## About Red Ed's Sudoku symmetry group

Everything about Sudoku that doesn't fit in one of the other sections
Red Ed wrote:So: what do you mean by "symmetry"

Hm, basically a symmetry in our sense is a non-trivial automorphisn a of a unique sudoku puzzle S, i.e. Sa=S and a is not equal to the identity. But we dont distinguish betwen equivalent automorphisms like a and tat' for some transformation t and its inverse t'.
So each symmetry can be mapped to a symmetry class of your paper (where invariants exist) and vice versa.
Now it turned out, that e.g. a puzzle has symmetry DBS (class 43) exactly, when it has both a symmetry D (class 37) and a second symmetry BS (class 22). Therefore udusuk does not call this a "basic" symmetry.
eleven

Posts: 2066
Joined: 10 February 2008

So "half turn" shouldn't be basic, either, since QT => HT, right?

More generally, if automorphism A is conjugate to a power of a (different) automorphism B, then A shouldn't be "basic", if I understand your intention correctly.
Red Ed

Posts: 633
Joined: 06 June 2005

No. 90° rotation implies 180° rotation, but 180° rotation is not equivalent to a combination of 2 or more 90° rotations (or of any other symmetries).
DBS is equivalent to a combination of D and BS (equivalence in the sense, that if a puzzle has DBS symmetry, it also has both D and BS symmetry and vice versa). I dont have a proof for the "vice versa" part. But also one direction is "basic" for our purposes. If i know that a symmetry is not there without 2 "easier" symmetries, we dont have to investigate it seperately for solving methods.
Last edited by eleven on Mon Jan 05, 2009 4:58 am, edited 1 time in total.
eleven

Posts: 2066
Joined: 10 February 2008

What?! 180° rotation is precisely equivalent to a combination of two 90° rotations.
Red Ed

Posts: 633
Joined: 06 June 2005

I meant 180° rotational symmetry, not rotation, sorry.
eleven

Posts: 2066
Joined: 10 February 2008

Now we're getting somewhere. Let <A,B,C,...> be the group of automorphisms generated by automorphisms A,B,C,... Then you're saying that an automorphism M is basic only if <M> does not equal <A,B,...> for any (at least two, non-identity, non M-conjugate) automorphisms A,B,... ?

I don't know how much terminology you're familiar with, so do ask if I've overdone the jargon.
Red Ed

Posts: 633
Joined: 06 June 2005

Yes, thats what i meant.
eleven

Posts: 2066
Joined: 10 February 2008

So aren't you potentially missing a trick by considering techniques that nail puzzles with D symmetry (only) and puzzles with BS symmetry (only), but not those with both at the same time?

Trivial analogy: guess the number: 4321_ . Hint 1: it's divisible by 5. Hint 2: it's divisible by 2. If I use any hint in isolation then I can't solve the puzzle. But if I use both together, I know that the last digit is zero.
Red Ed

Posts: 633
Joined: 06 June 2005

Red Ed wrote:So aren't you potentially missing a trick by considering techniques that nail puzzles with D symmetry (only) and puzzles with BS symmetry (only), but not those with both at the same time?

We are aware of that, e.g. the puzzles i recently posted, can best be solved with using the combination. The point is, that its easier to look at the simple symmetries (with cycles 2 and 3) and combine them sometimes, than to look at 6-cycles all the time.
eleven

Posts: 2066
Joined: 10 February 2008

Thanks for the nice conversation gentlemen, it's very enlightening.

As I mentioned it's arguable if we put HT & QT in the same group or not, but considering I've found "techniques" that are only for QT (e.g. windmill) I think they should each be listed separately in my analysis.

