The New Sudoku Players' Forum

by **DonM** » Tue Jan 07, 2014 3:30 am

Welcome to the occasional/sporadic 'Suggest A Move' (SAM) series, consisting of of a puzzle at a point where there are some interesting patterns and manual solvers are invited to come up with any interesting patterns/chains they discover. There is no hidden surprise or backdoor (at least not that I know of) or some other trick awaiting and I haven't made any exhaustive search for all possible interesting patterns. It is simply an interesting puzzle.

SAM#5 is a particularly interesting puzzle with rather amazing 'depth'. There is a lot to be learned from a puzzle like this. Newer manual solvers shouldn't have any trouble finding some interesting initial patterns. More advanced solvers will find it a nice challenge. As the 'Suggest A Move' name implies, please allow the first day of the puzzle's release for initial 'moves' (ie. avoid giving full solutions). After that, the puzzle is open to a full solution. (This is not a 'single-stepper'!)

[Note: SSTS means Simple Sudoku Techniques Set. Post-SSTS means that the original puzzle was solved by Simple Sudoku to the point where there is 'No Hint Available'. (This means that some basic patterns that solvers may be used to finding in simpler puzzles have already been solved.) ER means Explainer Rating whereby Sudoku Explainer assigns a general difficulty value to the puzzle.]

Code: Select all: *-----------* |4..|...|..3| |..5|...|2..| |.8.|3..|69.| |---+---+---| |7..|.49|...| |52.|.8.|.79| |...|27.|..6| |---+---+---| |.73|..1|.6.| |..9|...|5..| |8..|...|..2| *-----------*

Code: Select all: SAM#5: ER=7.3 at the Post-SSTS position: *--------------------------------------------------------------------* | 4 69 27 | 6789 1269 25678 | 178 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 25 247 | 6 9 457 | |----------------------+----------------------+----------------------| | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 146 | 16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | |----------------------+----------------------+----------------------| | 2 7 3 | 4589 59 1 | 489 6 48 | | 6 14 9 | 478 23 2478 | 5 1348 1478 | | 8 5 14 | 4679 369 467 | 13479 134 2 | *--------------------------------------------------------------------*

by **David P Bird** » Tue Jan 07, 2014 11:52 am

Withdrawn - mistook a chain for a continuous loop.

by **JC Van Hay** » Tue Jan 07, 2014 1:29 pm

Manual "Basics" show how to tackle the puzzle. Here is a possible first move (somewhat simplifying a complex one-stepper):

Code: Select all: +---------------+---------------------+------------------------+ | 4 69 27 | 6789 1269 25678 | 178 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 2(5) 27(4) | 6 9 7(45) | +---------------+---------------------+------------------------+ | 7 13 168 | 16(5) 4 9 | 138 2 158 | | 5 2 146 | 16 8 (36) | 134 7 9 | | 39 1349 148 | 2 7 (35) | 1348 13458 6 | +---------------+---------------------+------------------------+ | 2 7 3 | 489(5) (59) 1 | 48(9) 6 48 | | 6 14 9 | 478 23 278 | 5 1348 1478 | | 8 5 14 | 4679 369 (467) | 134-7(9) 134 2 | +---------------+---------------------+------------------------+

#1. Chain [8] : 7r9c6=*[5r7c4=5r4c4-(5=36)r56c6-(6=*4)r9c7-4r3c6=(4-5)r3c9=5r3c5]-(5=9)r7c5-9r7c7=9r9c7 :=> -7r9c7; 4 Singles
or
Kraken Cell (467)r9c6 :=> [9r9c7==7r9c6]-7r9c7; 4 Singles

