The New Sudoku Players' Forum

[udosuk]

please show a move or two that would not be possible without "using the symmetrical property"
thanks

We try to solve this puzzle:

Code: Select all

. . 3 . . . . . 2
. . . . . 4 . 5 .
9 . . 2 6 . 7 . .
. 1 . . . . . 8 .
. . . 9 5 . 3 . .
4 9 . . 1 . 2 . .
. . 2 . . . . . 6
. 7 . 1 . 9 . . .
. . . 8 . . 1 . .

After singles/box-line interactions:

Code: Select all

 *-----------------------------------------------------------*
 | 678   4568  3     | 57    789   157   | 4689  1469  2     |
 | 2678  268   1678  | 37    3789  4     | 689   5     1389  |
 | 9     458   148   | 2     6     135   | 7     134   1348  |
 |-------------------+-------------------+-------------------|
 | 3     1     567   | 467   47    2     | 569   8     579   |
 | 2678  268   678   | 9     5     78    | 3     1467  147   |
 | 4     9     5678  |*367   1     378   | 2     67    57    |
 |-------------------+-------------------+-------------------|
 | 1     348   2     | 3457  347   357   | 489   3479  6     |
 | 68    7     468   | 1     234   9     | 458   234   3458  |
 | 5     34    9     | 8     2347  6     | 1     2347  347   |
 *-----------------------------------------------------------*

r6c4 must not be 3 or 6 because these 2 digits must be opposite to each other across the "/" diagonal...

Therefore r6c4=7 and the rest are all naked singles...

Take a look at this thread about the technique we call "Gurth's Symmetrical Placement"...

[gsf]

please show a move or two that would not be possible without "using the symmetrical property"
thanks

After singles/box-line interactions:

Code: Select all

 *-----------------------------------------------------------*
 | 678   4568  3     | 57    789   157   | 4689  1469  2     |
 | 2678  268   1678  | 37    3789  4     | 689   5     1389  |
 | 9     458   148   | 2     6     135   | 7     134   1348  |
 |-------------------+-------------------+-------------------|
 | 3     1     567   | 467   47    2     | 569   8     579   |
 | 2678  268   678   | 9     5     78    | 3     1467  147   |
 | 4     9     5678  |*367   1     378   | 2     67    57    |
 |-------------------+-------------------+-------------------|
 | 1     348   2     | 3457  347   357   | 489   3479  6     |
 | 68    7     468   | 1     234   9     | 458   234   3458  |
 | 5     34    9     | 8     2347  6     | 1     2347  347   |
 *-----------------------------------------------------------*

r6c4 must not be 3 or 6 because these 2 digits must be opposite to each other across the "/" diagonal...

Therefore r6c4=7 and the rest are all naked singles...

but r6c4 is on the diagonal -- what other { 3 6 } are you basing the move on?

Take a look at this thread about the technique we call "Gurth's Symmetrical Placement"...

I already looked and need a concrete example -- thus the questions here

[ronk]

r6c4 must not be 3 or 6 because these 2 digits must be opposite to each other across the "/" diagonal...

Therefore r6c4=7 and the rest are all naked singles...

but r6c4 is on the diagonal -- what other { 3 6 } are you basing the move on?

r1c3 <--> r7c9, r3c5 <--> r5c7, r4c1 <--> r9c6 is what I see.

[udosuk]

but r6c4 is on the diagonal -- what other { 3 6 } are you basing the move on?

Ha! gsf, being so intelligent in backdoor theories and other sudoku topics, I see you finally stump on a field of which you're totally unfamiliar...

Symmetry-based moves are very intriguing indeed... You need to do more research to grasp the concept... I suggest you to use the search engine in this forum to search for terms such as "symmetrical placement" or "emerald" and read through the threads you find... Perhaps you'd understand it better...

I'd like to explain to you in details but right now it's 3:30am here and I've got a busy weekend ahead... So I could only write a more detailed explanation of that move next Monday... If you're lucky perhaps gurth or others would kindly explain to you and let you understand...