I'm still in the process of morphing Ed's example grids to expose the symmetries (more than half done already, should be finished soon) before I post my full analysis. Meanwhile here is a quick list of how eleven's listing of the 26 groups correspond to the 13 "basic symmetry" I listed earlier:

Code: Select all
`1. Fixed boxesR           8 : Mini-Column (MC)RC1         7 : MD @ c123 + MC @ c456789RC1C2       9 : MD @ c123456 + MC @ c789RC         10 : Mini-Diagonal (MD)2. Boxes move in bandsS          25 : Jumping-Row (JR)SR1        28 : GR @ r123 + JR @ r456789SR1R2      30 : GR @ r123456 + JR @ r789SR         32 : Gliding-Row (GR)SC1        27 : Full-Row (FR)SR1C1      26 : WR @ r123 + FR @ r456789SR1R2C1    29 : WR @ r123456 + FR @ r789SRC1       31 : Waving-Row (WR)3. Boxes move triangular (B 159, 267, 368)BS         22 : Jumping-Diagonal (JD)BSR1       24 : Broken-Column (BC)BSR1C1     23 : Full-Diagonal (FD)4. Rotational symmetriesDD2        79 : Half-Turn (HT)DBxRx      86 : Quarter-Turn (QT)5. Diagonal symmetriesD          37 : Diagonal-Mirror (DM)DBS        43 : DM + JDDRC        40 : DM + MD6. Sticks symmetriesBxCx      134 : Column-Sticks (CS)BxCxR     135 : CS + MCBxCxS     145 : CS + JRBxCxSR2   144 : CS + JR @ r123789 + GR @ r456BxCxSR    142 : CS + GRBxCxSR1R3 143 : CS + GR @ r123789 + JR @ r456`

udosuk

Posts: 2698
Joined: 17 July 2005

eleven wrote:Three puzzles with 2 symmetries, 180° plus RC ("minidiagonal").
The first one is easy.
Code: Select all
` +-------+-------+-------+ | . . 2 | 8 9 . | . . . | | 7 . . | . 1 3 | . . . | | . 6 . | 5 . 4 | . . . | +-------+-------+-------+ | 1 . 7 | . . . | 2 8 . | | 6 4 . | . . . | . 7 1 | | . 2 8 | . . . | 4 . 6 | +-------+-------+-------+ | . . . | 7 . 9 | . 1 . | | . . . | 3 6 . | . . 4 | | . . . | . 5 2 | 8 . . | +-------+-------+-------+ +-------+-------+-------+ | . 5 . | . . 8 | . 4 . | | . . 9 | 1 . . | . . 8 | | 3 . . | . 4 . | 1 . . | +-------+-------+-------+ | . . 2 | 9 . . | . 1 . | | 7 . . | . 3 . | . . 4 | | . 6 . | . . 5 | 8 . . | +-------+-------+-------+ | . . 6 | . 7 . | . . 3 | | 2 . . | . . 6 | 5 . . | | . 7 . | 2 . . | . 9 . | +-------+-------+-------+ +-------+-------+-------+ | . . . | 3 4 . | . 1 . | | . . . | . 5 8 | . . 4 | | . . . | 1 . 9 | 8 . . | +-------+-------+-------+ | 8 3 . | . . . | 7 . 5 | | . 1 5 | . . . | 9 6 . | | 9 . 4 | . . . | . 3 2 | +-------+-------+-------+ | . . 2 | 5 . 6 | . . . | | 7 . . | 2 9 . | . . . | | . 6 . | . 7 3 | . . . | +-------+-------+-------+`

The first one is indeed easy, so I won't bother posting my path for it.

This is the path for the 2nd one:

(After simple moves)

Code: Select all
`+----------------------+----------------------+----------------------+|*16     5      17     | 367    269    8      | 23679  4     -2679   || 46     24     9      | 1      256    237    | 2367   23567  8      || 3     #28     78     | 567    4      279    | 1     -2567   25679  |+----------------------+----------------------+----------------------+| 458    348    2      | 9      68     47     | 367    1      567    || 7      189    158    | 68     3      12     | 269    256    4      || 149    6      134    | 47     12     5      | 8      237    279    |+----------------------+----------------------+----------------------+| 14589 -1489   6      | 458    7      149    | 24    #28     3      || 2      13489  1348   | 348    189    6      | 5      78     17     ||-1458   7      13458  | 2      158    134    | 46     9     *16     |+----------------------+----------------------+----------------------+`