4r9c6-4r3c6=(4-5)r3c9=5r3c5-(5=9)r7c5-9r7c7=9r9c7
||
6r9c6-(6=35)r56c6-5r4c4=5r7c4-(5=9)r7c5-9r7c7=9r9c7
||
7r9c6

by **ArkieTech** » Tue Jan 07, 2014 4:27 pm

Code: Select all: *--------------------------------------------------------------------* | 4 69 27 | 6789 1269 25678 | 178 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 25 247 | 6 9 457 | |----------------------+----------------------+----------------------| | 7 13 *68-1 |*156 4 9 | 138 2 158 | | 5 2 *146 |*16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | |----------------------+----------------------+----------------------| | 2 7 3 | 4589 59 1 | 489 6 48 | | 6 14 9 | 478 23 2478 | 5 1348 1478 | | 8 5 14 | 4679 369 467 | 13479 134 2 | *--------------------------------------------------------------------* Hidden Rectangle 16r45c34 => -1r4c3

by **Luke** » Tue Jan 07, 2014 7:07 pm

Code: Select all: *--------------------------------------------------------------------* | 4 69 27 | 6789 1269 25678 | 178 158 3 | |*39 *369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 25 247 | 6 9 457 | |----------------------+----------------------+----------------------| | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 146 | 16 8 36 | 134 7 9 | |*39 *1349 148 | 2 7 35 | 1348 13458 6 | |----------------------+----------------------+----------------------| | 2 7 3 | 4589 59 1 | 489 6 48 | | 6 14 9 | 478 23 2478 | 5 1348 1478 | | 8 5 14 | 4679 369 467 | 13479 134 2 | *--------------------------------------------------------------------*

Another hidden rectangle (39)r26c12 ==>r6c2<>9

by **DonM** » Wed Jan 08, 2014 5:19 pm

SAM#5 is now open to full solving.

by **JC Van Hay** » Wed Jan 08, 2014 9:45 pm

Code: Select all: +-----------------+--------------------+----------------------+ | 4 69 2 | 689 169 568 | 7 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 148 | | 1 8 7 | 3 -5(2) 24 | 6 9 (45) | +-----------------+--------------------+----------------------+ | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 16(4) | 16 8 36 | 13(4) 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | +-----------------+--------------------+----------------------+ | 2 7 3 | 4589 59 1 | 89(4) 6 8(4) | | 6 4(1) 9 | 48 (23) 28 | 5 8(134) 7 | | 8 5 (14) | 4679 369 467 | 139(4) 13(4) 2 | +-----------------+--------------------+----------------------+

#2. Kraken 4B9 :=> [2r3c5==5r3c9]-5r3c5; ste

EmptyRectangle(4r5c37,4r79c7r9c8)-(4=1)r9c3-1r8c2=(1-3)r8c8=(3-2)r8c5=2r3c5
||
4r8c8-3r8c8=(3-2)r8c5=2r3c5
||
4r7c9-(4=5)r3c9

by **DonM** » Wed Jan 08, 2014 11:35 pm

JC, could you edit your post above and put Step#1 (from further above) in it? The solution is really worth seeing altogether. Thanks.

by **JC Van Hay** » Thu Jan 09, 2014 8:07 am

The complete solution.

#0. "Basics" = N-tuples + NxM-Fishes, N=1,2,3, ...

5 Singles; LC(1r45c4,3r56c6)-1r12c4,3r89c6
Jellyfish(4R357C2) : r8c2=r6c2-r5c3=r5c6-r7c7=*Skyscraper[r7c4=*r7c9-r3c9=r3c6] :=> -4r8c6
FXWing(5C68)-5r1c45=XWing(5R16)-5r3c5 [=> only 2 solutions for the digit 5]
HP(27-6)r13c3=LC(6r12c2)-6r4c2

Note : r6c6=5->4r5c3=4 AND r9c3=4 !? :=> r6c6=3, solving the 5s; ste

As the analysis of the puzzle shows that r9c7<>7 [=> 4 Singles] (the other implied placements in stack 1 playing no role) :