Meanwhile here is my effort on gurth's GC30 (btw where the heck is GC29? )...

Code: Select all

. . . . . . 9 2 .
8 . . . . . . 6 .
. . 4 . 8 . 1 . 3
. . 1 2 . . . 4 .
. 3 . 5 1 . . . .
. . . . . . . . .
9 . 2 . . . . . 4
. 5 3 6 . . . . .
1 6 . . 2 . . 8 .

c123 <-> c789:

9 2 . . . . . . .
. 6 . . . . 8 . .
1 . 3 . 8 . . . 4
. 4 . 2 . . . . 1
. . . 5 1 . . 3 .
. . . . . . . . .
. . 4 . . . 9 . 2
. . . 6 . . . 5 3
. 8 . . 2 . 1 6 .

c2345 -> c3254:

9 . 2 . . . . . .
. . 6 . . . 8 . .
1 3 . 8 . . . . 4
. . 4 . 2 . . . 1
. . . 1 5 . . 3 .
. . . . . . . . .
. 4 . . . . 9 . 2
. . . . 6 . . 5 3
. . 8 2 . . 1 6 .

9 8 2 4 7 5 3 1 6
4 7 6 3 1 2 8 9 5
1 3 5 8 9 6 7 2 4
8 6 4 9 2 3 5 7 1
7 2 9 1 5 4 6 3 8
5 1 3 6 8 7 2 4 9
6 4 7 5 3 1 9 8 2
2 9 1 7 6 8 4 5 3
3 5 8 2 4 9 1 6 7

c3254 -> c2345:

9 2 8 7 4 5 3 1 6
4 6 7 1 3 2 8 9 5
1 5 3 9 8 6 7 2 4
8 4 6 2 9 3 5 7 1
7 9 2 5 1 4 6 3 8
5 3 1 8 6 7 2 4 9
6 7 4 3 5 1 9 8 2
2 1 9 6 7 8 4 5 3
3 8 5 4 2 9 1 6 7

c789 <-> c123:

3 1 6 7 4 5 9 2 8
8 9 5 1 3 2 4 6 7
7 2 4 9 8 6 1 5 3
5 7 1 2 9 3 8 4 6
6 3 8 5 1 4 7 9 2
2 4 9 8 6 7 5 3 1
9 8 2 3 5 1 6 7 4
4 5 3 6 7 8 2 1 9
1 6 7 4 2 9 3 8 5

Again, more explanations next Monday... But I think I've found the consistent way to morph a puzzle grid to it's diagonally/rotationally symmetrical form...

[gsf]

but r6c4 is on the diagonal -- what other { 3 6 } are you basing the move on?

Ha! gsf, being so intelligent in backdoor theories and other sudoku topics, I see you finally stump on a field of which you're totally unfamiliar...

ok, now I get it
ST is a sudoku variant, something like sudoku X or DG
the added constraints are that the given puzzle solution has at least one permutation
that is invariant up to and including labelling with respect to some symmetry operation
solving such a puzzle is twofold
first find the perumtation(s) that expose the solution symmetry up to and including labelling
then use the solution symmetry as an added constraint
the key point is the upfront statement about solution symmetry

[Mauricio]

Your technique is the most beatiful technique I have ever seen, gurth.

ST is a sudoku variant, something like sudoku X or DG

No, it is not a variant, it is a uniqueness based technique, as I will prove with an example, but the same can be said in general:

Take the example given by udosuk:

Code: Select all

. . 3 . . . . . 2
. . . . . 4 . 5 .
9 . . 2 6 . 7 . .
. 1 . . . . . 8 .
. . . 9 5 . 3 . .
4 9 . . 1 . 2 . .
. . 2 . . . . . 6
. 7 . 1 . 9 . . .
. . . 8 . . 1 . .


