3 r8}c8 - {n9 n6}r3c9 - {n6 n1}r8c9 - n1r9{c8 c6} - n1r6{c6 c8} - n6r6{c8 c5} - n6{r6 r9}c8 - n2r6{c5 c2} - {n6 n2}r8c7 - n2{r8 r7}c1 - n1r7{c1 .}
4)r4c2<>4, Braid[16]  {n4 n9}r4c7 - n3r4{c2 c5} - n3b4{r4c2 r5c3} - n3r7{c5 c8} - n9r5{c3 c1} - n9r7{c8 c5} - n9r1{c5 c8} - n9r9{c8 c9} - n4r9{c9 c3} - n4r3{c3 c6} - {n4 n5}r5c6 - n9r3{c6 c3} - {n9 n8}r2c3 - n8r3{c2 c5} - n7{r3 r1}c5 - n5{r1 .}c5
5)r2c3<>8, gBraid[16]  n3r2{c3 c2} - n3{r2 r5}c3 - n8{r2 r1}c7 - n7r2{c2 c789} - n9{r5 r3}c3 - n7{r3 r8}c8 - {n9 n6}r3c9 - {n6 n1}r8c9 - n1r9{c8 c6} - n1r6{c6 c8} - n9{r9 r2}c6 - n9r1{c5 c8} - {n9 n5}r4c8 - {n5 n8}r4c2 - n8r9{c2 c5} - n8{r8 .}c4
6)r7c8<>1, gBraid[28]  n1r4{c8 c4} - n3r7{c8 c5} - n1r9{c9 c6} - n1{r7 r8}c1 - n3{r7 r8}c8 - n9r7{c5 c7} - n3{r8 r5}c6 - n6b7{r8c1 r9c2} - n7{r8 r123}c8 - {n6 n2}r9c8 - n7r2{c9 c2} - n2{r8 r2}c7 - n8{r2 r1}c7 - {n8 n4}r1c2 - n4r3{c3 c6} - n4r7{c2 c1} - n4{r1 r5}c4 - n9{r3 r2}c6 - {n4 n8}r9c3 - n2{r7 r5}c1 - n2r6{c3 c5} - n6b5{r6c5 r5c5} - n6r3{c5 c789} - {n6 n5}r2c9 - n5r5{c9 c3} - {n5 n8}r6c2 - n8r3{c2 c5} - n8r4{c5 .}
7)r7c1<>4, Whip[4]  n1r7{c1 c4} - n1r4{c4 c8} - n1r9{c8 c9} - n4r9{c9 .}
8)r5c3<>4, Whip[10]  n4{r4 r1}c1 - n4{r1 r4}c4 - n1r4{c4 c8} - n1b5{r4c4 r6c6} - n1r9{c6 c9} - n4r9{c9 c2} - n6r9{c2 c8} - {n6 n5}r6c8 - n5b4{r6c3 r4c2} - n3b4{r4c2 .}
9)r1c7<>9, Braid[14]  {n9 n4}r4c7 - n8{r1 r2}c7 - {n4 n2}r7c7 - n4{r6 r9}c9 - {n2 n1}r7c1 - n9{r9 r5}c9 - {n1 n5}r7c4 - n9r4{c8 c1} - {n5 n6}r2c4 - {n6 n2}r2c1 - {n2 n4}r5c1 - {n4 n2}r5c4 - n2r8{c4 c3} - n5r8{c3 .}
10)r7c8<>2, gBraid[14]  {n2 n1}r7c1 - n2{r8 r2}c7 - n3{r7 r8}c8 - {n1 n5}r7c4 - n8{r2 r1}c7 - n7{r8 r123}c8 - {n5 n4}r7c2 - n7r2{c9 c2} - {n7 n6}r1c2 - {n6 n4}r1c4 - n6{r2 r8}c1 - n2{r8 r5}c1 - {n2 n6}r5c4 - n6{r5 .}c7
11)r8c6<>1, gBraid[10]  n1{r7 r4}c4 - n3{r8 r5}c6 - n3b8{r8c6 r7c5} - {n3 n9}r7c8 - n5b8{r7c5 r789c4} - {n9 n5}r4c8 - n5r1{c8 c5} - n9r1{c5 c1} - n5r5{c5 c3} - n9{r5 .}c3
12)r9c8<>1, Whip[2]  n1r4{c8 c4} - n1{r6 .}c6
13)r4c8<>9, gBraid[12]  {n9 n4}r4c7 - {n9 n3}r7c8 - n1r4{c8 c4} - n3r8{c8 c6} - n1{r6 r9}c6 - n9{r9 r123}c6 - n9r1{c5 c1} - {n9 n3}r2c3 - n4{r1 r5}c1 - {n4 n5}r5c6 - n5r4{c5 c2} - n3{r4 .}c2
14)r6c6<>8, Whip[7]  n1b5{r6c6 r4c4} - {n1 n5}r4c8 - {n5 n3}r4c5 - n3r5{c6 c3} - {n3 n9}r2c3 - {n9 n5}r2c6 - n5r1{c5 .}
15)r2c9<>6, Braid[8]  n5b3{r2c9 r1c8} - {n5 n1}r4c8 - {n1 n6}r6c8 - n6r9{c8 c2} - n6r3{c2 c5} - n6{r8 r1}c1 - n9r1{c1 c5} - n7{r1 .}c5
