About Red Ed's Sudoku symmetry group

Everything about Sudoku that doesn't fit in one of the other sections

Postby StrmCkr » Thu Jan 01, 2009 8:46 am

deleted back to the drawing board... didnt think off patterns being found on same row in a house befor...

thanks red ed for the examples they help alot!
Last edited by StrmCkr on Fri Jan 02, 2009 10:12 am, edited 1 time in total.
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Postby eleven » Thu Jan 01, 2009 9:33 am

As udosuk already stated, all the "new" symmetries with 6/9-cycles, i.e. those we have not already handled in the symmetry threads, can be expressed also as a combination of 2 "classical" symmetries. I updated the table in the first post accordingly, also inserted the Bludo technique for the block symmetry and added the names, udosuk mentioned here.
In the bands group i changed the ops B(CR) to S(RC), so now the boxes really move in bands instead of the stacks.

From the solvers POV its easier to look at 2 simpler symmetries than at one complicated one with 6- or 9-cycles, so the new symmetries are better handled as combined ones.
We are left then with 13 classical symmetries, and after Bludo was found, only for those with fixed boxes and the Band symmetry no special techniques are known for solving (additionally probably the Gludo/Band symmetries, when 6 or all numbers are fixed).

I would like to add names for those i dont have one so far, maybe udosuk can help.

A word to the sticks symmetry. Though its the most common one, it is hard to find any challenging puzzles, if you dont have such a good program as Mauricio. And also his top rated puzzles solve with singles after applying the symmetry to the fixed cells. (A reason for this is, that each fixed cell "sees" 4 others, not only 2 like with diagonal symmetry, so the 9 number normally can be put in from the beginning)

Thanks to Red Ed for giving us an idea how the number of non-equivalent grids was calculated.
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Postby eleven » Thu Jan 01, 2009 4:36 pm

We now generated 28 puzzles with sticks symmetry with ER ratings between 9.0 and 9.2. 24 of them solved with singles after placing the numbers in the fixed cells (if you cant, the puzzle is not unique), 3 needed an xy-wing and only this one more, but i cant see a symmetry move.
Not very exciting ...

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


This is the list of the other puzzles:
..1.8..6...4..21....71..5..9187.6.........8.9....2....1...9..7.5..2....16....1..4
..1..96.436...1......65.3..6.7.2....8.9.......1....9.81..8..5.7.731.........47..3
.....169..4...6..26..25.8.4.3.5.4...9.8......5.4.1.......1...87.5.7..2....7.425.9
....9.1..9..5...4715...29.........3..2.7.6...4.5..........8...1..8..465..412....8
5...64..39.....1...72..3.8...........1.6.7.......3.8.9..457.3....8.....126.3...9.
..46...8537.4....2.....1.......1.6.76.75.48.9.2.......5....749..63..52.....1.....
...9..6..65...7..1..25....45.4.2.9.8.........938...........8..7.476..1..2....45..
2..7....936...45.......214....9.8.1.4.5.......2....7.6..2..68...735....4...2...51
..8..31....4..879.....41....2....5.47.6...8.9.......1.9..3....15..9...86...15....
..6..4..5..8.....25..271.8.......8.9...7.6.1.....2....7..5..4..9.....2....4162.9.
...3...4.7....68.38..19.....2....5.45.4...6.7.......3......3.5...67..3.9..9.81...
8....4..9.5.37.4.......12......3....6.7......5.47.6.1...95..8...4..63..5...1....2
.79..2.....65.37..3......42.1.8.9.........8.9...7.6.3.86.2.....7..3.4..6..3...25.
5..1.9..6..1.5.3..7....2..58.9.2.......5.49.8.......3...48.17..1...4...3..62..4..
.41.9.6....57.4..13....1...8.9.2....5.49.8.1.......9.815..8...74..5.61....31.....
6..3....4...8....23...951..5.4....2..1.7.6..............7..35.......92....348...1
6....1..82....97....52....3...5.4....2.......9.8716.....71..9....28....64....23..
...5.8..16....3.....27...84.2.......7.6....3.4.5...8.9...9.41....73.....2....659.
.7...95..3.....1..5.63...879.8.......3.6.7..........1..6.8....4..3.....17.4..369.
47..682....274....3.6..5..75.4....2.....3.9.8..........6597...22...56...7.34..6..
2....5.434..8....6.9.1..2...2.9.8.......1....7.6........24..35...5..97...8...1..2
38...2.4...7..31....178...5...4.5.1.....2.8.9..........932...5.6..3....11...964..
4......79...2..4..39..74......8.97.67.6.2.....1.........5...86......2..5.8356....
..4..56.......829...32....5...6.7...6.75.4.1..3.......5..4....7...9...823....24..
6..7.1..53.......1.5....29....8.9...8.94.5.2..3.........71.64....3...1...4.....82
.53.7.2..1.....9..7..8....5......5.4...637......4.5.2.34..6...2..1.....8..6..94..
3..9.57..17......2..8..14......1.8.9.3.6.7..............34.8..6.61...2..9..1....5
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Postby udosuk » Fri Jan 02, 2009 2:25 am

