The New Sudoku Players' Forum

by **champagne** » Thu Sep 20, 2007 2:18 pm

Hello,

My thread on "full tagging" is sleeping. I am coming back asking for external contribution.
The reason why I came here from time to time is the search for additional tools.

One simple way is for sure to get non T&E solutions for puzzles I (my solver) do not crack.

Here is the bottom list of unsolved puzzles in GSF list. I partially posted it in Denis Berthier’s thread.

Nobody answered, but I don’t know if this means that there is no known solution without T&E, which would be surprising.

Can anybody help me, either confirming the fact that there is no positive answer or bringing me to the right place to see the solution.

Code: Select all: 600000002090400050001000700050800000000942000000305040002000600030009080700000001,coloin-04-10_09716,106,1253, 600000002090040050001000700000895000030410080000300000007000600050080040200000001,jpf-04/07,0,_08737,110,1794, 600000002090400050001000700050308000000604000000095040002000600030009080700000001,coloin-04-10_08656,106,1489, 600000002090400050001000700050943000000800000000025040002000600030009080700000001,coloin-04-10_08499,107,1614, 600000002090400050001000700050100000000904000000853040002000600030009080700000001,coloin-04-10_07657,106,1230, 100000002030040050006000700000534000040810090000900000002000600080090030700000001,jpf-04/05,0,_07397,109,1657, 100000002030040050006000700000538000040960080000400000002000600090080030700000001,jpf-04/02,0,_07346,110,1711, 100000000006020700080005030300006800002070090040100000007030060000800500000009004,ravel-04/02,_07150,105,0790, 100000006020500040003000700040050000000810400000020080007000300050009020600000001,gsf-2007-05-_04519,104,1273, 003050000400009000080200000200004000090800000005060001000000217001000806000070503,rw-04-06,0,0_04220,106,0910, 100000002090400050006000700050903000000010009000800040700000600030009080002000001,gsf-2007-05-_03465,107,1059, 003050000400009000080200000200004000090800000005060001000000097001000806000070203,rw-04-06,0,0_03349,104,0900, 600000002090400050001000700050943000000105000000800040007000600030009080200000001,coloin-04-10_02805,098,1223,

Another way to crack these puzzles would be to combine forces. I understood than RONK have the possibility to start at any point of a puzzle in a specific solver.

I give hereafter the final status for puzzles my solver could crack partially.

The firs one is

Code: Select all: jpf-04/07,0,_08737,110,1794, clear 6r3c6 clear 6r9c6 clear 6r6c3 clear 6r6c9 == clear 7r1c6 clear 2r3c6 clear 2r6c3 clear 7r6c9 clear 7r6c1 clear 2r6c7 clear 2r7c6 clear 7r9c6 ending here 6 478 3458 |1579 357 1389 |13489 139 2 378 9 238 |1267 4 123678 |138 5 368 3458 248 1 |2569 2356 389 |7 369 34689 -------------------------------------------------------- 147 12467 246 |8 9 5 |1234 12367 3467 579 3 2569 |4 1 267 |259 8 5679 14589 124678 4589 |3 267 267 |1459 12679 459 -------------------------------------------------------- 13489 148 7 |1259 235 1349 |6 239 58 139 5 369 |12679 8 123679 |239 4 379 2 468 34689 |5679 3567 349 |58 379 1

Next one is

Code: Select all: coloin-04-10_08656,106,1489, clear 7r8c5 == clear 2r3c5 === clear 1r8c5 === clear 7r5c9 === clear 1r7c1 clear 1r5c1 === clear 1r4c1 === clear 2r8c5 === 6 478 34578 |15789 13578 137 |13489 139 2 238 9 378 |4 123678 12367 |138 5 368 23458 248 1 |2589 3568 236 |7 369 34689 --------------------------------------------------------- 249 5 4679 |3 127 8 |129 12679 679 2389 1278 3789 |6 127 4 |123589 12379 3589 1238 12678 3678 |127 9 5 |1238 4 3678 --------------------------------------------------------- 4589 148 2 |1578 134578 137 |6 379 34579 145 3 456 |1257 456 9 |245 8 457 7 468 45689 |258 234568 236 |23459 239 1

And the last one I propose here

Code: Select all: jpf-04/05,0,_07397,109,1657, clear 6r1c6 clear 6r9c6 clear 6r6c1 clear 6r6c9 === clear 2r6c7 clear 7r6c9 clear 7r1c6 clear 2r3c6 clear 7r7c6 clear 2r9c6 clear 2r6c1 clear 7r6c3 === 1 579 45789 |367 5678 3589 |3489 468 2 289 3 789 |1267 4 126789 |189 5 689 24589 259 6 |123 258 13589 |7 148 3489 -------------------------------------------------------- 2689 12679 1789 |5 3 4 |128 12678 678 2356 4 357 |8 1 267 |235 9 3567 358 12567 1358 |9 267 267 |13458 124678 3458 -------------------------------------------------------- 3459 159 2 |1347 578 1358 |6 478 45789 456 8 145 |12467 9 12567 |245 3 457 7 569 3459 |2346 2568 358 |24589 248 1

Thanks for help.

by **champagne** » Mon Oct 22, 2007 1:19 pm

Finally "Help yourself" seems to be the best way to go ahead.

I tried my last idea for the following step in my "full tagging" process. It works.

All theese puzzles but one (a remaining bug in my program) seem cracked by the new version of the solver.

I made a quick check, Eater Monster should be cracked as well, in a relatively easy way. Again, I feel that ranking proposed by gsf and jpf are not in line with "full tagging" capability to solve puzzles.

Having done the test in a quick and dirty mode, I have to stay cautious regarding the new capabilities, but for sure, full tagging have strong additional ones.

I have still to work on the process to find out to the "simplest" solution, which should be done within one or two weeks. Then, I'll post one or two detailed solutions.

by **champagne** » Wed Oct 24, 2007 5:35 pm

Hello,

I have now enough results to be confident in the new version of "full taging".

I tested it on the top list of jpf and gsf. In both cases; puzzles are cracked.

I have not that much time in the near future, so I start a full explanation of how it solves EasterMonster.

I start with the first clearing of candidates and I'll add other clearing progressively.

PRINTING RULES:

The rules have been described in the thread "full tagging". I have only to define how I will work with the new stuff.

I'll do it using the puzzle at the start

Code: Select all: 1 47d8 345f78 |3567y 3689e 5678 |34h89 36i9 2 2o38 9 37a8 |4 12368 1267A8 |1g38 5 36I8 2345F8 2P48 6 |1235 12389E 1258 |7 1G39 34H89 --------------------------------------------------------------- 2468 5 147x8 |9 124j6 3 |128 1267C 678 234689 12468 13489 |126 7 124J6 |1235b89 12369 35B689 2369 1267D 1379 |8 5 126 |1239 4 367x9 --------------------------------------------------------------- 7 1N48 14589k |1235 12348 12458 |6 2l39 3459 456v 3 1M45 |12567c 1246 9 |2L45 8 457C 45689K 46w8 2 |3567 3468 4567y8 |3459 37c9 1

I indicate "almost cell" using |*.
eg :
. r1c3 r3c1 is an "almost cell"
. 4(r1c3 r3c1)/2r3c1 |*r1c3r3c1
is the weak link 4/2 in that "almost cell".

Similarly, I indicate the new stuff ALS/"Almost cell" freedom 2 here after "AC2"(two digit TRUE and two digit FALSE) using ||*
eg : ||*(r1c2 r3c2).

We need as well a convention to identify a "couple tag".
eg : in ||*(r1c2 r3c2) the two valid digits can be 2 and 4.
. To avoid use of symbols, I'll write this {2;4||*r1c2r3c2)

A weak link using "couple tags" will be written
. (2;4)/(2;6)||*(r1c2 r3c2) inside ||*(r1c2 r3c2)
. 4(r1c3r3c1)/(2;4)||*r1c2r3c2 for a native weak link out of ||*r1c2r3c2
....
==================================
The first clearing move for Easter Monster uses two AC2 as Choices and two others as targets to prepare the final clearing:

Intermediate targets are:

||*r1c2r3c2 where we intend to cancel (4;8) => (2;7) in ||*r1c3r2c1r2c3r3c1 (complementary set)
nota: as in ||*r1c2r3c2 (2;7) is tagged (P;d) in ||*r1c3r2c1r2c3r3c1 (2;7) is tagged (p;D)

||*r2c1r2c3 where we intend to cancel {3;8} => (2;7) or (O;A) in ||*r1c2r1c3r3c1r3c2

AC2 used as choices are

||*r8c7r8c9 2r8c7-L 4(r8c7r8c9) 5(r8c7r8c9) 7r8c9-C
. "couple tags" (2;4) (2;5) (2;7) (4;5) (4;7) (5;7)

||*r7c8r9c8 2r7c8-l 3(r7c8r9c8) 7r9c8-c 9(r7c8r9c8)
. "couple tags" (2;3) (2;7) (2;9) (3;7) (3;9) (7;9)

==================================
XX= {(4;8)||*r1c2r3c2} ; {(2;7)||*r1c3r2c1r2c3r3c1} not valid => weak link p/D

one chain used several times

Code: Select all: #[]4(r7c3r9c1)/XX []4(r7c3r9c1)/8(r7c3r9c1)|*r7c3r9c1_8(r7c2r9c2)/XX ||*r1c2r3c2

#[]{(2,4)||*r8c7r8c9}/XX
[]4(r7c3r9c1)/XX|#
[]4(r8c1r8c3)/4(r8c7r8c9)/{(2,4)||*r8c7r8c9}
[]4(r7c2r9c2)/XX ||*r1c2r3c2