HT Symmetrical Pair: r1c1+r9c9={16}
=> r1c9+r9c1 can't have {16}
HT Symmetrical Pair: r3c2+r7c8={28}
=> r3c8+r7c2 can't have {28}
MD Symmetry: r1c9+r2c7+r3c8 can't be [627|762]
MD Symmetry: r7c2+r8c3+r9c1 can't be [481|814]

Code: Select all
`+----------------------+----------------------+----------------------+| 16     5     *17     | 367    269    8      |-23679  4      29     || 46    #24     9      | 1      256   #237    |#37    -23567  8      || 3      28     78     | 567    4      279    | 1      56     25679  |+----------------------+----------------------+----------------------+| 458    348    2      | 9      68     47     | 367    1      567    || 7      189    158    | 68     3      12     | 269    256    4      || 149    6     *134    | 47     12     5      | 8      237    279    |+----------------------+----------------------+----------------------+| 14589  19     6      | 458    7      149    | 24     28     3      || 2     -13489 *34     | 348    189    6      | 5      78     17     || 58     7     -13458  | 2      158    134    | 46     9      16     |+----------------------+----------------------+----------------------+`

r168c3 from {1347} must have {47}
=> r1c7+r9c3 can't be [74]
HT Symmetry: r1c7<>7, r9c3<>4
r2c267 from {2347} must have {47}
=> r2c8+r8c2 can't be [74]
HT Symmetry: r2c8<>7, r8c2<>4
MD Symmetry: r1c7+r2c8+r3c9 can't be [762|276]
MD Symmetry: r7c1+r8c2+r9c3 can't be [814|148]

Code: Select all
`+----------------+----------------+----------------+| 16   5    17   | 367  269  8    | 369  4    29   || 46   24   9    | 1    256  237  | 37   235  8    || 3    28   78   |#567  4    279  | 1   #56  #579  |+----------------+----------------+----------------+| 458  348  2    | 9    68   47   | 367  1   -567  || 7    189  158  | 68   3    12   | 269  256  4    ||-149  6    134  |*47   12   5    | 8   -237 *279  |+----------------+----------------+----------------+| 459  19   6    | 458  7    149  | 24   28   3    || 2    389  34   | 348  189  6    | 5    78   17   || 58   7    135  | 2    158  134  | 46   9    16   |+----------------+----------------+----------------+`

r3c489 from {5679} must have {79}
=> r4c9+r6c1489 can't be [74729]
HT+MD Symmetry: r4c9<>7, r6c1<>4, r6c8<>2
MD Symemtry: r4c2<>8, r5c3<>1, r5c7<>6

Code: Select all
`+----------------+----------------+----------------+| 16   5    17   | 367  269  8    | 369  4    29   || 46  #24   9    | 1    256 #237  |#37  -235  8    || 3    28   78   | 567  4    279  | 1    56   579  |+----------------+----------------+----------------+| 458 *34   2    | 9    68   47   | 367  1    56   || 7    189  58   | 68   3    12   | 29   256  4    || 19   6    134  | 47   12   5    | 8    37   279  |+----------------+----------------+----------------+| 459  19   6    | 458  7    149  | 24   28   3    || 2   -389  34   | 348  189  6    | 5    78   17   || 58   7    135  | 2    158  134  | 46   9    16   |+----------------+----------------+----------------+`

r2c267 from {2347} must have {34}
=> r2c8+r48c2 can't be [343]
HT Symmetry: r2c8<>3, r8c2<>3
MD Symmetry: r1c7<>9, r3c9<>5, r7c1<>9, r9c3<>5

The rest is easy.