We see that every 5 is diagonally (/) oposed to every 5, every 7 to 7, 2 to 2, 9 to 1, 8 to 4, and 3 to every 6 given, so if we reflect the sudoku with respect to / diagonal and then swap 5-5, 7-7, 2-2, 9-1, 8-4, 3-6, we will obtain the same sudoku.
So if we do the same thing (reflect and swap the same numbers) TO THE SOLUTION GRID we obtain the same solution grid as is, as if we don't, then we have 2 different solutions for that sudoku.
After singles/box-line interactions:

Code: Select all

 *-----------------------------------------------------------*
 | 678   4568  3     | 57    789   157   | 4689  1469  2     |
 | 2678  268   1678  | 37    3789  4     | 689   5     1389  |
 | 9     458   148   | 2     6     135   | 7     134   1348  |
 |-------------------+-------------------+-------------------|
 | 3     1     567   | 467   47    2     | 569   8     579   |
 | 2678  268   678   | 9     5     78    | 3     1467  147   |
 | 4     9     5678  |*367   1     378   | 2     67    57    |
 |-------------------+-------------------+-------------------|
 | 1     348   2     | 3457  347   357   | 489   3479  6     |
 | 68    7     468   | 1     234   9     | 458   234   3458  |
 | 5     34    9     | 8     2347  6     | 1     2347  347   |
 *-----------------------------------------------------------*


r6c4 must not be 3 or 6 because these 2 digits must be opposite to each other across the "/" diagonal...

The argument is right, but he did not give a proof. The proof of that fact is above, ie, if r6c4 is 3 (6), then after reflecting and swapping numbers we get that r6c4 is 6(3), and we will get a different solution grid.

Therefore r6c4=7 and the rest are all naked singles...

That is right.

So the techique is this: In a given sudoku, identify a set of moves (swap rows, columns, permute, reflect) that leave the sudoku invariant (the same clues in the same position and number). Now if we do the same moves to the solution grid we get the same grid as is, as if we don't, then the sudoku has at least 2 solutions. With that extra information we could solve easily the puzzle, the hard part will be then to find the moves (if they exist for the sudoku given).

Mauricio.
(I am still amazed of this technique, gurth, congratulations!)

[gsf]

Your technique is the most beatiful technique I have ever seen, gurth.

ST is a sudoku variant, something like sudoku X or DG

No, it is not a variant, it is a uniqueness based technique, as I will prove with an example, but the same can be said in general:

this technique can only be used on puzzles with the added assertion that a symmetric solution exists

here's an arbitrary puzzle with diagonal symmetry
can the technique be used on it?

Code: Select all

. 1 7 | . . . | . . 2
. 9 . | . . 8 | . . .
. . 2 | 6 . 4 | . . .
------+-------+------
. . . | . 9 . | 4 1 .
5 . . | . . 3 | . . .
. . . | 2 . . | 6 . .
------+-------+------
8 . . | . . . | 3 . 5
. . . | . . . | . 6 8
2 . 1 | . 6 . | . . .


[RW]

ST is a sudoku variant, something like sudoku X or DG

No, it is not a variant

I'm not sure which side to take in this debate... I know it's not a variant in the sence that i needs some added constraints, it can be applied in any standard sudoku that happens to fit the requirements. However, the requirements are not very likely to be met in a random sudoku, which is why this technique most likely can be used only in puzzles constructed for that specific purpose - which makes it kind of a variant...

this technique can only be used on puzzles with the added assertion that a symmetric solution exists

That's not enough. The initial clues must also be picked according to that symmetry. At least I think so, or can you use it on this puzzle with a symmetrical solution:

Code: Select all

. . .|. . .|7 . .
. . 9|. 4 .|. . 6
. 2 .|5 . .|. 4 .
-----+-----+-----
4 . .|. . 3|1 . .
. 8 .|6 . .|. . 2
5 . 1|. 9 .|. . .
-----+-----+-----
. . .|. 3 .|. . 9
. . 7|. . .|2 8 .
6 . .|2 . .|. 3 .