eleven wrote:... and only this one more, but i cant see a symmetry move.
Not very exciting ...

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

After easy moves:

Code: Select all
+-------------------+-------------------+-------------------+
| 7     5689  389   |*256   569  *25    | 1     4     23    |
| 159   459   149   | 1245  4579  3     | 27    6     8     |
| 13    46    2     | 8     467   147   | 5     9     37    |
+-------------------+-------------------+-------------------+
| 4     3     5     | 9     1     8     | 67    2     67    |
| 6     2     7     |*45    3    *45    | 8     1     9     |
| 89    1     89    | 7     2     6     | 4     3     5     |
+-------------------+-------------------+-------------------+
| 389   4789  6     |*24    478  *247   | 23    5     1     |
| 158   458   148   | 3     4568  1245  | 9     7     26    |
| 2     57    13    | 156   567   9     | 36    8     4     |
+-------------------+-------------------+-------------------+

Actually there is a simple symmetry-BUG-lite move (I'm not that hot on "classical uniqueness techniques" like BUG-lite but since all symmetry-based moves are also uniqueness-based I guess here the "ethical issue" is diluted a little bit:?: ):

r157c46 can't form the deadly pattern {245}
Stick Symmetry: r1c4+r7c6=[67]

Unfortunately, that doesn't crack the puzzle directly - you still need a chain-like move later on, which can actually be applied right away. So let's forget the move above and proceed from the previous pencilmark state:

Code: Select all
+-------------------+-------------------+-------------------+
| 7     5689  389   |#256   569   25    | 1     4    *23    |
| 159   459   149   | 1245  4579  3     | 27    6     8     |
| 13    46    2     | 8     467   147   | 5     9     37    |
+-------------------+-------------------+-------------------+
| 4     3     5     | 9     1     8     | 67    2     67    |
| 6     2     7     |#45    3     45    | 8     1     9     |
| 89    1     89    | 7     2     6     | 4     3     5     |
+-------------------+-------------------+-------------------+
|-389   4789  6     |#24    478   247   |*23    5     1     |
| 158   458   148   | 3     4568  1245  | 9     7     26    |
| 2     57   *13    |#156   567   9     |-36    8     4     |
+-------------------+-------------------+-------------------+

r1579c4 from {12456} must have 1 or 2 or both
Stick Symmetry: r1c9+r7c7+r9c3 can't be [221]
=> r7c7+r9c3 can't be [21], must be [23|31|33] having 3
=> r7c1+r9c7, seeing r7c7+r9c3, can't have 3

:idea:
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Re: About Red Ed's Sudoku symmetry group

Postby udosuk » Fri Jan 02, 2009 3:08 am

eleven wrote:
Code: Select all
    M         C         N        L    S
1. Fixed boxes
R           8    107.495.424   3    N  Mini-row symmetry
RC1         7     21.233.664   3    N 
RC1C2       9      4.204.224   3    N 
RC         10      2.508.084   3    N  Mini-diagonal symmetry