Code: Select all: +---------------+---------------------+------------------------+ | 4 69 27 | 6789 1269 25678 | 178 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 2(5) 27(4) | 6 9 7(45) | +---------------+---------------------+------------------------+ | 7 13 168 | 16(5) 4 9 | 138 2 158 | | 5 2 146 | 16 8 (36) | 134 7 9 | | 39 1349 148 | 2 7 (35) | 1348 13458 6 | +---------------+---------------------+------------------------+ | 2 7 3 | 489(5) (59) 1 | 48(9) 6 48 | | 6 14 9 | 478 23 278 | 5 1348 1478 | | 8 5 14 | 4679 369 (467) | 134-7(9) 134 2 | +---------------+---------------------+------------------------+

#1. Chain [8] : 7r9c6=*[5r7c4=5r4c4-(5=36)r56c6-(6=*4)r9c6-4r3c6=(4-5)r3c9=5r3c5]-(5=9)r7c5-9r7c7=9r9c7 :=> -7r9c7; 4 Singles
or
Kraken Cell (467)r9c6 :=> [9r9c7==7r9c6]-7r9c7; 4 Singles

4r9c6-4r3c6=(4-5)r3c9=5r3c5-(5=9)r7c5-9r7c7=9r9c7
||
6r9c6-(6=35)r56c6-5r4c4=5r7c4-(5=9)r7c5-9r7c7=9r9c7
||
7r9c6

Solving the 5s :

Code: Select all: +-----------------+--------------------+----------------------+ | 4 69 2 | 689 169 568 | 7 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 148 | | 1 8 7 | 3 -5(2) 24 | 6 9 (45) | +-----------------+--------------------+----------------------+ | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 16(4) | 16 8 36 | 13(4) 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | +-----------------+--------------------+----------------------+ | 2 7 3 | 4589 59 1 | 89(4) 6 8(4) | | 6 4(1) 9 | 48 (23) 28 | 5 8(134) 7 | | 8 5 (14) | 4679 369 467 | 139(4) 13(4) 2 | +-----------------+--------------------+----------------------+

#3. Chain [7] : (5=4)r3c9-4r7c9=*[1r8c8=1r8c2-(1=4)r9c3-EmptyRectangle(4r5c37,4r79c7r9c8*)=4r8c8]-3r8c8=(3-2)r8c5=2r3c5 :=> -5r3c5; ste
or
Kraken 4B9 :=> [2r3c5==5r3c9]-5r3c5; ste

EmptyRectangle(4r5c37,4r79c7r9c8)-(4=1)r9c3-1r8c2=(1-3)r8c8=(3-2)r8c5=2r3c5
||
4r8c8-3r8c8=(3-2)r8c5=2r3c5
||
4r7c9-(4=5)r3c9

by **blue** » Thu Jan 09, 2014 11:28 am

Hi JC,

Nice solution :!:

Here's an alternate ending.
(I'm trying to mimic your style here ...)

Code: Select all: +---------------+---------------------+------------------+ | 4 69 2 | 689 19(6) (568) | 7 158 3 | | 39 369 5 | 46789 19(6) 4678 | 2 148 148 | | 1 8 7 | 3 (25) 24 | 6 9 45 | +---------------+---------------------+------------------+ | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 146 | 16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | +---------------+---------------------+------------------+ | 2 7 3 | 4589 59 1 | 489 6 48 | | 6 14 9 | 48 -2(3) (28) | 5 1348 7 | | 8 5 14 | 4679 9(36) 467 | 1349 134 2 | +---------------+---------------------+------------------+

#3. Chain[5] : 3r8c5=(3-6)r9c5=r12c5-6r1c6=*XYWing[(2=8)r8c6-(8=*5)r1c6-(5=2)r3c5] :=> -2r8c5; ste
or
Kraken cell (568)r1c6 :=> [3r8c5==2r8c6==2r3c5]-2r8c5; ste

6r1c6 - r12c5 = (6-3)r9c5 = 3r8c5
||
8r1c6 - (8=2)r8c6
||
5r1c6 - (5=2)r3c5

by **eleven** » Thu Jan 09, 2014 11:56 am

[Edit:] Ah, i missed blues' nice almost xy-wing !

Nice step, JC (almost almost empty rectangle).

However, to find such a solution, it looks easier to me to follow this chain and observe, that it kills all 4's in box 9.