(I know, it's not very hard without this technique either)

Anyway, interesting discussion and interesting technique! Keep the puzzles coming!

RW

[gurth]

udosuk,
Your solution is the first and it gets full marks. Congratulations on winning this CHALLENGE, just as you were the first to solve the EMERALD CHALLENGE.

There are several techniques you can develop for yourselves to get expertise in recognising potentially symmetric sudokus. One is this: remember there has to be a certain minimum quota of symmetric boxes, and two boxes can only be symmetric to each other if they contain the same number of BRICKS, because no scrambling can ever change the number of bricks in a box.

There are many replies from many people, and I need time to study these before I can give answers on all points.

Mauricio, thanks for your appreciation. That will inspire me in days ahead to strive for beauty.
___________________________________________________________

[gsf]

this technique can only be used on puzzles with the added assertion that a symmetric solution exists

That's not enough. The initial clues must also be picked according to that symmetry.

right -- the first step is to transform the puzzle to match a/the symmetric solution,
then the technique can be applied
(sudoku permutations were implied in "a symmetric solution exists" -- thanks for the clarification)

[Mauricio]

I searched for example to use the new technique, and I obtained better results than I expected!

Code: Select all

1 0 0|0 0 7|0 8 0
0 5 0|8 0 0|6 0 0
0 0 0|0 4 0|0 0 7
-----+-----+-----
0 2 0|1 0 0|7 0 0
0 0 6|0 5 0|0 3 0
3 0 0|0 0 9|0 0 4
-----+-----+-----
0 4 0|3 0 0|1 0 0
2 0 0|0 7 0|0 5 0
0 0 3|0 0 6|0 0 9 ER 10.0,

now make r3c3=9, and we get

Code: Select all

1 0 0|0 0 7|0 8 0
0 5 0|8 0 0|6 0 0
0 0 9|0 4 0|0 0 7
-----+-----+-----
0 2 0|1 0 0|7 0 0
0 0 6|0 5 0|0 3 0
3 0 0|0 0 9|0 0 4
-----+-----+-----
0 4 0|3 0 0|1 0 0
2 0 0|0 7 0|0 5 0
0 0 3|0 0 6|0 0 9 ER=7.2

Other one,

Code: Select all

0 0 0|0 0 7|0 2 0
0 0 0|8 0 0|3 0 0
0 0 5|0 6 0|0 0 7
-----+-----+-----
0 2 0|1 0 0|0 0 0
0 0 4|0 5 0|0 7 0
3 0 0|0 0 9|0 0 4
-----+-----+-----
0 7 0|0 0 0|1 0 0
8 0 0|0 3 0|0 5 0
0 0 3|0 0 6|0 0 9 ER=10.6, first SE step rating=10.6, gsfr=gsfr-X=99950

make the naked double {1,9} at r1c1 and r2c2 and we get ER =9.2, can
you do more logical steps in this sudoku using your technique, gurth?

One interesant sudoku,

Code: Select all

0 0 0|0 0 7|0 6 0
0 5 0|3 0 0|8 0 0
0 0 9|0 4 0|0 0 3
-----+-----+-----
0 7 0|1 0 0|4 0 0
0 0 6|0 5 0|0 2 0
3 0 0|0 0 0|0 0 8
-----+-----+-----
0 2 0|6 0 0|1 0 0
4 0 0|0 8 0|0 0 0
0 0 7|0 0 2|0 0 5 ER=10.4, gsfr=99849, gsfr-X=99859

now make r1c1=1, r6c6=9 and r8c8=9, to get

Code: Select all

1 0 0|0 0 7|0 6 0
0 5 0|3 0 0|8 0 0
0 0 9|0 4 0|0 0 3
-----+-----+-----
0 7 0|1 0 0|4 0 0
0 0 6|0 5 0|0 2 0
3 0 0|0 0 9|0 0 8
-----+-----+-----
0 2 0|6 0 0|1 0 0
4 0 0|0 8 0|0 9 0
0 0 7|0 0 2|0 0 5 ER=3.8

and we can start with a swordfish of 9's and then singles!