Just some thoughts here:

"Mini-row symmetry" should be given by the mapping "C" instead. As a result I think the table should be like this:

Code: Select all
M           C         N        L    S
1. Fixed boxes
C           8    107.495.424   3    N  Mini-row symmetry
R1C         7     21.233.664   3    N  MDS @ band 1, MRS @ band 2,3
R1R2C       9      4.204.224   3    N  MDS @ band 1,2, MRS @ band 3
RC         10      2.508.084   3    N  Mini-diagonal symmetry



Also, since you asked me for names, here are some proposals:

C -> Mini-Row Symmetry (MRS)
R -> Mini-Column Symmetry (MCS)
RC -> Mini-Diagonal Symmetry (MDS)
S -> Band Symmetry (BAS)
B -> Stack Symmetry (SAS)
BS -> Block Symmetry (BOS) (Bludo applicable)
SR -> Horizontal Glide Symmetry (HGS) (Gludo applicable)
BC -> Vertical Glide Symmetry (VGS) (Gludo applicable)
DD2 or BxRxSxCx-> Half-Turn (180 Degree Rotation) Symmetry (HTS)
DBxRx -> Quarter-Turn (90 Degree Rotation) Symmetry (QTS)
D -> Leading-Diagonal (Reflection) Symmetry (D\S)
D2 -> Non-Leading-Diagonal (Reflection) Symmetry (D/S)
SxRx -> Horizontal Stick Symmetry (HSS)
BxCx -> Vertical Stick Symmetry (VSS)

(All names/abbreviations are subjected to rectifying.)

Also note that MRS, MCS, HGS, VGS sometimes can be applied locally in the grid (i.e. in certain bands/stacks) instead of globally to the whole grid.



The next item on my wishlist is an example puzzle/solution grid for each of the 26 groups.:)
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Postby eleven » Fri Jan 02, 2009 9:56 am

udusuk wrote:r1579c4 from {12456} must have 1 or 2 or both
Stick Symmetry: r1c9+r7c7+r9c3 can't be [221]
=> r7c7+r9c3 can't be [21], must be [23|31|33] having 3

Nice solution using the symmetry, though not easy needing a 4-cell ALS. Because i misinterpreted the notation first: The second line follows from the first (not from the symmetry).
Since i doubt that many sticks puzzles will be generated, that need a second interesting step, its a historical solution:)

Now to the last post. I think, we should be careful here not to repeat the problems we have with this flood of horribly bad and partially contradictory names for solving techniques. How often was the y/Y/w-wing [styles]/SRP theme discussed ? And why does everyone call a unique rectangle, what definitely is the opposite ?
We are only a few people and should manage to have the same name for the same thing - and the name should make sense. (Also here i already saw discrepances, which lead to a discussion about Eels vs. EUR's, later on NEP's - i really want to avoid that from beginning).

So thanks for the suggestions, and this is my opinion:

I not happy with "Block symmetry", because i suppose, most people would more expect it to be the "Band symmetry" (a stack - a block of boxes - moves right in the symmetry) than what is meant. Didn't Mauricio originally call it the "Threefold symmetry" ? For me a better name.

"Mini-row symmetry" is no good name either: It is true, that its analogue to the Band symmetry, where the boxes move in the band, here the cells in the minirow. But when i think of mini-row and symmetry, i am interested, where it goes to by the symmetry in first line, only in second line, what happens with the cells inside. Thats why i had as mini-row symmetry, what you would call mini-column symmetry. So my proposal is "Mini-band symmetry".

"Mini-diagonal symmetry" is definitely a wrong name: The cells are not diagonally reflected like the cells in the diagonal boxes, when you have "Diagonal symmetry". Unfortunately "Mini-threefold symmetry" is a very long name.