Code: Select all: 4r3c9 -> 2r3c6->8r8c6 -> 4r8c4 -> 4r9c3 -> 4r5c7 r7c9 r8c8 r9c78 r7c7 <> 4 => contradiction, r3c9<>4

Btw, also the 1 in r5c3 is killed by the UR 16r45c34, but the strong ling for 1 then in c3 was no help for me.

by **DonM** » Thu Jan 09, 2014 8:33 pm

JC Van Hay wrote:
Code: Select all
+-----------------+--------------------+----------------------+ | 4 69 2 | 689 169 568 | 7 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 148 | | 1 8 7 | 3 -5(2) 24 | 6 9 (45) | +-----------------+--------------------+----------------------+ | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 16(4) | 16 8 36 | 13(4) 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | +-----------------+--------------------+----------------------+ | 2 7 3 | 4589 59 1 | 89(4) 6 8(4) | | 6 4(1) 9 | 48 (23) 28 | 5 8(134) 7 | | 8 5 (14) | 4679 369 467 | 139(4) 13(4) 2 | +-----------------+--------------------+----------------------+

#3. Kraken 4B9 :=> [2r3c5==5r3c9]-5r3c5; ste

EmptyRectangle(4r5c37 ,4r79c7r9c8) -(4=1)r9c3-1r8c2=(1-3)r8c8=(3-2)r8c5=2r3c5
||
4r8c8-3r8c8=(3-2)r8c5=2r3c5
||
4r7c9-(4=5)r3c9

JC's move here is very clever (and the notation makes it about as easy to understand as possible). You won't find an almost-almost Empty Rectangle that often, because you have to both know and find the pattern and then you have to apply it. But the concept behind it is relatively (though deceptively) simple- so for those who may not be sure how it works:

The core of the move is based on the fact that if 4r8c8 and 4r7c9 were not true then the existence of the Empty Rectangle pattern would eliminate 4r9c3 which, when you follow the inferences, eliminates 5r3c5 and places 2r3c5 and the puzzle falls apart.

JC first declares the empty rectangle pattern: (4r5c37 ,4r79c7r9c8). Take a look at the conjugate pair: 4r5c37: if 4r5c3 were true then 4r9c3 is eliminated. But if 4r5c7 is true then 4r79c7 in box 9 would be eliminated and, now, if 4r9c3 were also true then 4r79c7 would be eliminated which would mean that, again if 4r8c8 and 4r7c9 weren't true, then box 9 would be emptied of 4s which is impossible. Thus, ordinarily, no matter whether 4r5c3 is true or 4r5c7 is true, 4r9c3 could be eliminated.

But to make this work, there is still 4r8c8 and 4r7c9 to deal with: JC's chains show that if you assume that 4r8c8 and 4r7c9 are not true, then if you follow the inferences, 5r3c5 can be eliminated and 2r3c5 placed. Thus, all three premises, the ER pattern, not 4r8c8 and not 4r8c8 all result in support of the elimination of 5r3c5 and the placement of 2r3c5.

by **daj95376** » Thu Jan 09, 2014 8:41 pm

_

Not a manual solution ... that starts off with an almost cannibalistic Kraken Cell. (at least, one way to view it!)

Code: Select all: +-----------------------------------------------------------------------+ | 4 69 27 | 6789 1269 25678 | 178 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 25 247 | 6 9 457 | |-----------------------+-----------------------+-----------------------| | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 146 | 16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | |-----------------------+-----------------------+-----------------------| | 2 7 3 | 4589 59 1 | 489 6 48 | | 6 14 9 | 478 23 2478 | 5 1348 1478 | | 8 5 14 | 4679 369 467 | 13479 134 2 | +-----------------------------------------------------------------------+ # 102 eliminations remain JC's step as a discontinuous (network) loop: (7)r9c7 - (9)r9c7 = r7c7 - (9=5)r7c5 - r3c5 = (5-4)r3c9 = r3c6 - (4)r9c6 \ - r7c4 = r4c4 - (35=6)r56c6 - (6)r9c6 = (7)r9c6 - (7)r9c7 Three additional discontinuous loops: (1)r9c7 - (1=4)r9c3 - r5c3 = (4-3)r5c7 = r5c6 - (3=5)r6c6 - r1c6 = r3c5 - (5=9)r7c5 - r7c7 = (9-1)r9c7 (3)r9c7 - (3)r5c7 = r5c6 - (3=5)r6c6 - r1c6 = r3c5 - (5=9)r7c5 - r7c7 = (9-3)r9c7 (4)r9c7 - (4)r7c79 = (4-5)r7c4 = r4c4 - r4c9 = (5-4)r6c8 = (4)r56c7 - ( 4)r9c7 (9)r9c7 resulting Single