#[]5(r8c1r8c3r8c7)/{(2,5)||*r8c7r8c9}
[]7r8c9-C/(2,5)||*r8c7r8c9
[]4r8c9/(2,5)||*r8c7r8c9}
[]5r8c9/5(r8c1r8c3r8c7)

#[]2r5c2/{(2,7)||*r8c7r8c9}
[]2r7c8-l/(2,7)||*r8c7r8c9}
[]2r4c8/7r4c8-C_c/(2,7)||*r8c7r8c9}
[]2r5c8/2r5c2

#[]2(r3c2r6c2)/XX
[]2(r4c1r5c1r6c1)-P/XX
[]2r6c2/7r6c2-D_d/XX
2r5c2

#[]m/{(4,5)||*r8c7r8c9}
1r8c3-M
[]4r8c3/(4,5)||*r8c7r8c9
[]5r8c3/(4,5)||*r8c7r8c9

#[]M/XX
[]1r7c2/1(r7c4r7c5r7c6)-M
[]4r7c2/XX
[]8r7c2/XX

#[]{(4,7)||*r8c7r8c9}/XX
[]4(r7c3r9c1)/XX|#
[]4(r8c1r8c3)/(4,7)||*r8c7r8c9}
[]4(r7c2r9c2)/XX ||*r1c2r3c2

#[]5(r5c7r9c7)/{(5,7)||*r8c7r8c9}
[]2r8c7-L/(5,7)||*r8c7r8c9}
[]4r8c7/(5,7)||*r8c7r8c9}
5r8c7

#[]XX thru||*r8c7r8c9 (2;4) (2;5) (2;7) (4;5) (4;7) (5;7)
[](2;4)/XX|#
[](2;5)/5(r8c1r8c3r8c7)|#_5(r8c4r8c9)/5(r8c1r8c3)_5(r7c3r9c1)/8(r7c3r9c1)|*r7c3r9c1 _8(r7c2r9c2)/XX
[](2;7)/2r5c2|#_2(r3c2r6c2)/XX|#
[](4;5)/m|#_M/XX|#
[](4;7)/XX|#
[](5;7)/5(r5c7r9c7)|#_5r8c7/5(r8c1r8c3)_5(r7c3r9c1)/8(r7c3r9c1)|*r7c3r9c1 _8(r7c2r9c2)/XX
==========================================
YY= {{3;8}||*r2c1r2c3} or {(2;7) (O;A)||*r1c2r1c3r3c1r3c2} not valid => weak link O/A
in a similar way

a chain reused

Code: Select all: #[]3(r1c7r3c9)/YY []3(r1c7r3c9)/8(r1c7r3c9)|*r1c7r3c9 _8(r2c7r2c9)/YY

#[]g/||{(3;9)||*r7c8r9c8}
1r3c8-G
[]9r3c8/(3;9)||*r7c8r9c8
[]3r3c8/(3;9)||*r7c8r9c8

#[]G/YY
1r2c7-g
[]8r2c7/YY
[]3r2c7/YY

#[]2r2c5/{(2;7)||*r7c8r9c8}
[]2r8c7-L/(2;7)||*r7c8r9c8}
[]2r8c4/7r8c4-c_ C/(2;7)||*r7c8r9c8}
[]2r8c5/2r2c5

#[]2(r2c1r2c6)/YY
[]2r2c1-o/YY
[]2r2c6/7r2c6-A_a/YY
2r2c5

#[]YY/{(2;3)||*r7c8r9c8}
[]3(r1c7r3c9)/YY|#
[]3(r2c7r2c9)/YY
[]3(r1c8r3c8)/(2;3)||*r7c8r9c8

#[]9(r9c1r9c7)/{(2;9)||*r7c8r9c8}
[]7r9c8-c/(2;3)||*r7c8r9c8
[]3r9c8/(2;3)||*r7c8r9c8
9r9c8

#[]YY/{(3;7)||*r7c8r9c8}
[]3(r1c7r3c9)/YY|#
[]3(r2c7r2c9)/YY
[]3(r1c8r3c8)/(3;7)||*r7c8r9c8}

#[]9(r7c3r7c9)/{(7;9)||*r7c8r9c8}
[]2r7c8-l/(7;9)||*r7c8r9c8
[]3r7c8/(7;9)||*r7c8r9c8
9r7c8

#[]YY thru ||*r7c8r9c8 (2;3) (2;7) (2;9) (3;7) (3;9) (7;9)
[](2;3)/YY|#
[](2;7)/2r2c5|#_2(r2c1r2c6)/YY|#
[](2;9)/9(r9c1r9c7)|#_9r9c8/9(r1c8r3c8)_9(r1c7r3c9)/8(r1c7r3c9)|*r1c7r3c9 _8(r2c7r2c9)/YY
[](3;7)/YY|#
[](3;9)/g|#_G/YY|#
[](7;9)/9(r7c3r7c9)|#_9r7c8/... as above
======================
Having derived two weak links p/D and O/A, we have now a Nice loop

[]ad Dp Po OA []7r2c3-a/7r1c2-d_ /p_2(r4c1r5c1r6c1)-P/2r2c1-o_O/A

clearing thru
=7r2c3 =7r1c2
and =2(r4c1r5c1r6c1) =2r2c1
====> start edit one

I will not cover the remaining part of the "Kurzhals loop" cleaning for two reasons:

- I will likely introduce the "virus pattern" analysis and may be the "Kurzhals loop" identification, so the rest is obsolete,
- I have a small problem in backtracking of clearing due to incomplete handling of "couple tags" clearing and I need time to solve it.

I have no problem with the next steps, so I can comment the solver proposal

Here is the situation at the end of "Kurzhals loop" cleaning.

Code: Select all: 1 47A8 345h8 |3567Ã 389f 5678 |34j89 36k9 2 2g38 9 37a8 |4 126 1267A |1i38 5 36K8 345H8 2G48 6 |1235 389F 1258 |7 1I39 34J89 ------------------------------------------------------- 24b68 5 14B7Â8 |9 126 3 |128 1267D 678 389 126 389 |126 7 4 |35c89 126 35C89 2369 1267a 1379 |8 5 126 |1239 4 367Â9 ------------------------------------------------------- 7 1l48 4589n |1235 34e8 1258 |6 2o39 3459 456m 3 1L45 |1267d 126 9 |2O45 8 457D 4589N 46M8 2 |3567 34E8 567Ã8 |3459 37d9 1

This is the status after basic tagging. The "Kurzhals loop" is still there with a lot of new strong links.

In fact, "G" is "a"; "K" is "i"; "o" is "d"; "M" is "l" for the visible part of it.
Unless I missed something, Loop effect is of no use in the following sequence

We intend to show that 1r6c7 can be cleared.

Two chains re used later

#[]6r5c2/6r8c4 []6r5c2/6r9c2_6r8c1/6r8c4
#[]1r2c5/1r4c8 []1r2c5/1r2c7-i_1r3c8-I/1r4c8

And we start deriving weak links

#[]1r8c4/1r6c7 []1r5c4/1r8c4 []1r5c2/1r7c2_1r8c3/1r8c4
[]1r5c8/1r6c7
#[]1r8c4/1r2c5 []1r5c8/1r3c8_1r2c7/1r2c5 +..
#[]1r2c6/8r3c9 []7r1c2-==7r2c6-A/1r2c6 []4r1c2/4r1c7_4r3c9/8r3c9
[]8r1c2/8(r2c1r2c3)_8(r2c7r2c9)/8r3c9
#[]6r2c5/6r8c4 []6r5c8/6r1c8_6r2c9/6r2c5
[]6r5c4/6r8c4 []6r5c2/6r8c4|#
#[]Ù/1r4c8 []1r8c5/2r8c5-Ù []1r4c5/1r4c8 []1r2c5/1r4c8|#
#[]Ù/6r4c8 []6r8c5/2r8c5-Ù []6r4c5/6r4c8 []6r2c5/6r2c9_6r1c8/6r4c8
#[]2r2c5/2r8c4 []2r5c8/2r7c8_2r8c7/2r8c4-Ö []2r5c4/2r8c4-Ö []2r5c2/2r3c2_2r2c1/2r2c5
#[]2r5c8/6r8c4 []6r5c8/2r5c8 []6r5c4/6r8c4 []6r5c2/6r8c4|#
#[]1r2c5/1r4c3 []1r8c3/1r4c3 []1r8c5/1r2c5 []1r8c4/1r2c5|#
#[]d/Ù 7r4c8-D []1r4c8/Ù|# []2r4c8/2r7c8_2r8c7/2r8c5-Ù []6r4c8/Ù|#
#[]2r8c4-Ö/2r4c7 []2r8c5-Ù/2r8c4-Ö []2r4c5/2r4c7 []2r2c5/2r8c4-Ö|#
#[]2(r4c7r6c7)-D/2r2c5 2r8c7-d []2r8c5-Ù/2r2c5 []2r8c4-Ö/2r2c5|#
#[]2(r4c8r5c8)-d/6r8c4 2r7c8-D []2r5c8/6r8c4|# []2r4c8/7r4c8-D_7r8c4-d/6r8c4
#[]1r2c5/1r6c7 []1r4c5/1r2c5 []1r4c3/1r2c5|# []1r4c8/1r2c5|# []1r4c7/1r6c7
#[]d/2r4c7 []2r8c7/2r4c7 []2r8c5-Ù/d|# []2r8c4/7r8c4-d