Alternatively, there is a 1-move step to replace the last 2 steps, but it makes use of an AALS (n+2 candidates for n cells) instead of an ALS (n+1 candidates for n cells):

Code: Select all
`+----------------+----------------+----------------+| 16   5    17   | 367  269  8    | 369  4    29   ||#46   24   9    | 1   #256  237  | 37  #235  8    || 3    28   78   | 567  4    279  | 1    56   579  |+----------------+----------------+----------------+|-458  348  2    | 9    68   47   | 367  1    567  || 7    189  158  | 68   3    12   | 269 -256  4    || 149  6   -134  | 47  #12   5    | 8   #237 -279  |+----------------+----------------+----------------+| 459  19   6    | 458  7    149  | 24   28   3    || 2    389  34   | 348  189  6    | 5    78   17   || 58   7    135  | 2    158  134  | 46   9    16   |+----------------+----------------+----------------+`

r2c158 from {23456} must have {234}
=> r4c1+r5c8+r6c3589 can't be [421237]
HT+MD Symmetry: r4c1<>4, r5c8<>2, r6c3<>1, r6c9<>7
MD Symmetry: r4c7<>6, r5c2<>8

The rest is easy.

For the 3rd one, I can use an AALS for a single-step solving path :

(After simple moves)

Code: Select all
`+-------------------+-------------------+-------------------+| 56    2589  789   | 3     4     27    |*256   1     679   ||#136   29    1379  |#67    5     8     |#236  *279   4     || 3456  245   37    | 1     26    9     | 8     257  *367   |+-------------------+-------------------+-------------------+| 8     3     6     | 9     12    12    | 7     4     5     || 2     1     5     | 47    3     47    | 9     6     8     || 9     7     4     | 68    68    5     | 1     3     2     |+-------------------+-------------------+-------------------+|*134   489   2     | 5     18    6     | 34    789   1379  || 7     458   138   | 2     9     14    | 3456  58    136   || 145   6     189   | 48    7     3     | 245   2589  19    |+-------------------+-------------------+-------------------+`

r2c147 from {12367} must have {127}
=> r1c7+r2c8+r3c9+r7c1 can't be [2761]
MD Symmetry: r1c7<>2, r2c8<>7, r3c9<>6
HT Symmetry: r7c1<>1, r8c2<>4, r9c3<>8

The rest is easy.
udosuk

Posts: 2698
Joined: 17 July 2005

Nice, that someone solved the puzzles, and i am not surprised, who it was

I had similar steps for the second puzzle (grid after the 2 easy moves).
Code: Select all
` *--------------------------------------------------------------* | 16     5     *17     |#367  269  8    | 3679   4      29     | | 46     24     9      | 1    256  237  | 37     2356   8      | | 3      28     78     | 567  4    279  | 1      56     2579   | |----------------------+----------------+----------------------| | 458    348    2      | 9    68   47   | 367    1      567    | | 7      189    158    | 68   3    12   | 269    256    4      | | 149    6     *134    | 47   12   5    | 8      237    279    | |----------------------+----------------+----------------------| | 4589   19     6      | 458  7    149  | 24     28     3      | | 2      1389  *34     | 348  189  6    | 5      78     17     | | 58     7     *1345   | 2   #158 #134  | 46     9      16     | *--------------------------------------------------------------*The * cells (out of 13457) must have 7 in r1c3 or 5 in r9c31. like udosuk: r1c6=7 -> r9c3=4 => r1c6<>7          HT        MD2. r1c4=7 -> r9c6=4 -> r9c5=5 => r1c4<>7`

Then i finished it with a pure MD step (note that cells in a minirow/column must have numbers from different cycles)
Code: Select all
` *--------------------------------------------------* | 1    5    7    |#36   269  8    |#369  4   #29   | | 6    4    9    | 1    25   237  | 37   235  8    | | 3    2    8    | 567  4    79   | 1    56  #579  | |----------------+----------------+----------------| | 458  38   2    | 9    68   47   | 367  1    57   | | 7    189  15   |#68   3    12   | 69   256  4    | | 49   6    134  |#47   12   5    | 8    23  #279  | |----------------+----------------+----------------| | 459  19   6    | 45   7    149  | 2    8    3    | | 2    389  34   | 348  89   6    | 5    7    1    | | 58   7    135  | 2    158  13   | 4    9    6    | *--------------------------------------------------*       MD                                     MDr1c7=3 -> (r3c9=9&r1c9=2) -> r6c9=7 -> r6c4=4 -> r5c4=6 -> r1c4=3`