Code: Select all: +-----------------------------------------------------------------------+ | 4 69 27 | 6789 1269 25678 | 178 158 3 | | 39 369 5 | 46789 169 4678 | 2 148 1478 | | 1 8 27 | 3 25 247 | 6 9 457 | |-----------------------+-----------------------+-----------------------| | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 146 | 16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | |-----------------------+-----------------------+-----------------------| | 2 7 3 | 4589 59 1 | 489 6 48 | | 6 14 9 | 478 23 2478 | 5 1348 1478 | | 8 5 14 | 4679 369 467 | 9 134 2 | +-----------------------------------------------------------------------+ # 98 eliminations remain Basics

Code: Select all: +--------------------------------------------------------------+ | 4 69 2 | 689 1 568 | 7 58 3 | | 39 369 5 | 478 69 478 | 2 48 1 | | 1 8 7 | 3 25 24 | 6 9 45 | |--------------------+--------------------+--------------------| | 7 13 168 | 156 4 9 | 138 2 58 | | 5 2 146 | 16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 45 6 | |--------------------+--------------------+--------------------| | 2 7 3 | 59 59 1 | 48 6 48 | | 6 14 9 | 48 23 248 | 5 13 7 | | 8 5 14 | 467 36 467 | 9 13 2 | +--------------------------------------------------------------+ # 60 eliminations remain r45c34 <16> UR via s-link <> 1 r4c3 sufficient -- 1 of 3 r45c34 <16> UR via s-link <> 1 r5c3 sufficient -- 2 of 3 r46c37 <18> UR via s-link + (1=3)r4c2 <> 1 r6c3 sufficient -- 3 of 3

by **DonM** » Fri Jan 10, 2014 5:47 am

Just for giggles, here is a ridiculously long non-net-based solution. I don't think I've ever come across a puzzle of this relatively lower difficulty rating that had all sorts of non-net based moves available, but which just wouldn't die.

Ordinarily, with puzzles like this, most chains/moves will result in useful exclusions and/or placements. This thing not only didn't follow those rules, it also offered up a number of exclusions and even a few placements that led nowhere. I'll post a net-based solution later.

The final move here is of some interest since a BW pattern offers up a simple SIS.

1. (7)r3c3=r1c3-r1c7=(7-9)r9c7=r7c7-(9=5)r7c5-(5=2)r3c5 => r3c3<>2=7, r1c3=2
2. (5)r1c6=r3c5-(5=9)r7c5-r7c7=(9-7)r9c7=(7)r1c7 => r1c6<>7
-> Kite: (7)r1c47, (7)r28c9 => r8c4<>7
3. (6=3)r5c6-(3=5)r6c6-r4c4=r7c4-(5=9)r7c5-als(9=16)r12c5 => r12c6<>6
4. (8=4)r8c4-r8c2=r6c2-als(4=16)r5c34-(6=3)r5c6-(3=5)r6c6-(5=8)r1c6 => r12c4<>8, r8c6<>8
5. (4)r8c2=r6c2-als(4=16)r5c34-(6)r5c6=grp(6)r45c4-als(6=479)r129c4 => r8c4<>4=8
6. (4)r8c2=r6c2-r5c3=(4-3)r5c7=r5c6-(3=5)r6c6-r4c4=(4-5)r7c4=grp(4)r7c79 => r8c89<>4
7. (7)r8c9=(7-2)r8c6=r3c6-(2=5)r3c5-r7c5=(5-4)r7c4=grp(4)r7c79-als(4=13)r89c8 => r8c9<>1=7, r1c7=7
-> np(16)r1c24: r1c5=1, np(24)r38c6: r29c6<>4
8. (1=3)r8c8-(2=3)r5c5-(2=4)r8c6-r4c4=grp(4)r7c79-als(4=13)r89c8 => r2c8<>1 -> r2c9=1