#[]2r8c4-Ö/1r6c7[]2r8c7/2r8c4-Ö []2r6c7/1r6c7 []2r4c7/2r8c4-Ö|#
#[]þ/6r8c4 1r2c5 (þ is any counterpart of 1r2c5)
[]2r2c5/2(r4c7r6c7)-D|#_2(r4c8r5c8)-d/6r8c4|# []6r2c5/6r8c4|#
#[]6r8c4/1r6c7 []6r8c4/þ|#_1r2c5/1r6c7|#
#[]1r6c7/8r3c9 []1r2c7-i/1r6c7 []1r2c5/1r6c7|# []1r2c6/8r3c9|#
#[]1r6c7/2r4c7 []7r8c4-d/2r4c7|# []1r8c4/1r6c7|# []2r8c4-Ö/1r6c7|# []6r8c4/1r6c7|#
#[]D/1r6c7 7r8c4-d
[]1r8c4/1r6c7|# []2r8c4-Ö/1r6c7|# []6r8c4/1r6c7|#
#[]É/1r6c7 []1r4c7/1r6c7 []2r4c7/1r6c7|# []8r4c7/8(r1c7r2c7)-É
#[]1r6c7/6r4c9 []7r8c9-D/1r6c7|# []7r6c9-Â/1r6c7|*r5c7r5c9r6c7r6c9 []7r4c9/6r4c9
#[]K/1r6c7 []8(r1c7r2c7)-É/1r6c7|# []8r2c9/6r2c9-K []8r3c9/1r6c7|#
#[]k/1r6c7 6r2c9-K
[]6r6c9/1r6c7|*r5c7r5c9r6c7r6c9 []6r4c9/1r6c7|#
#[]1r6c7/1r6c7 []1r6c7/K|#_k/1r6c7|#

Nota: the sequence as such is valid. However, on can notice that the computer has tagged -Ö 2r8c4 and 2r5c2.
This is not used in the best way in the sequence.

The refreshed tagging situation

Code: Select all: 1 47A8 345i8 |3567Å 389f 5678 |34j89 36k9 2 2h38 9 37a8 |4 126 1267A |1g38 5 36K8 345I8 2H48 6 |1235 389F 1258 |7 1G39 34J89 ------------------------------------------------------- 24b68 5 14B7Ä8 |9 126 3 |1G28 1267D 678 389 126 389 |126 7 4 |35c89 1Q26 35C89 2369 1267a 1379 |8 5 1P26 |239 4 367Ä9 ------------------------------------------------------- 7 1l48 4589n |1235 34e8 1258 |6 2o39 3459 456m 3 1L45 |1267d 126 9 |2O45 8 457D 4589N 46M8 2 |3567 34E8 567Å8 |3459 37d9 1

and showing 1r2c6 is not valid

#[]L/1r2c6 1r7c2-l []1r5c2/1(r6c2r6c3)_1r6c6/1r2c6 []1r6c2/7r6c2_7r2c6/1r2c6
#[]2r5c8/1r8c4 []1r5c8/2r5c8 []1r5c4/1r8c4 []1r5c2/1r7c2_1r8c3/1r8c4
#[]2r5c8/6r8c4 []6r5c8/2r5c8 []6r5c4/6r8c4 []6r5c2/6r9c2_6r8c1/6r8c4
#[]1r8c5/6r4c8 []6r8c5/1r8c5 []6r4c5/6r4c8 []6r2c5/6r2c9_6r1c8/6r4c8
#[]1r5c2/2r2c5 []1r8c5/1r8c3_1r7c2/1r5c2 []1r4c5/1r6c6_1(r6c2r6c3)/1r5c2 []1r2c5/2r2c5
#[]D/2r5c8 7r8c4-d []1r8c4/2r5c8|# []2r8c4/2r8c7_2r5c8/2r7c8-D []6r8c4/2r5c8|#
#[]l/6r4c8 1r8c3-L []1r8c5/6r4c8|# []1r8c4/7r8c4_7r4c8/6r4c8
#[]6r5c8/2r2c5 []1r5c2/2r2c5|# []2r5c2/2r3c2_2r2c1/2r2c5 []6r5c2/6r5c8
#[]2r5c8/1r2c6 []7r4c8-D/2r5c8|# []1r4c8/1r4c7_1r2c7/1r2c6
[]2r4c8/2r5c8 []6r4c8/l|#_L/1r2c6|#
#[]6r5c8/1r2c6 []1r2c5/1r2c6 []2r2c5/6r5c8|# []6r2c5/6r2c9_6r1c8/6r5c8
#1r2c6 []1r5c8/1r4c7-G_1r2c7-g/1r2c6 []2r5c8/1r2c6|# []6r5c8/1r2c6|#

and the refresed situation

Code: Select all: 1 47B8 345i8 |3567Å 389g 5678 |34j89 36k9 2 2h38 9 37b8 |4 1a26 267B |1A38 5 36K8 345I8 2H48 6 |1235 389G 1258 |7 1a39 34J89 ------------------------------------------------------- 24c68 5 14C7Ä8 |9 126 3 |1a28 1267E 678 389 126 389 |126 7 4 |35d89 1Q26 35D89 2369 1267b 1379 |8 5 1P26 |239 4 367Ä9 ------------------------------------------------------- 7 1l48 4589n |1235 34f8 1258 |6 2o39 3459 456m 3 1L45 |1267e 126 9 |2O45 8 457E 4589N 46M8 2 |3567 34F8 567Å8 |3459 37e9 1

Proving 4r3c1 and 4r7c2 are not valid is interessing.
It is the last very difficult clearing, still far from the end.

The first step is to show they have the same tag.
We will show that 4r3c1 forces 4r7c2 and 4r7c2 forces r3c1, which is enough to have the same tag for both.

Common chains

#[]4r7c9/4r3c1 []4r3c1/4(r1c2r3c2)_4(r7c2r9c2)/4(r8c1r8c3)_4(r8c7r8c9)/4r7c9
#[]2r4c8/2r8c5 []2r8c5/2r8c7_2r7c8/2r4c8

1) 4r3c1 forces 4r7c2
#[]E/4r3c1 []4r3c9-J/4r3c1 []4r8c9/7r8c9-E []4r7c9/4r3c1|#
#[]2r8c5/1r4c8 []1r2c5==1r4c7-a/1r4c8 []1r8c5/2r8c5 []1r4c5/1r4c8
#[]2r8c5/6r4c8 []6r8c5/2r8c5 []6r4c5/6r4c8 []6r2c5/6r2c9_6r1c8/6r4c8
#[]1r8c5/4r3c1 []1r8c3/1r8c5 []5r8c3/5r1c3_5r3c1/4r3c1 []4r8c3/4r4c3-4r4c1/4r3c1
#[]2r8c5/4r3c1 []7r4c8-E/4r3c1|# []1r4c8/2r8c5|# []2r4c8/2r8c5|# []6r4c8/2r8c5|#
#[]4r9c2/4r3c1 []1r8c5/4r3c1|# []2r8c5/4r3c1|# []6r8c5/6r8c1_6r9c2/4r9c2
and Final 4r7c2 []4(r1c2r3c2)/4r3c1 []4r9c2/4r3c1|#

2) 4r7c2 forces 4r3c1

#[]4r8c1/4r1c3 []4r8c1/4r4c1_4r4c3/4r1c3
#[]5r8c1/4r1c3 []5r8c1/5r3c1_5r1c3/4r1c3
#[]6r8c4/4r1c3 []6r8c1/6r8c4 []5r8c1/4r1c3|# []4r8c1/4r1c3|#
#[]6r8c4/4r3c1 []4r3c9/4r3c1 []4r8c9/6r8c4|*r8c4r8c9 []4r7c9/4r3c1|#
#[]6r8c4/4r7c2 []4(r1c2r3c2)/4r7c2 []4r3c1/6r8c4|# []4r1c3/6r8c4|#
2r8c5 forces 'E'
#[]e/2r8c5 7r4c8-E []1r4c8/2r8c5|# []2r4c8/2r8c5|# []6r4c8/2r8c5|#
#[]6r8c5/4r1c3 []6r8c1/6r8c5 []5r8c1/4r1c3|# []4r8c1/4r1c3|#
#[]2r8c5/4r7c2 []7r8c4-e/2r8c5|# []1r8c4/1r8c3_1r7c2/4r7c2 []2r8c4/2r8c5 []6r8c4/4r7c2|#
#[]4r7c2/4r1c3 []1r8c5/1r8c3_1r7c2/4r7c2 []2r8c5/4r7c2|# []6r8c5/4r1c3|#
and final 4r3c1 []4(r1c2r3c2)/4r7c2 []4r1c3/4r7c2|#

We can now justify the clearing with 4r7c2 == 4r3c1 (same tag here XX)
We will re use from previous step
#[]e/2r8c5 2r8c5 forces 'E'
#[]E/XX
#[]6r8c4/XX
#[]2r8c5/XX
#[]2r8c5/1r4c8
#[]2r8c5/2r4c8
#[]2r8c5/6r4c8

And the sequence startng with chains

#[]1r4c5/1r5c2 []1r5c2/1(r6c2r6c3)_1r6c6/1r4c5
#[]1r8c5/1r5c2 []1r5c2/1r7c2_1r8c3/1r8c5
#[]1r8c5/XX []1r8c5/1r8c3_1r7c2/4r7c2-XX
#[]5r1c4/XX []5r1c4/5r1c3_5r3c1/4r3c1-XX
#[]5r8c1/4r1c3 []5r8c1/5r3c1_5r1c3/4r1c3
#[]4r8c1/4r1c3 []4r8c1/4r4c1_4r4c3/4r1c3
#[]2r2c5/2r5c2 []2r5c2/2r3c2_2r2c1/2r2c5
#[]6r8c5/6r5c2 []6r5c2/6r9c2_6r8c1/6r8c5
#[]6r2c5/6r4c8 []6r2c5/6r2c9_6r1c8/6r4c8
#[]4r7c9/XX []4r3c1-XXX/4(r1c2r3c2)_4(r7c2r9c2)/4(r8c1r8c3)_4(r8c7r8c9)/4r7c9
#[]6r6c1/6r1c4 []6r6c6/6r6c1 []6r2c6/6r1c4 []6r9c6/6r9c2_6r8c1/6r6c1 []6r1c6/6r1c4