16)r6c6<>5, gBraid[10]  n1{r6 r9}c6 - n1b5{r6c6 r4c4} - {n1 n5}r4c8 - n5r5{c9 c3} - n5r8{c3 c4} - n3b4{r5c3 r4c2} - {n3 n8}r4c5 - n8b8{r9c5 r8c6} - n8r3{c6 c123} - n8{r2 .}c1
17)r7c5<>5, Braid[9]  n3r7{c5 c8} - n9r7{c8 c7} - n4b9{r7c7 r9c9} - n1r9{c9 c6} - {n1 n4}r6c6 - n1{r8 r4}c4 - {n1 n5}r4c8 - n5r1{c8 c4} - n4{r1 .}c4
18)r4c4<>5, gWhip[8]  n5{r6 r1}c5 - n5r2{c6 c9} - n5r5{c9 c3} - n9{r5 r123}c3 - n9r1{c1 c8} - {n9 n3}r7c8 - n3r8{c8 c6} - n5r8{c6 .}
19)r2c2<>6, gBraid[9]  n6{r1 r8}c1 - n7r2{c2 c789} - n7{r3 r8}c8 - {n7 n1}r8c9 - n3r8{c8 c6} - n1r9{c9 c6} - n8{r9 r123}c6 - {n8 n5}r2c4 - n5b8{r8c4 .}
20)r4c2<>8, Braid[9]  n3r4{c2 c5} - n8r6{c3 c5} - n3{r4 r2}c2 - n5r4{c5 c8} - {n3 n9}r2c3 - n5{r6 r2}c9 - {n5 n8}r2c6 - n8r3{c6 c3} - n8r9{c3 .}
21)r8c6<>8, Braid[9]  n3{r8 r5}c6 - n5{r5 r2}c6 - n5r1{c5 c8} - {n5 n1}r4c8 - {n1 n6}r6c8 - n1r6{c9 c6} - {n1 n9}r9c6 - {n9 n2}r9c5 - {n2 .}r9c8
22)r8c8<>1, gBraid[9]  {n1 n5}r4c8 - n3r8{c8 c6} - {n5 n3}r4c2 - {n5 n6}r6c8 - n5b8{r8c6 r789c4} - n5r1{c4 c5} - n7{r1 r3}c5 - n6{r3 r5}c5 - n3r5{c5 .}
23)r456c8==1
24)r9c9<>6, Whip[5]  n1r9{c9 c6} - {n1 n4}r6c6 - {n4 n5}r6c9 - {n5 n1}r4c8 - n1r6{c8 .}
25)r1c8<>6, Braid[9]  n6r9{c8 c2} - n6r3{c2 c5} - n6r6{c5 c9} - n7{r3 r1}c5 - {n7 n8}r1c7 - {n8 n4}r1c2 - n4r3{c3 c6} - n4r6{c6 c3} - n4{r5 .}c1
26)r5c4<>5, Braid[9]  n5r7{c4 c2} - n5r4{c2 c8} - n1r4{c8 c4} - {n1 n2}r7c4 - n1{r6 r9}c6 - {n2 n8}r8c4 - {n8 n9}r9c5 - {n9 n4}r9c9 - n4r7{c7 .}
27)r9c9<>9, Whip[9]  n1r9{c9 c6} - {n1 n4}r6c6 - n1r6{c6 c8} - {n1 n5}r4c8 - {n5 n6}r6c9 - {n6 n7}r3c9 - n7{r3 r1}c5 - n5r1{c5 c4} - n4{r1 .}c4
28)r3c8<>9, Braid[6]  n9{r2 r5}c9 - {n9 n4}r4c7 - n9{r5 r2}c3 - n9{r2 r9}c6 - n1r9{c6 c9} - n4{r9 .}c9
29)r1c7<>7, gWhip[8]  n7r5{c7 c9} - n9{r5 r123}c9 - {n9 n5}r1c8 - {n5 n1}r4c8 - n1r6{c8 c6} - n1r9{c6 c9} - n4{r9 r6}c9 - n5{r6 .}c9
30)r1c5<>8, Braid[7]  {n8 n6}r1c7 - n7{r1 r3}c5 - n6r3{c5 c2} - n8r3{c2 c3} - n8r6{c3 c2} - n8{r4 r8}c1 - n6{r8 .}c1
31)r2c7<>7, gWhip[8]  n7r5{c7 c9} - n9{r5 r123}c9 - {n9 n5}r1c8 - {n5 n1}r4c8 - n1r6{c8 c6} - n1r9{c6 c9} - n4{r9 r6}c9 - n5{r6 .}c9
32)r1c8<>9, gWhip[7]  {n9 n3}r7c8 - n3r8{c8 c6} - n5b8{r8c6 r789c4} - n5r1{c4 c5} - n7r1{c5 c2} - n7r2{c2 c9} - n5r2{c9 .}
33)r789c8==9
34) r4c7==9,  n9{. r4}c7
35)r7c5<>2, Whip[2]  n3r7{c5 c8} - n9r7{c8 .}
36)r2c2<>3, Whip[4]  {n3 n9}r2c3 - n9r1{c1 c5} - {n9 n3}r7c5 - n3r4{c5 .}
37) r2c3==3,  n3r2{. c3}
38) r4c2==3,  n3{. r4}c2
39)r6c5<>5, Braid[4]  n5b4{r6c3 r5c3} - n9{r5 r3}c3 - n5{r5 r2}c9 - n9{r2 .}c9
40)r6c8<>5, Whip[4]  n5{r5 r2}c9 - n9{r2 r3}c9 - n9{r3 r5}c3 - n5b4{r5c3 .}