A 5-sided pattern of conjugate 4s is impossible so either (4)r7c7 or (4)r9c4 have to be true.

9. Broken Wing (aka Guardian) pattern:(4)r2c48, (4)r37c9, (4)r7c4, sis: (4)r7c7=(4)r9c4
(4)r5c3=r5c7-[(4)r7c7=(4)r9c4] => r9c3<>4=1
stte

by **JC Van Hay** » Fri Jan 10, 2014 11:49 am

Thanks to Blue's post, here is a short complex one-stepper using only 10 SIS:

Code: Select all: +----------------+------------------------+------------------------+ | 4 69 (27) | 6789 19(26) (25678) | 18(7) 158 3 | | 39 369 5 | 46789 19(6) 4678 | 2 148 148(7) | | 1 8 27 | 3 -5(2) 247 | 6 9 45(7) | +----------------+------------------------+------------------------+ | 7 13 168 | 156 4 9 | 138 2 158 | | 5 2 146 | 16 8 36 | 134 7 9 | | 39 1349 148 | 2 7 35 | 1348 13458 6 | +----------------+------------------------+------------------------+ | 2 7 3 | 4589 (59) 1 | 48(9) 6 48 | | 6 14 9 | 478 (23) (278) | 5 1348 148(7) | | 8 5 14 | 4679 9(36) 467 | 134(79) 134 2 | +----------------+------------------------+------------------------+

Chain[10] : Kraken Cell (25678)r1c6 :=> [5r7c5==2r3c5==5r1c6]-5r3c5

The Forbidding Matrix :

Code: Select all: 5r3c5 5r7c5=9r7c5 9r7c7=9r9c7 7r9c7=7r1c7 7r23c9=7r8c9 7r1c3========2r1c3 2r3c5==========================2r1c5=2r8c5 7r8c6=======2r8c6=8r8c6 3r8c5=======3r9c5 6r9c5=6r12c5 5r1c6=============7r1c6========2r1c6=======8r1c6=======6r1c6

Rows 2->7 allow to write the following derived SIS :

5r7c5==7r1c7
5r7c5==7r8c9
5r7c5==2r1c3
[5r7c5==2r3c5]=2r8c5

Reading the FM from the last row :

2r1c6-2r1c3==5r7c5
||
5r1c6
||
6r1c6-6r12c5=(6-3)r9c5=3r8c5-2r8c5=[2r3c5==5r7c5]
||
7r1c6-7r1c7==5r7c5
||
8r1c6-8r8c6=*[5r7c5==7r8c9-(7=*2)r8c6-2r8c5=[2r3c5==5r7c5]]

As the analysis of the puzzle shows r1c7=7, the FM can be simplified by removing rows 2->6 and columns 2->6 :

Kraken Cell (568)r1c6 :=> [2r3c5==5r1c6]-5r3c5

Code: Select all: 5r3c5 2r3c5=2r8c5 2r8c6=8r8c6 3r8c5=======3r9c5 6r9c5=6r12c5 5r1c6=======8r1c6=======6r1c6

5r1c6
||
6r1c6-6r12c5=(6-3)r9c5=3r8c5-2r8c5=2r3c5
||
8r1c6-(8=2)r8c6-2r8c5=2r3c5

or, on a single line :

5r1c6=*[(2=8)r8c6-(8=*6)r1c6-6r12c5=(6-3)r9c5=3r8c5]-2r8c5=2r3c5 :=> -5r3c5

which is equivalent but not as pretty as Blue's step.

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