#[]A/1r5c2 1r2c5-a []1r8c5/1r5c2|# []1r4c5/1r5c2|#
#[]7r1c4/XX []4r3c9/4r3c1-XX []4r8c9/7r8c9_7r8c4/7r1c4 []4r7c9/XX|#
== []5r8c1/4r1c3|# []4r8c1/4r1c3|#
#[]6r8c5/4r1c3 []6r8c1/6r8c5
#[]6r8c4/4r1c3 []6r8c1/6r8c4

#[]1r5c2/2r2c5 []1r2c5/2r2c5 []1r8c5/1r5c2|# []1r4c5/1r5c2|#
#[]6r5c2/6r4c8 []6r8c5/6r5c2|# []6r4c5/6r4c8 []6r2c5/6r4c8|#

#[]6r8c4/XX []4(r1c2r3c2)/4r7c2-XX []4r3c1-XX/6r8c4|# []4r1c3/6r8c4|#
#[]6r8c5/2r2c5 []1r5c2/2r2c5|# []2r5c2/2r2c5|# []6r5c2/6r8c5|#
#[]6r8c5/3r3c8 []1r2c5==1r3c8-a/3r3c8 []2r2c5/6r8c5|# []6r2c5/6r8c5

#[]6r2c5/6r4c8 []6r2c5/6r2c9_6r1c8/6r4c8
#[]m/6r4c8 []6r8c5/6r8c1-m []6r4c5/6r4c8 []6r2c5/6r4c8|#
#[]2r5c2/2r4c8 []2r8c5/2r4c8|# []2r4c5/2r4c8 []2r2c5/2r5c2|#
#[]a/2r4c8 []2r8c5/2r4c8|# []2r4c5/2r4c8 []2r2c5/1r2c5-a
#[]6r5c2/6r1c4 []6r1c8/6r1c4 []6r5c8/6r5c2 []6r4c8/6r5c2|#
#[]6r4c8/6r1c4 []6r8c1/6r4c8|# []6r4c1/6r4c8 []6r6c1/6r1c4|#
#[]ø/2r4c8 6r5c2 []1r5c2/A|#_a/2r4c8|# []2r5c2/2r4c8|# (ø complementary to 6r5c2)
#[]l/XX 1r8c3-L []5r8c3/5r1c3_5r3c1/4r3c1-XX []4r8c3/4r4c3_4r4c1/4r3c1-XX
#[]L/1r4c8 []1r2c5==1r4c7-a/1r4c8 []1r8c5/1r8c3-L []1r4c5/1r4c8
#[]6r1c4/XX []7r4c8-E/4r3c1-XX|# []1r4c8/L|#_l/4r3c1-XX|# []2r4c8/ø|#_6r5c2/6r1c4|# []6r4c8/6r1c4|#
#[]3r1c8/XX []7r1c4/4r3c1-XX|# []3r1c4/3r1c8 []5r1c4/4r3c1-XX|# []6r1c4/4r3c1-XX|#

#[]8r7c5/XX []7r1c4/XX|# []3r1c4/3(r7c4r9c4)_8(r1c5r3c5)/8r7c5 []5r1c4/4r3c1-XX|# []6r1c4/4r3c1-XX|#
#[]XX/4r1c3-§Q []1r8c5/XX|# []2r8c5/XX|# []6r8c5/4r1c3|#
#[]T/XX []4r7c5/4r7c2-XX []3r7c5/3(r7c8r7c9)-T []8r7c5/XX|#
#[]XX/3r3c8 []1r8c5/XX|# []2r8c5/XX|# []6r8c5/3r3c8|#
#[]3r9c8/XX []4r3c9/4r3c1-XX []4r8c9/3r9c8|*r8c9r9c8 []4r7c9/XX|#
#[]t/XX []3r9c8/XX|# []3r7c8/3(r7c4r7c5)-t []3r3c8/XX|# []3r1c8/XX|#
T/XX t/XX XX is dead clear 4r3c1 4r7c2

For the following steps, as the program will likely change the sequence taking benefit of the virus pattern structures, I'll give the clearing sequence.
If anybody wants more details, i'll give them.
Next steps (short sequences unless specified

- clear ch=2r2c5
- clear ch=6r2c6
- clear ch=6r6c9
- clear ch=2r6c6
- clear ch=8r7c3
- clear ch=8r1c2
- .clear ch=7r1c6
.clear ch=2r7c4
two medium size sequences
- clear ch=6r4c5 clear ch=6r8c4 medium size sequence
-. clear ch=8r1c7 clear ch=8r2c1
. clear ch=4r9c7
-.clear ch=3r6c3
.clear ch=3r2c7
- clear ch=3r3c1
-.clear ch=2r4c1
.clear ch=8r5c9 clear ch=7r4c9
.clear ch=8r3c6s
-.clear ch=8r9c5
.clear ch=1r5c2 clear ch=6r1c6 clear ch=6r4c8 clear ch=6r6c1
.clear ch=8r4c3
.clear ch=3r9c4
-.clear ch=9r6c3
.clear ch=5r8c1

we are there

Code: Select all: 1 4B7b 3S4Z5K8Ã |356a7B 389o 5p8P |34c9 36A9 2 2b3B 9 37B8Ä |4 1a6A 2B7b |1A8a 5 3S6a8 5k8K 2B4c8Æ 6 |12j35 38Ã9O 125 |7 1a39 34C8Ä9 -------------------------------------------------------------------- 4e6a8Å 5 1i4E7f |9 1D2d 3 |1a2D8 1f7F 6A8a 389 2q6Q 3B8ç9m |1g2D6i 7 4 |35h8Å9 1G2j6a 35H9 2B3U9è 1l26i7B 1r7R |8 5 1i6I |2q39 4 37f9 -------------------------------------------------------------------- 7 1L8l 459M |13V5 34n8P 12j58 |6 2J39 34y59 4a6A 3 1l45À |127f 12D6a 9 |2j4C5 8 457F 45K8Æ9m 4Z6a8 2 |567 3n4N 56i7B8l |359 37f9 1

Three steps are missing, in the next n, the tagging sequence clears 5r9c4 (saturation)
In the next one, similarly 5r3c6 and 1r7c4 are cleared
in addition, 3r7c8is cleared thru a chain

The last step is here

Code: Select all: 1 4B7b 3U4à5D8Ã |356a7B 389o 5p8P |34c9 36A9 2 2b3B 9 37B8Ä |4 1a6A 2B7b |1A8a 5 3U6a8 5d8D 2B4c8Æ 6 |12l35D 38Ã9O 1q2Q |7 1a39 34C8Ä9 --------------------------------------------------------------------- 4f6a8Å 5 1j4F7g |9 1E2e 3 |1a2E8 1g7G 6A8a 389 2r6R 3B8ç9m |1h2E6j 7 4 |35i8Å9 1H2l6a 35I9 2B3W9è 1K26j7B 1s7S |8 5 1j6J |2r39 4 37g9 --------------------------------------------------------------------- 7 1k8K 459M |3P5p 34n8P 1K2l58 |6 2L9l 3n4z59 4a6A 3 1K45Á |127g 12E6a 9 |2l4C5 8 457G 45D8Æ9m 4à6a8 2 |6t7T 3n4N 5â6j7B8K |359 3X7g9 1

several possibilities,

'B' 'g' 'D' 'a' are cleared, the puzzle is cracked

Edit one 2007 10 31 clearing 1r6c7 1r2c6
Edit 2 2007 11 01 clearing 4r3c1 4r7c2 and after

by **champagne** » Sat Oct 27, 2007 2:58 pm

I mark a pause in the printing of my solution to Easter Monster to make a comparison with another solution.

Thanks to him Glyn brought me on that post from Ronk at the start of Easter Monster.

Code: Select all: 1 A478 34578 | 3567 3689 5678 | 3489 I369 2 L238 9 L378 | 4 K12368 K12678 |J138 5 J368 23458 A248 6 | 1235 12389 1258 | 7 I139 3489 --------------------------+-------------------------+----------------------- 2468 5 1478 | 9 1246 3 | 128 H1267 678 234689 B12468 13489 | 126 7 1246 | 123589 H12369 35689 2369 B1267 1379 | 8 5 126 | 1239 4 3679 --------------------------+-------------------------+----------------------- 7 C148 14589 | 1235 12348 12458 | 6 G239 3459 D456 3 D145 |E12567 E1246 9 |F245 8 F457 45689 C468 2 | 3567 3468 45678 | 3459 G379 1 (27)r13c2=(27-16)r56c2=(16)r79c2-(16)r8c13=(16-27)r8c45=(27)r8c79-(27)r79c8 =(27- 16)r45c8=(16)r13c8-(16)r2c79=(16-27)r2c56=(27)r2c13 - loop Note that nodes marked A, C, D, F, G, I, J and L are all AALSs..

I don’t know if Ronk has a short explanation to tell why that loop is operative, but for sure if you start with the assumption that one of 2r3c2_7r1c2 is valid, we have that kind of loop

The first step in my solution was precisely to show that 2r3c2_7r1c2 is a strong link.