"MDS @ band 1, MRS @ band 2,3" is much too complicated for a simple thing. I have no suggestion yet, it also depends on the names we can find for the "Gludo applicable" symmetries, which are similar. "Horizontal Glide Symmetry" is fine for me, so we just have to add something to say that 1, 2 or 3 bands have this glide.

I wouldn't need new names for the rotational symmetries, but if everything needs a TLA ...:)

Hope that others also have suggestions here.

udusuk wrote:The next item on my wishlist is an example puzzle/solution grid for each of the 26 groups.:)

The time my friend and i reserved for this thread is over. But the program is here and we may adopt it for generating other puzzles step by step. But for the others out of the symmetries in the table, where no examples exist so far, i dont see much worth. E.g. you will never find a good puzzle with a 9-cycle symmetry, i suppose. Same for the exotic stick symmetries. What we got was trivial even without the stick symmetry included. But here are 3 grids, we have calculated:
Code: Select all
 +-------+-------+-------+
 | 1 7 6 | 2 5 4 | 3 9 8 |
 | 3 9 4 | 1 7 8 | 2 5 6 |
 | 2 8 5 | 3 6 9 | 1 4 7 |
 +-------+-------+-------+
 | 4 1 7 | 8 2 5 | 6 3 9 |
 | 5 2 8 | 9 3 6 | 7 1 4 |
 | 6 3 9 | 4 1 7 | 8 2 5 |
 +-------+-------+-------+
 | 9 4 1 | 7 8 2 | 5 6 3 |
 | 7 6 3 | 5 4 1 | 9 8 2 |
 | 8 5 2 | 6 9 3 | 4 7 1 |
 +-------+-------+-------+
Sticks (1)(2)(3)(47)(58)(69)
S (123)(486)((597)
BxCxS (123)(456789)
 +-------+-------+-------+
 | 8 7 2 | 1 6 5 | 9 4 3 |
 | 6 5 3 | 2 4 9 | 7 8 1 |
 | 4 9 1 | 3 8 7 | 5 6 2 |
 +-------+-------+-------+
 | 7 2 4 | 9 1 6 | 8 3 5 |
 | 5 3 8 | 7 2 4 | 6 1 9 |
 | 9 1 6 | 5 3 8 | 4 2 7 |
 +-------+-------+-------+
 | 2 4 5 | 8 9 1 | 3 7 6 |
 | 3 8 9 | 6 7 2 | 1 5 4 |
 | 1 6 7 | 4 5 3 | 2 9 8 |
 +-------+-------+-------+
Sticks (1)(2)(3)(47)(58)(69)
R (123)(486)(597)
BxCxR (123)(456789)
 +-------+-------+-------+
 | 6 9 1 | 4 7 2 | 8 5 3 |
 | 5 4 3 | 9 8 1 | 7 6 2 |
 | 2 8 7 | 3 6 5 | 1 4 9 |
 +-------+-------+-------+
 | 9 2 6 | 5 1 8 | 4 3 7 |
 | 8 1 5 | 7 3 4 | 9 2 6 |
 | 7 3 4 | 6 2 9 | 5 1 8 |
 +-------+-------+-------+
 | 1 6 9 | 2 4 7 | 3 8 5 |
 | 3 7 8 | 1 5 6 | 2 9 4 |
 | 4 5 2 | 8 9 3 | 6 7 1 |
 +-------+-------+-------+
Sticks (1)(2)(3)(47)(58)(69)
SR2 (123)(486)(597)
BxCxSR2 (123)(456789)
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Re: About Red Ed's Sudoku symmetry group

Postby Red Ed » Fri Jan 02, 2009 10:24 am

udosuk wrote:The next item on my wishlist is an example puzzle/solution grid for each of the 26 groups.:)
I now have a catalogue of all of them (automorphic solution grids, that is) which I am in the process of canonicalising and deduping.