For the third puzzle i later saw, that only simple diagonal coloring was needed:
Code: Select all
` *-----------------------------------------------------* | 56    2589  789   | 3   4   27  | 256   1     679   | | 136   29    1379  | 67  5   8   | 236   279   4     | | 3456  245   37    | 1   26  9   | 8     257   367   | |-------------------+-------------+-------------------| | 8     3     6     | 9   12  12  | 7     4     5     | | 2     1     5     | 47  3   47  | 9     6     8     | | 9     7     4     | 68  68  5   | 1     3     2     | |-------------------+-------------+-------------------| | 134   489   2     | 5   18  6   | 34    789   1379  | | 7     458   138   | 2   9   14  | 3456  58    136   | | 145   6     189   | 48  7   3   | 245   2589  19    | *-----------------------------------------------------*`

Either r7c1=3 or (r8c3=3 -> r2c7=3) => r2c1<>3
Then either r7c1=3 or (r3c1=3 -> r7c9=3) => r7c7<>3
eleven

Posts: 2066
Joined: 10 February 2008

Alrighty folks, finally "deciphered" all 26 of Red Ed's example grids. Really tough work collectively! (Especially
the "Full-Diagonal", "Half-Turn" & "Quarter-Turn" grids. I had to apply logical techniques to find the correct isomorphs. Definitely recommended to all who like challenges! )

So here I'll list them, including the original grid (in line format), morphing ops, new grid, and the details of the symmetry. I'll also colour some of the cells in the new grid to highlight the "symmetrical effect".

1a. C: 123456789987123546645987132312564897879312465456879213231645978798231654564798321

c123456798

123456798
987123564
645987123
312564879
879312456
456879231
231645987
798231645
564798312

Mini-Rows: (123)(456)(798)

1b. CR1: 123456789897123456645978123312564978978312564456897312581249637734681295269735841

r987123456

269735841
734681295
581249637
123456789
897123456
645978123
312564978
978312564
456897312

Mini-Diagonals @ r123, Mini-Rows @ r456789: (123)(456)(789)

1c. CR1R2: 123456789456789123789231645247395861691842537835617294374168952518924376962573418

r546789123

691842537
247395861
835617294
374168952
518924376
962573418
123456789
456789123
789231645

Mini-Diagonals @ r123456, Mini-Rows @ r789: (123)(456)(789)

1d. CR: 123456789456789123789123456214865937865937214937214865371542698542698371698371542

123456789
456789123
789123456
214865937
865937214
937214865
371542698
542698371
698371542

Mini-Diagonals: (159)(267)(348)

2a. S: 123456789789123456645798123312564897897312564456879312231645978978231645564987231

r<->c (Transpose)

176384295
284195376
395276184
417538629
529617438
638429517
741853962
852961743
963742851

Jumping-Rows: (132)(456)(789)

2b. SR1: 123456789456789123789123456234891567591267834867534291348915672612378945975642318

123456789
456789123
789123456
234891567
591267834
867534291
348915672
612378945
975642318

Gliding-Rows @ r123, Jumping-Rows @ r456789: (174)(285)(396)

2c. SR1R2: 123456789456789123789123456231897564567234891894561237315648972648972315972315648

r789123456

315648972
648972315
972315648
123456789
456789123
789123456
231897564
567234891
894561237

Gliding-Rows @ r123456, Jumping-Rows @ r789: (174)(285)(396)

2d. SR: 123456789897123654654879132312564978978312465546798321231645897789231546465987213

c132465789, r<->c

186395274
374286195
295174386
418537629
639428517
527619438
761943852
853762941
942851763

Gliding-Rows: (123)(4)(5)(6)(7)(8)(9)