The solver does not recognize the loop as such. In place, it clears individually

8r2c6 3r2c5 8r2c5 3r5c8 9r5c8 1r7c3 6r9c1
4r8c5 5r8c4 4r5c2 8r5c2

Just following what I have already shown. (will be the first edit of the post)

Ronk made a fantastic job discovering that loop.

Clearly, it’s following another logic that what I made in step four.

The stuff is more or less the same, Ronk is working on couples in OR condition (may be exclusive OR) while I am working on couples in And condition.

There is likely some room for improvement using Ronk way in addition of what I made.

This meet another topic I wanted to raise after having published the first step of the solution.

In AC2 (here ||*r1c2r3c2 ), if you have identified a strong link 2r3c2_7r1c2, then the rest is also a strong link (4(r1c2r3c2)_8(r1c2r3c2).

This has also to be included in the processing. We see that here it’s part of the powerful analysis made by Ronk.

by **ronk** » Sat Oct 27, 2007 4:23 pm

champagne wrote:... itâ€™s part of the powerful analysis made by Ronk.

The credit for that deduction goes to Stephen Kurzhals. See his hidden pair loop topic on the Eureka! forum. [20120331 edit: repaired link]

a long ago champagne wrote:I understood than RONK have the possibility to start at any point of a puzzle in a specific solver.

Sorry, I'm sure some posters here have solvers with that capability, but I'm not one of them. Been wishing though.

by **champagne** » Sat Oct 27, 2007 4:57 pm

Ronk wrote

champagne wrote:
... it’s part of the powerful analysis made by Ronk.

The credit for that deduction goes to Stephen Kurzhals. See his hidden pair loop topic on the Eureka! forum.

Glyn pushed me on the right place. Your post in that thread was much closer to my presentation and in a quick reading I neglected the rest of the thread.

EDIT one
Just a remark regarding "Kurzhals loop".

You can find it at the start of these three puzzles at the top list of jpf.

11.4 # 500000009020100070008000300040600000000050000000207010003000800060004020900000005 # m_b_metcalf
11.4 # 500000009020100070008000300040002000000050000000706010003000800060004020900000005 # StrmCkr
11.4 # 100000002090400050006000700050903000000070000000850040700000600030009080002000001 # JPF 04/07/01 (Easter Monster)

You have it as well in the three puzzles shown at the top of this thread.

It is not in dml55 I used to introduce step four, but it's not so far.

Code: Select all: 1 56u7 3B568 |9 34A7 2f3O78 |24568 4e678 25678 789 2 6S89 |14a78 5 178 |4A689â 3 1L67 3b578 579â 4 |12F78 13B7 6 |2589 1m79å 12578 ------------------------------------------------------------- 6 4D59 2h8H |2457 4a79 23c579 |1 7Z89 3C578 4d59 3 1g59 |14D56t7 8 1579 |569 2 567w 2H8h 1G59 7 |1256 13O6V9 12359 |3c5689 689 4 ------------------------------------------------------------- 259 14e569 12569 |3 169ä 15i8À9 |7 14E68 12j68 34D79 1467y9 1369 |1678á 2 1789Ã |34e68 5 13c68 2357X 8 12356 |15I67 167 4 |2J3n6 1k6K 9

No surprise, some of these extreme puzzle have a common basis.

by **champagne** » Tue Nov 06, 2007 7:19 pm

Example of a level 2 puzzle.

Easter Monster is not the best way to show full tagging capabilities.
I have choosen a puzzle not exceeding level 2 as first example.
A second will follow requesting a short step in level3
This puzzle has been published on Gould forum in 2005.

Code: Select all: 600040003010000070005000800000502000300090002000103000008000900070000050200030004

Code: Select all: 6 289 279 |2789 4 15a789 |125 129 3 489 1 2349 |23689 2568 689 |246 7 569 479 2349 5 |23679 1267 1679 |8 12469 169 ----------------------------------------------------------- 14789 4689 14679 |5 678 2 |13467 134689 16789 3 568 167 |4678 9 4678 |1567 168 2 45789 24689 24679 |1 678 3 |467 4689 56789 ---------------------------------------------------------- 145 346 8 |2467 12567 1467 |9 23 167 149 7 13469 |24689 1268 14689 |23 5 168 2 569 169 |6789 3 156789 |167 168 4

Here basic taging doesn't help. The only "group tag" used is the Empty Rectangle pattern in box 9 6(r9c78)/6(r78c9)

As shown in another thread, the solver starts with that Nice Loop

Code: Select all: []6(r9c78)/6(r9c23)_6(r7c2r8c3)/4(r7c2r8c3)|*r7c2r8c3_4(r78c1)/4(r23c1)_4(r2c3r3c2)/2(r2c3r3c2)|*r2c3r3c2 _2(r1c23)/2(r1c78)_2(r2c7r3c8)/6(r2c7r3c8)|*r2c7r3c8_6(r23c9)/6(r78c9)

As you can see, weak links requesting level 2 are generated in AHS |*r7c2r8c3 |*r2c3r3c2 and |*r2c7r3c8.
Strong links ALS_AHS are also generated in level 2.

Code: Select all: 6 28f9 27e9 |789 4 1j5a789 |125A 129Æ 3 48F9 1 23b4 |23B689 25A68 689 |24g6 7 5a69 47E9 23B4 5 |23b679 1m267 1679 |8 24G6 1j69 ---------------------------------------------------------------- 1l789 4689 14679 |5 678 2 |13B467 13b4689 1789 3 5A68 1K67 |4c678 9 4C678 |15a67 168 2 5a789 2d4689 2D4679 |1 678 3 |467 4689 5A789 ---------------------------------------------------------------- 145A 3b46 8 |2467 125a67 1467 |9 2b3B 167h 149Ä 7 3B46 |24689 1268 14689 |2B3b 5 168i 2 5a69 1l69 |789 3 15A789 |167H 168I 4

Four similar ternary chains (symetry).

Code: Select all: #2r3c8 []3r2c3==2r7c8-b/2r3c8 []4r2c3/4r2c7-g_4r3c8-G/2r3c8 []2r2c3/2(r1c23)-o_2(r1c78)-O/2r3c8

Code: Select all: #4r2c3 []4r2c7-g/4r2c3 []2r2c7/2r8c7-B_3r2c3-b/4r2c3 []6r2c7/6(r23c9)-o_4(r23c1)-O/4r2c3

Code: Select all: #2r2c7 []3r3c2==2r8c7-B/2r2c7 []4r3c2/4r3c8-G_4r2c7-g/2r2c7 []2r3c2/2(r1c23)-o_2(r1c78)-O/2r2c7

Code: Select all: #4r3c2 []4r3c8-G/4r3c2 []2r3c8/2r7c8-b_3r3c2-B/4r3c2 []6r3c8/6(r23c9)-o_4(r23c1)-O/4r3c2

Now in level 1

Code: Select all: 6 89 79 |789 4 1B5b |125B 12 3 489 1 23 |23689 25B68 689 |46 7 5b9B 479 23 5 |23679 1267 1679 |8 46 1B9b --------------------------------------------------------- 1n789 4689 14679 |5 678 2 |13467 134689 178 3 5B68 167 |4678 9 4678 |15b67 168 2 5b789 24689 24679 |1 678 3 |467 4689 5B78 --------------------------------------------------------- 1b5B 34 8 |2467 125b67 1467 |9 23 167 19 7 34 |24689 1268 14689 |23 5 168 2 5b69 1n69 |789 3 15B789 |17 18 4

[]np PB bn []1r9c3-n/1(r9c7r9c8)_1(r7c9r8c9)/1r3c9-B_1r7c1-b/1r9c3-n

nota []np PB bn
is our prefered way to show that clearing of 1r9c3 1r4c1
P,p layer is not visible and needs additionnal comment in preferred mode

Code: Select all: 6 89 79 |789 4 15 |12a5 1a2A 3 489 1 2A3a |23A689 2568 689 |46 7 59 479 2a3A 5 |23a679 1267 1679 |8 46 19 ----------------------------------------------------------- 789 4689 14679 |5 678 2 |13A467 13a4689i 178 3 568 167 |4678 9 4678 |1567 168 2 5789 2A4689 2a4679 |1 678 3 |467 4689I 578 ----------------------------------------------------------- 15 3a4A 8 |2467 12567 1467 |9 2a3A 167 19 7 3A4a |24689 1268 14689 |2A3a 5 168 2 569 69 |789 3 15789 |17 18 4

Here we have (level 1)

Clearing inside the Bb layer thru
=1r7c1 =1r1c6
=1r7c1 =1r3c9
=9r8c1 =9r2c9

Clearing out of the weak link []ai => OR {AI}
=2r6c2 =9r6c8

locked set 15 r7c15

Refreshing tagging leads to 2r7c4-A
2r3c4 is cleared

And we restart a level 2

Code: Select all: 6 8d9D 7c9C |7C8D9S 4 1B5b |12a5B 1a2A 3 4d8D 1 2A3a |2m3A689 25B68 689 |4D6d 7 5b9B 4D7C9S 2a3A 5 |3a679 1l2A67p 1679 |8 4d6D 1B9b -------------------------------------------------------------------- 789 4689 1g4679 |5 678 2 |13A467 13a4689h 178 3 5B68 1G67 |4e678 9 4E678 |15b67 168 2 5b789 2A468 2a4679 |1 678 3 |467 4689H 5B78 -------------------------------------------------------------------- 1b5B 3a4A 8 |2A467 1B5b 467 |9 2a3A 6i7I 1B9b 7 3A4a |24689 12m68 14689 |2A3a 5 1k6I8j 2 5b6f9 6F9f |789 3 1k5B789 |1I7i 1j8J 4