Here's one solution grid from each class. I've borrowed eleven's terminology, since that's what you seem to have got used to already.
Code: Select all
1. Fixed boxes
R           8 : 123456789987123546645987132312564897879312465456879213231645978798231654564798321
RC1         7 : 123456789897123456645978123312564978978312564456897312581249637734681295269735841
RC1C2       9 : 123456789456789123789231645247395861691842537835617294374168952518924376962573418
RC         10 : 123456789456789123789123456214865937865937214937214865371542698542698371698371542

2. Boxes move in bands
S          25 : 123456789789123456645798123312564897897312564456879312231645978978231645564987231
SR1        28 : 123456789456789123789123456234891567591267834867534291348915672612378945975642318
SR1R2      30 : 123456789456789123789123456231897564567234891894561237315648972648972315972315648
SR         32 : 123456789897123654654879132312564978978312465546798321231645897789231546465987213
SC1        27 : 123456789456789123789123456234567891567891234891234567345678912678912345912345678
SR1C1      26 : 123456789456789123789231645231897456645123978897564231312978564564312897978645312
SR1R2C1    29 : 123456789456789123789231645231897456645123978897564231312645897564978312978312564
SRC1       31 : 123456789456789123789231645231564978645897231897123456312645897564978312978312564

3. Boxes move triangular (B 159, 267, 368)
BS         22 : 123456789456789123789123456235964817817235964964817235392641578578392641641578392
BSR1       24 : 123456789456789231789123645231564897564897312978312564312645978645978123897231456
BSR1C1     23 : 123456789457289163689173452235741896816392574974568231392817645568924317741635928

4. Rotational symmetries
DD2        79 : 123456789869127435457938126216374958785219643934685217671543892392861574548792361
DBxRx      86 : 123456789456789231789312456214938567375641892698527314561273948837194625942865173

5. Diagonal symmetries
D          37 : 123456789579128643684739125412673958958214376367985214831562497796841532245397861
DBS        43 : 123456789457189236968372514291738465374265198685941327546813972732694851819527643
DRC        40 : 123456789456789231789123645215348976398672514674915823541897362832564197967231458

6. Sticks symmetries
BxCx      134 : 123456789978213465546798231217864593835129674469375128752931846694582317381647952
BxCxR     135 : 123456789456789123789123456234591867567834291891267534378942615612375948945618372
BxCxS     145 : 123456789457189326689327154291635847745891632836742591318264975574918263962573418
BxCxSR2   144 : 123456789456789123789132465218967534564213978937548216391875642645321897872694351
BxCxSR    142 : 123456789456789123789123456214937865865214937937865214342678591591342678678591342
BxCxSR1R2 143 : 123456789456789123789132465248573916537961248961248537394815672672394851815627394

In some sense, this is just a rehash of work that I did with Frazer three years ago!
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Postby udosuk » Fri Jan 02, 2009 11:07 am

Thanks Red Ed for all the example grids. Saved and studying.:)

Also I agree with you eleven that my names need a lot of rectifying. I'm no good in the art of nomenclature and just name them with my instincts, which are probably too inconsistent and I just wish someone else can systematically name them properly.

Meanwhile I have this project to briefly describe the "instinctive structure" of each of the 14 "classical" symmetries, i.e. the way digit cycles/pairings generally appear in the grid.

(The 14 "classical" symmetries are actually of 8 essentially different ones as besides the 2 rotational symmetries all other 12 are actually 6 pairs, each in 2 orientations.)

:idea:
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Postby StrmCkr » Fri Jan 02, 2009 4:10 pm

thanks red ed that helps me alot
i can see all those symmetries better with the complteted grids.
Some do, some teach, the rest look it up.
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Postby eleven » Fri Jan 02, 2009 8:59 pm

udosuk wrote:(The 14 "classical" symmetries are actually of 8 essentially different ones as besides the 2 rotational symmetries all other 12 are actually 6 pairs, each in 2 orientations.)
I counted 13. In fact all possible symmetries are built up very simple.