2e. SC1: 123456789456789123789123456234567891567891234891234567345678912678912345912345678

r<->c

147258369
258369471
369471582
471582693
582693714
693714825
714825936
825936147
936147258

Full-Rows: (123456789)

2f. SR1C1: 123456789456789123789231645231897456645123978897564231312978564564312897978645312

c123456897

123456897
456789231
789231456
231897564
645123789
897564312
312978645
564312978
978645123

Waving-Rows @ r123, Full-Rows @ r456789: (174285396)

2g. SR1R2C1: 123456789456789123789231645231897456645123978897564231312645897564978312978312564

r789123456,c123456897

312645978
564978123
978312645
123456897
456789231
789231456
231897564
645123789
897564312

Waving-Rows @ r123456, Full-Rows @ r789: (174285396)

2h. SRC1: 123456789456789123789231645231564978645897231897123456312645897564978312978312564

c123456897

123456897
456789231
789231456
231564789
645897312
897123564
312645978
564978123
978312645

Waving-Rows: (174285396)

3a. BS: 123456789456789123789123456235964817817235964964817235392641578578392641641578392

r123465798

123456789
456789123
789123456
235964817
964817235
817235964
392641578
641578392
578392641

Jumping-Diagonals: (195)(267)(348)

3b. BSR1: 123456789456789231789123645231564897564897312978312564312645978645978123897231456

r<->c

147259368
258367149
369148257
471583692
582691473
693472581
726835914
834916725
915724836

Broken-Columns: (159267348)

3c. BSR1C1: 123456789457289163689173452235741896816392574974568231392817645568924317741635928

r132564978, c132465879, r<->c

164892735
397645128
285173496
412357689
639281574
578964312
856739241
741528963
923416857

Full-Diagonals: (132986547)

4a. DD2: 123456789869127435457938126216374958785219643934685217671543892392861574548792361

r879213546, c654879123

168754392
345982671
297631548
721345869
654879123
839216457
912463785
473598216
586127934

Half-Turn: (7)(14)(25)(36)(89)

4b. DBxRx: 123456789456789231789312456214938567375641892698527314561273948837194625942865173

r987123456, c465879321

856713249
149265738
237498165
465879321
798321654
321546987
983657412
614982573
572134896

Quarter-Turn: (2)(1437)(5896)

5a. D: 123456789579128643684739125412673958958214376367985214831562497796841532245397861

r456789123

412673958
958214376
367985214
831562497
796841532
245397861
123456789
579128643
684739125

Diagonal-Mirror: (4)(5)(7)(19)(23)(68)

5b. DBS: 123456789457189236968372514291738465374265198685941327546813972732694851819527643

r789213654, c132456879

564813792
723694581
891527463
475189326
132456879
986372154
658941237
347265918
219738645

Diagonal-Mirror: (1)(2)(5)(39)(48)(67)
Jumping-Diagonals: (125)(368)(497)

5c. DRC: 123456789456789231789123645215348976398672514674915823541897362832564197967231458

r789213456, c123456879

541897632
832564917
967231548
456789321
123456879
789123465
215348796
398672154
674915283

Diagonal-Mirror: (3)(5)(7)(19)(26)(48)
Mini-Diagonals: (186)(294)(375)

6a. BxCx: 123456789978213465546798231217864593835129674469375128752931846694582317381647952

r978123456, c312456897

138647529
275931468
469582173
312456897
897213654
654798312
721864935
583129746
946375281

Column-Sticks: (1)(5)(9)(23)(46)(78)

6b. BxCxR: 123456789456789123789123456234591867567834291891267534378942615612375948945618372

r897123456

612375948
945618372
378942615
123456789
456789123
789123456
234591867
567834291
891267534

Column-Sticks: (2)(5)(8)(13)(46)(79)
Mini-Columns: (147)(258)(369)

6c. BxCxS: 123456789457189326689327154291635847745891632836742591318264975574918263962573418

r321456789, r<->c

641278359
852943176
973156842
314687295
285394617
796512483
137865924
528439761
469721538