From now on we have to take care of additionnal tagging

-1r7c5-B 1r9c6-k 1(r8c56)
-1r1c6-B 1r9c6-k 1(r38c6)
-1r3c9-B 1r8c9-k 1r4c9 1r4c9==1(r38c6)==1(r8c56) (same tag)

-1r7c5-B 1r3c5-l 1r8c5
-1r1c6-B 1r3c5-l 1r3c6 1r3c6==1r8c5

and six candidates are cleared

Code: Select all: #6r3c5 []5r5c2-5r1c7-B/2r1c7-a_2r3c5-A/6r3c5 []8r5c2/8r1c2-d_6r3c8-D/6r3c5 []6r5c2/6(r5c46)-o_6(r46c5)-O/6r3c5 #8r6c2 []6r5c8/6r3c8-D_8r1c2-d/8r6c2 []1r5c8/1r1c8-a_2r6c2-A/8r6c2 []8r5c8/8r9c8-J_1r9c8-j/1r1c8-a_2r6c2-A/8r6c2 #6r4c7 []6r5c8/6r4c7 []1r5c8/1r1c8-a_3r4c7-A/6r4c7 []8r5c8/8r9c8-J_1r9c8-j/1r1c8-a_3r4c7-A/6r4c7 #6r2c4 []6r5c8/6r3c8-D_6r2c7-d/6r2c4 []1r5c8/1r1c8-a_3r2c4-A/6r2c4 []8r5c8/8r9c8-J_1r9c8-j/1r1c8-a_3r2c4-A/6r2c4 #6r5c7 []7(r5c46)==7r3c5-p/2r3c5-A_2r1c7-a/5r1c7-B_5r5c7-b/6r5c7 []7r5c7/6r5c7 []7r5c3/7r1c3-c_7r1c4-C/8r1c4-D_6r2c7-d/6r5c7 #1r8c6 []7(r5c46)==7r3c5-p/1r3c5-l_1(r78c5)-L/1r8c6 []7r5c7/5r5c7-b_1r7c5-B/1r8c6 []7r5c3/1r5c3-G_1r4c3-g/1r4c9==1(r38c6)_1(r19c6)/1r8c6

Code: Select all: 6 8d9D 7c9C |7C8D9R 4 1B5b |12a5B 1a2A 3 4d8D 1 2A3a |2l3A89 25B68 689 |4D6d 7 5b9B 4D7C9R 2a3A 5 |3a679 1k2A7o 1679 |8 4d6D 1B9b ----------------------------------------------------------------- 789 4689 1g4679 |5 678 2 |13A47 13a4689h 178 3 5B68 1G67 |4e678 9 4E678 |15b7 168 2 5b789 2A46 2a4679 |1 678 3 |46D7 4689H 5B78 ----------------------------------------------------------------- 1b5B 3a4A 8 |2A467 1B5b 467 |9 2a3A 6i7I 1B9b 7 3A4a |24689 12l68 4689 |2A3a 5 1k6I8j 2 5b6f9 6F9f |789 3 1k5B789 |1I7i 1j8J 4

still
1r4c9==1(r3c6)==1(r8c5)
and now
-2r3c5-A 2r2c4-l 2r2c5
-2r7c4-A 2r2c4-l 2r8c4 2r8c4==2r2c5

from AHS/ALS r367c2 we have 4r4c2==6r6c2
both are cleared thru

Code: Select all: []5r5c2==5r1c7-B/2r1c7-a_2r6c2-A/6r6c2 []6r5c2/6r6c2 []8r5c2/8r1c2-d_6r6c7-D/6r6c2

Code: Select all: 6 8d9D 7c9C |7C8D9R 4 1B5b |12a5B 1a2A 3 4d8D 1 2A3a |2l3A89 25B68 689 |4D6d 7 5b9B 4D7C9R 2a3A 5 |3a679 1k2A7o 1679 |8 4d6D 1B9b ------------------------------------------------------------------ 789 689 1g4m679 |5 678 2 |13A47 13a4689h 178 3 5B68 1G67 |4e678 9 4E678 |15b7 168 2 5b789 2A4a 2a4679 |1 678 3 |46D7 4689H 5B78 ------------------------------------------------------------------ 1b5B 3a4A 8 |2A467 1B5b 467 |9 2a3A 6i7I 1B9b 7 3A4a |24689 12l68 4689 |2A3a 5 1k6I8j 2 5b6f9 6F9f |789 3 1k5B789 |1I7i 1j8J 4

still
1r4c9==1(r3c6)==1(r8c5)
2r8c4==2r2c5

Code: Select all: #k []4r4c3/4r6c2-a_2r3c5-A/1r3c5-k []4r4c7==7r6c7-ä/7r9c7-i_6r8c9-I/1r8c9-k []4r4c8/3r4c8-a_2r3c5-A/1r3c5-k #7r7c4 []4r4c3/4r6c2-a_2r7c4-A/7r7c4 []4r4c7==7r6c7-ä/7r9c7-i_7r7c9-I/7r7c4 []4r4c8/3r4c8-a_2r7c4-A/7r7c4 #1r4c7 []4r4c3/4r6c2-a_3r4c7-A/1r4c7 []4r4c7-ä/1r4c7 []4r4c8/3r4c8-a_3r4c7-A/1r4c7 #7r4c7 []4r4c3/4r6c2-a_3r4c7-A/7r4c7 []4r4c7-ä/7r4c7 []4r4c8/3r4c8-a_3r4c7-A/7r4c7

Code: Select all: 6 89 79 |789 4 1B5b |12a5B 1a2A 3 48 1 2A3a |2j3A89 25B68 689 |46 7 5b9B 479 2a3A 5 |3a679 2A7a 1b679 |8 46 1B9b ---------------------------------------------------------- 789 689 14679 |5 678 2 |3A4a 13a4689 1b78 3 5B68 167 |4678 9 4678 |15b7 168 2 5b789 2A4a 2a4679 |1 678 3 |467 4689 5B78 ---------------------------------------------------------- 1b5B 3a4A 8 |2A46 1B5b 467 |9 2a3A 67 1B9b 7 3A4a |24689 1b2j68 4689 |2A3a 5 68 2 5b69 69 |789 3 5B789 |17 18 4

[]jb Ba Aj []2r8c5-j/1r8c5-b_5r1c7-B/2r1c7-a_3r2c4-A/2r2c4-j

Code: Select all: 6 8d9D 7c9C |7C8D9N 4 1B5b |12a5B 1a2A 3 4d8D 1 2A3a |3A89 2a5B68 689 |4D6d 7 5b9B 4D7C9N 2a3A 5 |3a679 2A7a 1b679 |8 4d6D 1B9b -------------------------------------------------------------- 789 689 1h4j679 |5 678 2 |3A4a 13a4689i 1b78 3 5B68 1H67 |4e678 9 4E678 |15b7 168 2 5b789 2A4a 2a4679 |1 678 3 |46D7 4689I 5B78 -------------------------------------------------------------- 1b5B 3a4A 8 |2A46 1B5b 467f |9 2a3A 6f7F 1B9b 7 3A4a |2a4689 1b68 4689 |2A3a 5 6F8f 2 5b6g9 6G9g |7l89 3 5B789 |1F7f 1f8F 4

still
1r4c9==1(r3c6)==1(r8c5)
2r8c4==2r2c5

and now
-5r5c7-b 1r5c7 7r5c7
-1r4c9-b 1r5c7 1(r45c8) 7r5c7==1(r45c8)
followed by
-1r1c8-a 1(r45c8) 1r9c8-f
-7(r5c46)-a 7r5c7 7r5c3 7r5c3 is 'f'

-4r6c2-a 4r4c3-j 4r6c3
-4r4c7-a 4r4c3-j 4r4c8 4r4c8==4r6c3

a sub chain common to the three chains

[]fH hb []7r5c3-f/1r5c3-H_1r4c3-h/1r4c9-b

Code: Select all: []1r1c7/1r9c7-F_fb_5r1c7-B/1r1c7 []8r9c6/8r9c8-F_fb_5r9c6-B/8r9c6 []7r6c9/7r7c9-F_fb_5r6c9-B/7r6c9

Code: Select all: 6 8c9C 7b9B |7B8C9M 4 1a5A |2A5a 1A2a 3 4c8C 1 2a3A |3a89 2A5a68 689 |4C6c 7 5A9a 4C7B9M 2A3a 5 |3A679 2a7A 1A679 |8 4c6C 1a9A --------------------------------------------------------------- 789 689 1g4i679 |5 678 2 |3a4A 13A4689h 1A7e8 3 5a68 1G67 |4d678 9 4D678 |1e5A7 168 2 5A789 2a4A 2A4679 |1 678 3 |46C7 4689H 5a8A --------------------------------------------------------------- 1A5a 3A4a 8 |2a46 1a5A 467e |9 2A3a 6e7E 1a9A 7 3a4A |2A4689 1A68 4689 |2a3A 5 6E8e 2 5A6f9 6F9f |7k8e9 3 5a79 |1E7e 1e8E 4

5r2c5-a 2r2c5-A clearing 68(r2c5)

[]AC 4r4c7-A/4r2c7-C =5r5c2 =8r1c2 clearing 8r5c2

ending with two Nice loops

Code: Select all: 6 89 79 |789 4 1a5A |2A5a 1A2a 3 48 1 2a3A |3a89 2A5a 689 |46 7 5A9a 479 2A3a 5 |3A679 2a7A 1A679 |8 46 1a9A ---------------------------------------------------------- 789 689 14679 |5 678 2 |3a4A 13A4689 1A7e8 3 5a6A 167 |4678 9 4678 |1e5A7 168e 2 5A789 2a4A 2A4679 |1 678 3 |467 4689 5a8A ---------------------------------------------------------- 1A5a 3A4a 8 |2a46 1a5A 467e |9 2A3a 6e7E 1a9A 7 3a4A |2A4689 1A68E 4689 |2a3A 5 6E8e 2 5A69 69 |78e9 3 5a79 |1E7e 1e8E 4

[]em ME []8r9c4-e/8r8c5-m_8r5c8-M/8r9c8-E

[]AE eA []1r8c5-A/8r8c5-E_1r9c8-e/1r1c8-A

1 r1c6=1 B
Nothing special to the end

by **champagne** » Tue Nov 06, 2007 10:00 pm

example of level 3 puzzle in full tagging method

This puzzle N°52 in the top 1465 list is a good example for level 3
One advantage to choose a top1465 puzzle is that they are well known by most SUDOKU players, giving a better chance to compare with other methods.