We have the 2-cycled symmetries
- 180° rotational
- diagonal
- sticks, described here

the 4-cycled symmetry
- 90° rotational

and the 3-cycled symmetries. They are built with only 2 patterns:

Code: Select all
 A1 A2 A3         A1 C2 B3
 B1 B2 B3   and   B1 A2 C3   
 C1 C2 C3         C1 B2 A3


where A1 goes to A2 goes to A3 goes to A1, same for B, C

These patterns can be
- pattern 1 or 2 in the cells of all boxes of a band (fixed boxes symmetries)
- the mini-rows in a band have pattern 1 or 2 (Band symmetries)
- boxes have pattern 2 (Block/Threefold symmetry)

Thats all.

[Added:] I suspect, that to get all possible combinations of symmetries, you just have to add the 90° rotated versions (bands/rows/columuns -> stacks/columns/rows) of the Diagonal, Band and fixed boxes symmetries and try to combine them.
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Postby ronk » Fri Jan 02, 2009 10:00 pm

eleven wrote:In fact all possible symmetries are built up very simple.

We have the 2-cycled symmetries
- 180° rotational
- diagonal
- sticks, described here

the 4-cycled symmetry
- 90° rotational

and the 3-cycled symmetries.

How would you correlate the above to gfroyle's Combinatorial Concepts With Sudoku I: Symmetry, Gordon Royle, March 29, 2006 ?

Additionally, discussion of this paper occurred at Sudoku Symmetry - Formalized.
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Postby udosuk » Fri Jan 02, 2009 11:05 pm

ronk wrote:How would you correlate the above to gfroyle's Combinatorial Concepts With Sudoku I: Symmetry, Gordon Royle, March 29, 2006 ?

Additionally, discussion of this paper occurred at Sudoku Symmetry - Formalized.

Ron, the 2 links you cited are concerning the clue-pattern/shape symmetry of puzzles. In this thread Red Ed, eleven, I & others are talking about the automorphism symmetry of puzzles/solutions. Two different topics. (Although all automorphic puzzles should be in theory morphable into a form with a certain clue-pattern/shape symmetry, we generally aren't putting much interest in that aspect.):idea:



eleven wrote:I counted 13.

You probably didn't count diagonal reflection symmetry in 2 different orientations.

I generally agree with your view that all symmetries can be built up from various basic ones, but am still working hard to write out all 26 groups in details. The 5 "flavoured" stick symmetries are especially tricky to analyse.



Red Ed, I've just realised most of the example grids you listed are normalised. Which means I have my work cut out to work out the morphing operations to "expose" the symmetries. Very tough work!:(
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Postby ronk » Fri Jan 02, 2009 11:37 pm

udosuk wrote:the 2 links you cited are concerning the clue-pattern/shape symmetry of puzzles. In this thread Red Ed, eleven, I & others are talking about the automorphism symmetry of puzzles/solutions. Two different topics.

I don't think they're as different as you make it sound.

In order to have "clue value symmetry", you first need "clue positional symmetry". And the symmetry doesn't need to be perfect, i.e., the "symmetric distance" may be greater than zero.
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Postby Red Ed » Sat Jan 03, 2009 1:41 am

ronk wrote:How would you correlate the above to gfroyle's Combinatorial Concepts With Sudoku I: Symmetry, Gordon Royle, March 29, 2006 ?

Gordon was just looking at isometries -- that is, the different forms you can get by rotating and turning-over a puzzle drawn on a piece of tracing paper. Other validity-preserving operations such as band-cycling are, I suppose, not so pleasing to the human eye. That's why his list of symmetries is so much shorter than our list of "symmetries". To be fair, "symmetry" is probably not quite the right word for what we're doing.
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Postby Red Ed » Sat Jan 03, 2009 1:44 am

udosuk wrote:Red Ed, I've just realised most of the example grids you listed are normalised. Which means I have my work cut out to work out the morphing operations to "expose" the symmetries. Very tough work!:(

Ah yes, sorry about that. I've done one of those "expositions" myself. It was tough, sure, but satisfying ... just like a good Sudoku puzzle:D
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