Column-Sticks: (1)(8)(9)(25)(34)(67)
Jumping-Rows: (189)(236)(475)

6d. BxCxSR2: 123456789456789123789132465218967534564213978937548216391875642645321897872694351

r789123456, c312546879

139785462
564231987
287964531
312546879
645879213
978312645
821697354
456123798
793458126

Column-Sticks: (1)(4)(7)(23)(56)(89)
Jumping-Rows @ r123789, Gliding-Rows @ r456: (174)(295)(386)

6e. BxCxSR: 123456789456789123789123456214937865865214937937865214342678591591342678678591342

r456789132, c132465798

241973856
856241973
973856241
324687519
519324687
687519324
132465798
798132465
465798132

Column-Sticks: (1)(2)(8)(34)(59)(67)
Gliding-Rows: (1)(2)(3)(4)(5)(6)(7)(8)(9)

6f. BxCxSR1R3: 123456789456789123789132465248573916537961248961248537394815672672394851815627394

r456789312, c312546879

824753196
753691428
196428357
439185762
267934581
581267934
978312645
312546879
645879213

Column-Sticks: (3)(6)(8)(15)(27)(49)
Gliding-Rows @ r123789, Jumping-Rows @ r456: (174)(295)(386)

PS: nice moves eleven, I think you've found a great technique in "cells in a minirow/column must have numbers from different cycles"!

Will explore to see if this idea can be developed into more powerful tricks.
udosuk

Posts: 2698
Joined: 17 July 2005

eleven wrote:For the third puzzle i later saw, that only simple diagonal coloring was needed:
Code: Select all
` *-----------------------------------------------------* | 56    2589  789   | 3   4   27  | 256   1     679   | | 136   29    1379  | 67  5   8   | 236   279   4     | | 3456  245   37    | 1   26  9   | 8     257   367   | |-------------------+-------------+-------------------| | 8     3     6     | 9   12  12  | 7     4     5     | | 2     1     5     | 47  3   47  | 9     6     8     | | 9     7     4     | 68  68  5   | 1     3     2     | |-------------------+-------------+-------------------| | 134   489   2     | 5   18  6   | 34    789   1379  | | 7     458   138   | 2   9   14  | 3456  58    136   | | 145   6     189   | 48  7   3   | 245   2589  19    | *-----------------------------------------------------*`

Either r7c1=3 or (r8c3=3 -> r2c7=3) => r2c1<>3
Then either r7c1=3 or (r3c1=3 -> r7c9=3) => r7c7<>3

Sometimes it just puzzles me how the easiest move manages to escape from our eyes.

Code: Select all
`+-------------------+-------------------+-------------------+| 56    2589  789   | 3     4     27    | 256   1     679   || 136   29    1379  | 67    5     8     |#236   279   4     || 3456  245  -37    | 1     26    9     | 8     257  #367   |+-------------------+-------------------+-------------------+| 8     3     6     | 9     12    12    | 7     4     5     || 2     1     5     | 47    3     47    | 9     6     8     || 9     7     4     | 68    68    5     | 1     3     2     |+-------------------+-------------------+-------------------+| 134   489   2     | 5     18    6     |-34    789   1379  || 7     458   138   | 2     9     14    | 3456  58    136   || 145   6     189   | 48    7     3     | 245   2589  19    |+-------------------+-------------------+-------------------+`

3 @ b3 locked @ r2c7+r3c9
=> r3c3+r7c7 can't be [33]
HT Symmetry: r3c3<>3, r7c7<>3
MD Symmetry: r1c1<>5, r2c2<>9, r8c8<>5, r9c9<>9

The rest is easy.

Hopefully there is something similarly easy in the 2nd one.
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:Alrighty folks, finally "deciphered" all 26 of Red Ed's example grids.

Good work done.

I am very busy these days, so it will take some time to update the first page (want to add more links also).
eleven

Posts: 2066
Joined: 10 February 2008

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