It has been solved in 5.3 seconds by my PC.
NOTA: Again, time is given more to show than this excludes any recursive search of chains.

....3..715..4.2............2..6..4...38.7..............7..8..1.6..5..2..........._52

Start with level 2

Code: Select all: 489 2a4689 2A469 |8l9L 3 5b689 |5B689 7 1 5 1c68Å9 7 |4 1C69 2 |3689 3689 3689 1C3e89 689 3E69 |7 5z69 1c568é9 |5b689 2d4D 2D4d ---------------------------------------------------------------------- 2 159 1k59 |6 1O5W9ë 3f8F |4 3F8f 7 1O4g9 3 8 |12r9 7 14G5x9 |1n69 2569 2569 7 456h9 456H9 |123t8L9 2m4v59 134589 |1N38è9 23589 23589 ---------------------------------------------------------------------- 349 7 2i345j9 |2I39 8 346Á9 |369 1 345J69 6 148Æ9 1K3s49 |5 1P49 7 |2 3489 3489 1P348ç9 2A45Y89 23459 |1Q239 2M46â9 13469 |7 345à689 345689

Code: Select all: #4r9c6 []4r7c3/2r7c3-i_2r7c4-I/2r9c5-M_2r6c5-m/4r6c5-v_4(r89c5)-V/4r9c6 []4r7c1/4r5c1-g_4r5c6-G/4r9c6 []4r7c9/5r7c9-J_5r7c3

-j/2r7c3-i... as above
[]4r7c6/4r9c6

then we are forced to level 3 Here with only one loop, which means that all clearings done here are indirects.

We start with the clearing of 'O' =1r4c5 =1r5c1
£w is the tag for 3r2c7, £W is the tag for complementary sets 3(r2c89); 3(r79c7)
This is the logic applied by the solver

Code: Select all: #[]O/£W derived weak link 3r2c7 []3r6c7/1r6c7-N_1r5c7-n/1r5c1-O []3r7c7/3(r8c89)-S_3r8c3/1r8c3-K_1r4c3-k/1r4c5-O

Code: Select all: #[]O/3r2c7 derived weak link []8r2c2/1r2c2-c_1r2c5-C/1r4c5-O []8r6c7/1r6c7-N_1r5c7-n/1r5c1-O []8r2c7/3r2c7

#[]O/O []O/£W|#_3r2c7/O|# Nice loop applied to derived weak links.

We could say as well : 'O'
- forces 3r2C7
- 'O' and 3r2c7 can not be both in the solution
- so 'O' is out.

We have here the two tools for level 3 (in fact it is nearly the same) :

In a very similar way, are cleared at the same moment

Code: Select all: clear 3r8c3 #[]P/3r8c3 []8(r1c12)/8(r1c46)-É_8r3c6/1r3c6-c_1r2c5-C/1r8c5-P []8(r2c2r3c1)/3r3c1-e|*r2c2r3c1_3r3c3-E/3r8c3 []8r3c2/8(r13c1)-Ç_8r9c1-ç/1r9c1-P #[]p/3r8c3 1r8c5-P []1r2c5==1r3c1-C/3r3c1-e_3r3c3-E/3r8c3 []1r4c5-O/1r4c3-k_1r8c3-K/3r8c3

here 8(r2c2r3c1)/3r3c1-e|*r2c2r3c1 is a weak link directly generated by the "Almost cell" r2c2r3c1

Code: Select all: clear 'v' =4r7c6 =4r6c5 #[]v/6(r2c89) []6(r1c23)/4(r1c23)|*r1c23 _4r1c1/4r5c1-g_4r5c6-G/4r7c6-v []6r1c6/6(r23c5)-Â_6r9c5-â/2r9c5-M_2r6c5-m/4r6c5-v []6r1c7/6(r2c89) #[]v/6(r123c7) []6r7c6-Á/4r7c6-v []6r7c9/2r7c3-i|*r7c39_2r7c4-I/2r9c5-M_2r6c5-m/4r6c5-v []6r7c7/6(r123c7)

small difference here, 'v' is in weak link with candidates strongly linked.
two weak links out of "almost cells"

Code: Select all: clear ch=8r1c7 #[]I/8r1c7 []5r4c5==5(r6c23)-W/4(r6c23)-G|*r6c23_4r5c1-g/4r1c1_4(r1c23)-ï/8(r1c267)|*r1c2367_8(r1c14)/8r1c7 []5r3c5-z/5r1c6-b_5r1c7-B/8r1c7 []5r6c5/2r6c5-m_2r9c5-M/2r7c4-I #[]i/8r1c7 []4r7c6-v/4r5c6-G_... as above []4(r7c3r7c9)/2r7c3-i|*r7c3r7c9 []4r7c1/4r1c1_... as above

Code: Select all: clear ch=9r1c7 #[]I/9r1c7 []5r4c5==5(r6c23)-W/4(r6c23)-G|*r6c3_4r5c1-g/4r1c1_4(r1c23)-ï/9(r1c2367)|*r1c2367_9(r1c14)/9r1c7 []5r3c5/5r1c6-b_5r1c7-B/9r1c7 []5r6c5/2r6c5-m_2r9c5-M/2r7c4-I #[]i/9r1c7 []4r7c6-v/4r5c6-G_4r5c1-g/4r1c1_.. asz above []4(r7c39)/2r7c3-i|*r7c3r7c9 []4r7c1/4r1c1_.. as above

In that example, all cases have generated weak links ending (in the process) with a two terms chain..
What is typical to level 3 is generation of weak links.
The end can be a longer chain, but as well a choice in weak link with the target.[/color]

Following steps are mostly in level 2.

Code: Select all: 489 2a4689 2A469 |8m9M 3 5b689 |5B6b 7 1 5 1c68Æ9 7 |4 1C69 2 |3q689 3689 3689 1C3e89 689 3E69 |7 5z69 1c568Å9 |5b68Æ9ì 2d4D 2D4d ----------------------------------------------------------------------- 2 1f59 1F59 |6 5W9w 3g8G |4 3G8g 7 4h9H 3 8 |12s9 7 14H5x9 |1p69 2569 2569 7 456i9 456I9 |123u8M9 2n59 134h589 |1P3Q8é9 23589 23589 ----------------------------------------------------------------------- 349 7 2j345k9 |2J39 8 36Á9 |6Î9 1 4V5K69 6 148Ç9 1f49 |5 1c4o9 7 |2 3l489 3L489 1c348è9 2A45Y89 23459 |1R239 2N4O6â9 1369 |7 45à689 45689

Code: Select all: []3r3c1-e/3r7c1_3(r39c1)-ÿ/8(r39c1)-ý|*r39c1_8r1c1/8(r1c46)-å_8r3c6-Å/1r3c6-c_1r3c1-C/3r3c1-e []3r7c6/3r7c1_3(r39c1)-ÿ/8(r39c1)-ý|*r39c1_8r1c1/8r1c4-m_8r6c4-M/8r4c6-G_3r4c6-g/3r7c6 []9(r379c1)/8(r39c1)-ý|*r379c1_8r1c1/8r2c2-Æ_8(r26c7)-æ/1r6c7-P|*r26c7_1(r6c46)-p/4r6c6-h|*r4c646_9r5c1-H/9(r379c1) []2r6c4/8r6c4-M_8r1c4-m/8r1c1_8(r39c1)-ý/3(r39c1)-ÿ|*r39c1 _3r7c1/3(r7c46)-T_3(r7c13)-t/2r7c3-j|*r7c139_2r7c4-J/2r6c4 []9r2c5/9r4c5-w_9(r4c23)-W/9r5c1-H_4r6c6-h/1(r6c46)-p|*r4c6r6c46_1r6c7-P/8(r26c7)-æ|*r26c7_8r2c2-Æ/1r2c2-c_1r2c5-C/9r2c5 []9(r6c46)/9r4c5-w_9(r4c23)-W/9r5c1-H_4r6c6-h/9(r6c46)|*r4c6r6c46 []9r6c5/9r4c5-w_9(r4c23)-W/9r5c1-H_4r5c1-h/4r7c1_4(r7c39)-£O/2r7c3-j|*r7c39 _2r7c4-J/2r9c5-N_2r6c5-n/9r6c5

7 r3c3=3 B

Code: Select all: 489g 2a4689 2A46h9 |8m9M 3 5b689 |5B6b 7 1 5 1c68Â9È 7 |4 1C6c 2 |3q689 3689 3689 1C8c 689 3 |7 5w69 1c568Á9 |5b68Â9È 2d4D 2D4d ------------------------------------------------------------------- 2 1e59 1E59 |6 5T9t 3f8F |4 3F8f 7 4g9G 3 8 |12N9 7 14G5u9 |1p69 2569 2569 7 456h9 456H9 |13s8M 2n5N 134g58 |1P3Q8å9 23589 23589 ------------------------------------------------------------------- 3I4i 7 2i45j |2I3i 8 6k9K |6K9k 1 4j5J 6 148Ã9 1e49 |5 1c4o9 7 |2 3l489 3L489 1c3i48ä 2A45V89 2459 |1R239 2N4O6Y9 13s69 |7 45X689 45689