41)r1c4<>5, Whip[4]  n5{r1 r4}c8 - n1r4{c8 c4} - {n1 n4}r6c6 - n4{r5 .}c4
42)r5c5<>5, Whip[2]  n5r1{c5 c8} - n5r4{c8 .}
43)r5c7<>6, Braid[4]  {n6 n1}r6c8 - {n1 n5}r4c8 - {n1 n4}r6c6 - {n4 .}r6c9
44)r5c9<>6, Braid[4]  {n6 n1}r6c8 - {n1 n5}r4c8 - {n1 n4}r6c6 - {n4 .}r6c9
45)r5c456==6
46)r1c5<>5, Whip[4]  n7{r1 r3}c5 - n6{r3 r5}c5 - n3r5{c5 c6} - n5b5{r5c6 .}
47) r1c8==5,  n5r1{. c8}
48) r4c8==1,  {. n1}r4c8
49) r6c8==6,  {. n6}r6c8
50) r4c5==5,  n5r4{. c5}
51) r6c6==1,  n1r6{. c6}
52) r9c9==1,  n1r9{. c9}
53) r9c2==6,  n6r9{. c2}
54) r9c3==4,  n4r9{. c3}
55) r7c7==4,  n4r7{. c7}
56) r5c7==7,  {. n7}r5c7
57)r9c456==8
58)r3c5<>7, Whip[3]  n6r3{c5 c9} - {n6 n7}r8c9 - n7{r8 .}c8
59) r1c5==7,  n7{. r1}c5
60) r1c1==9,  n9r1{. c1}
61) r5c3==9,  n9r5{. c3}
62) r5c9==5,  n5r5{. c9}
63) r6c9==4,  {. n4}r6c9
64) r2c1==6,  n6{. r2}c1
65)r1c2<>8, Whip[3]  {n8 n2}r3c3 - n2r2{c2 c7} - n8{r2 .}c7
66) r1c2==4,  {. n4}r1c2
67) r3c6==4,  n4r3{. c6}
68) r5c6==3,  {. n3}r5c6
69) r8c6==5,  {. n5}r8c6
70) r2c4==5,  n5r2{. c4}
71) r7c2==5,  n5r7{. c2}
72) r6c3==5,  n5r6{. c3}
73) r8c8==3,  n3r8{. c8}
74) r7c8==9,  {. n9}r7c8
75) r7c5==3,  {. n3}r7c5
76) r9c8==2,  {. n2}r9c8
77) r3c8==7,  {. n7}r3c8
78) r2c9==9,  {. n9}r2c9
79) r2c6==8,  {. n8}r2c6
80) r1c4==6,  {. n6}r1c4
81) r1c7==8,  {. n8}r1c7
82) r2c7==2,  {. n2}r2c7
83) r2c2==7,  {. n7}r2c2
84) r3c5==9,  {. n9}r3c5
85) r3c9==6,  {. n6}r3c9
86) r8c7==6,  {. n6}r8c7
87) r8c9==7,  {. n7}r8c9
88) r9c5==8,  {. n8}r9c5
89) r6c5==2,  {. n2}r6c5
90) r5c4==4,  {. n4}r5c4
91) r4c4==8,  {. n8}r4c4
92) r4c1==4,  {. n4}r4c1
93) r5c1==2,  {. n2}r5c1
94) r5c5==6,  {. n6}r5c5
95) r6c2==8,  {. n8}r6c2
96) r3c2==2,  {. n2}r3c2
97) r3c3==8,  {. n8}r3c3
98) r7c1==1,  {. n1}r7c1
99) r7c4==2,  {. n2}r7c4
100) r8c1==8,  {. n8}r8c1
101) r8c3==2,  {. n2}r8c3
102) r8c4==1,  {. n1}r8c4
103) r9c6==9,  {. n9}r9c6
Most difficult rule: gBraid[28]
Time elapsed: 2220.706 seconds
You can use my solver to obtain the braids solution in less that 20 seconds on any respectable PC.
Mauricio
 
Posts: 1174
Joined: 22 March 2006

Re: g-whips and g-braids

Postby denis_berthier » Mon Oct 08, 2012 10:19 am

Hi Maurico,

I've been away for some time.

You're really good at finding examples with exceptional properties.
Err, do you want the resolution paths with g-whips ?
denis_berthier
2010 Supporter
 
Posts: 1253
Joined: 19 June 2007
Location: Paris

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