Code: Select all: []5r6c3/5(r4c23)-t_9(r4c23)-T/9r5c1-G_9r1c1-g/9(r1c46)-ç_9(r1c123)-Ç/6(r1c23)|*r1c123_6(r23c2)-£C/6r6c2-h_6r6c3-H/5r6c3 []9r6c3/6r6c3-H_6r6c2-h/6(r23c2)-£C_6(r1c23)/9(r1c123)-Ç|*r1c123_9(r1c46)-ç/9r1c1-g_9r5c1-G/9r6c3

Code: Select all: 489g 2a4689 2A46h9 |8m9M 3 5b689 |5B6b 7 1 5 1c68Â9È 7 |4 1C6c 2 |3q689 3689 3689 1C8c 689 3 |7 5w69 1c568Á9 |5b68Â9È 2d4D 2D4d ------------------------------------------------------------------- 2 1e59 1E5V9 |6 5T9t 3f8F |4 3F8f 7 4g9G 3 8 |12N9 7 14G5u9 |1p69 2569 2569 7 45T6h9É 4h6H |13s8M 2n5N 134g58 |1P3Q8å9 23589 23589 ------------------------------------------------------------------- 3I4i 7 2i45j |2I3i 8 6k9K |6K9k 1 4j5J 6 148Ã9 1e49 |5 1c4o9 7 |2 3l489 3L489 1c3i48ä 2A45V89 2459 |1R239 2N4O6Y9 13s69 |7 45X689 45689

17 sequences with small variations (each one is supposed to be done independantly)

Code: Select all: #9(r89c5) []5r1c6==5r3c7-b/1r6c7-P|*r236c7_1(r6c46)-p/4r6c6-g|*r4c6r6c46_9r5c1-G/9(r4c23)-T_9r4c5-t/9(r89c5) []8r1c6/8r1c4-m_9r1c4-M/9r1c1-g_..as above []9r1c6/9r1c1-g_.. as above []6r1c6/6r7c6-k_9r7c6-K/9(r89c5) #9r5c6 []5r1c6==5r3c7-b/1r6c7-P|*r236c7_1(r6c46)-p/4r6c6-g|*r4c6r6c46_4r5c6-G/9r5c6 []8r1c6/8r1c4-m_9r1c4-M/9r1c1-g_4r5c6-G/9r5c6 []9r1c6/9r5c6 []6r1c6/6r7c6-k_9r7c6-K/9r5c6 #c []5r3c5==5(r5c6r6c6)-w/1(r5c6r6c46)-£O|*r45c6r6c46_1r5c4/1r5c7-p_1r6c7-P/6(r123c7)|*r1236c7_6(r2c89)-Z/6r2c5-c []6r3c5/6r2c5-c []9r3c5/9r4c5-t_9(r4c23)-T/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_.. as above #6r3c7 []5r3c5==5(r56c6)-w/1(r5c6r6c46)-£O|*r456c6r6c4 _1r5c4/1r5c7-p_1r6c7-P/6(r123c7)|*r1236c7_6(r2c89)-Z/6r3c7 []6r3c5/6r3c7 []9r3c5/9r4c5-t_9(r4c2r4c3)-T/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_.. as above #8r6c7 []2r5c4==2r9c5-N/2r7c4-I_4r7c1-i/4(r15c1)_8r1c1/8r1c4-m_8r6c4-M/8r6c7 []1r5c4/1r5c7-p_1r6c7-P/8r6c7 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_1r6c7-P/8r6c7 #1r6c4 []2r5c4==2r9c5-N/2r7c4-I_4r7c1-i/4(r1c1r5c1)_8r1c1/8r1c4-m_8r6c4-M/1r6c4 []1r5c4/1r6c4 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_1r6c7-P/1r6c4 #4(r8c89) []2r5c4==2r9c5-N/4r9c5-O_4r8c5-o/4(r8c89) []1r5c4/1r5c7-p_1r6c7-P/8(r26c7)-â|*r26c7_8r2c2-Â/8r8c2-Ã_8(r8c89)-ã/4(r8c89)|*r8c89 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_.. as above #4r9c1 []2r5c4==2r9c5-N/2r7c4-I_3r9c1-i/4r9c1 []1r5c4/1r5c7-p_1r6c7-P/8(r26c7)-â|*r26c7_8r2c2-Â/8(r13c1)-Ä_8r9c1-ä/4r9c1 []9r5c4/9r5c1-G_4r5c1-g/4r9c1 #8r1c2 []2r5c4==2r9c5-N/2r9c2-A_2r1c2-a/8r1c2 []1r5c4/1r5c7-p_1r6c7-P/8(r26c7)-â|*r26c7_8r2c2-Â/8r1c2 []9r5c4/9r1c4-M_8r1c4-m/8r1c2 #6r2c7 []2r5c4==2r9c5-N/2r7c4-I_3r9c1-i/8r9c1-ä_8(r13c1)-Ä/8r2c2-Â_8(r26c7)-â/6r2c7|*r26c7 []1r5c4/1r5c7-p_1r6c7-P/6r2c7|*r26c7 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_.. as above #8r3c2 []2r5c4==2r9c5-N/2r7c4-I_3r9c1-i/8r9c1-ä_8(r13c1)-Ä/8r3c2 []1r5c4/1r5c7-p_1r6c7-P/8(r2c7r6c7)-â|*r26c7_8r2c2-Â/8r3c2 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_..as above #2r9c4 []2r5c4==2r9c5-N/2r9c4 []1r5c4/1r5c7-p_1r6c7-P/8(r26c7)-â|*r26c7_8r2c2-Â/8(r13c1)-Ä_8r9c1-ä/2r9c4|*r9c146 []9r5c4/9r5c1-G_4r5c1-g/4r7c1-i_2r7c4-I/2r9c4 #9r9c4 []2r5c4==2r9c5-N/2r7c4-I_4r7c1-i/4(r15c1)_8r1c1/8r1c4-m_9r1c4-M/9r9c4 []1r5c4/1r5c7-p_1r6c7-P/8(r26c7)-â|*r26c7_8r2c2-Â/8(r13c1)-Ä_8r9c1-ä/9(r9c46)|*r9c146 _9(r9c23589)/9r9c4 []9r5c4/9r9c4 #9(r26c7) []2r5c4==2r9c5-N/2r7c4-I_3r9c1-i/8r9c1-ä_8(r13c1)-Ä/8r2c2-Â_8(r26c7)-â/9(r26c7)|*r26c7 []1r5c4/1r5c7-p_1r6c7-P/9(r26c7)|*r26c7 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_.. as above #8r9c8 []2r5c4==2r9c5-N/2r7c4-I_4r7c1-i/4(r15c1)_8r1c1/8r1c4-m_8r6c4-M/8r4c6-F_8r4c8-f/8r9c8 []1r5c4/1r5c7-p_1r6c7-P/8(r26c7)-â|*r26c7_8r2c2-Â/8r8c2-Ã_8(r8c89)-ã/8r9c8 []9r5c4/9r5c1-G_4r6c6-g/1(r6c46)-p|*r4c6r6c46_..as above #u []9r1c4-M/9r1c1-g_4r5c6-G/5r5c6-u []9r9c4/9(r9c23589)_9(r9c46)/8r9c1-ä|*r9c146_8(r13c1)-Ä/8r2c2-Â_8(r26c7)-â/1r6c7-P|*r26c7_1(r6c46)-p/4r6c6-g|*r4c6r6c46_4r5c6-G/5r5c6-u []9r5c4/9r4c5-t_5r4c5-T/5r5c6-u #É []9r1c4-M/9r1c1-g_9r5c1-G/9(r5c789)-É []9r9c4/9(r9c23589)_9(r9c46)/8r9c1-ä|*r9c146 _8(r13c1)-Ä/8r2c2-Â_8(r26c7)-â/1r6c7-P|*r26c7 _1(r6c46)-p/4r6c6-g|*r4c6r6c46_9r5c1-G/9(r5c789)-É []9r5c4/9(r5c789)-É

8 r2c5=1 B
9 r3c1=1 B
10 r8c5=4

ending in level 2 with two Nice loops

Code: Select all: 48D9c 2a46 2A46h |8c9C 3 5b6p8 |5B6b 7 1 5 6b8d9K 7 |4 1 2 |3d8D 369 369 1 6p9P 3 |7 5o6D9c 568D |5b8d9K 2e4E 2E4e ----------------------------------------------------------------- 2 1f5F 1F5N9c |6 5c9C 3C8c |4 3c8C 7 4c9C 3 8 |1M2D9c 7 1c4C |1D6d 25g6 25G6 7 45c6h 4h6H |3c8C 2d5D 1D4c5o |1d3D 2389i 238R9I ----------------------------------------------------------------- 3d4D 7 2D45j |2d3D 8 6k9K |6K9k 1 4j5J 6 1F8q9 1f9F |5 4 7 |2 3l8R9 3L89 3D8d 2A45N89 2459 |1m3M 2D6d 1M3c69k |7 4e5G69 4568q9

in |* r56c6 we can see that 'o' is 'M'. We have consequently the two Nice loops

[]Km MD dk []9r3c7-K/5(r3c67)-m|*r3c6789_1r9c6-M/1r6c6-D_6r9c5-d/6r7c6-k

[]pk Km MD dk KP []6r1c6-p/6r7c6-k_Km MD dk_9r3c7-K/9r3c2-P

a lot of clearings, the puzzle is finished

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