This one too, make the new technique, then a swordfish of 9's and then singles!

Code: Select all

0 0 0|0 0 7|0 2 0
0 5 0|6 0 0|3 0 0
0 0 9|0 3 0|0 0 8
-----+-----+-----
0 4 0|1 0 0|8 0 0
0 0 7|0 0 0|0 4 0
3 0 0|0 0 9|0 0 6
-----+-----+-----
0 7 0|2 0 0|1 0 0
8 0 0|0 6 0|0 0 0
0 0 2|0 0 4|0 0 5 ER=9.9

This one is not rated by gsfr!

Code: Select all

1 0 0|0 6 0|8 0 0
0 0 6|0 0 2|0 0 4
0 4 0|3 0 0|0 0 0
-----+-----+-----
0 0 7|5 0 0|0 0 2
4 0 0|0 0 0|0 3 0
0 8 0|0 0 0|7 0 0
-----+-----+-----
2 0 0|0 0 3|9 0 0
0 0 0|0 7 0|0 1 0
0 6 0|8 0 0|0 0 5 ER=10.6

make the naked double reductions in the diagonal and we get ER=8.3

Other one not rated by gsfr

Code: Select all

1 0 0|0 0 7|0 2 0
0 9 0|4 0 0|0 0 8
0 0 0|0 3 0|4 0 0
-----+-----+-----
0 6 0|1 0 0|0 0 4
0 0 7|0 0 0|2 0 0
3 0 0|0 0 5|0 6 0
-----+-----+-----
0 0 6|0 8 0|9 0 0
8 0 0|0 0 4|0 0 0
0 2 0|6 0 0|0 0 5 ER=10.6, first step rating=10.6

the new technique and then 2 singles lead to

Code: Select all

1 0 0|0 0 7|0 2 0
0 9 0|4 0 0|0 0 8
0 0 5|0 3 0|4 0 0
-----+-----+-----
0 6 0|1 0 0|0 0 4
0 0 7|0 9 6|2 0 0
3 0 0|0 4 5|0 6 0
-----+-----+-----
0 0 6|0 8 0|9 0 0
8 0 0|0 0 4|0 1 0
0 2 0|6 0 0|0 0 5 ER=3.8

where there is a swordfish of 9's and then singles
Gurth, your technique is phenomenal!

[RW]

Other one,

Code: Select all

0 0 0|0 0 7|0 2 0
0 0 0|8 0 0|3 0 0
0 0 5|0 6 0|0 0 7
-----+-----+-----
0 2 0|1 0 0|0 0 0
0 0 4|0 5 0|0 7 0
3 0 0|0 0 9|0 0 4
-----+-----+-----
0 7 0|0 0 0|1 0 0
8 0 0|0 3 0|0 5 0
0 0 3|0 0 6|0 0 9 ER=10.6, first SE step rating=10.6, gsfr=gsfr-X=99950

make the naked double {1,9} at r1c1 and r2c2 and we get ER =9.2, can
you do more logical steps in this sudoku using your technique, gurth?

Certainly there's more logical steps using this technique. This technique comes with some new interesting Nishio applications:

Code: Select all

 *--------------------------------------------------------------------*
 | 19     3468   68     | 3459   149    7      | 45689  2      1568   |
 | 2467   19     267    | 8      1249   1245   | 3      1469   156    |
 | 24     348    5      | 2349   6      1234   | 489    1489   7      |
 |----------------------+----------------------+----------------------|
 | 5679   2      6789   | 1      478    348    | 5689   3689   3568   |
 | 169    1689   4      | 236    5      238    | 2689   7      12368  |
 | 3      1568   1678   | 267    278    9      | 2568   168    4      |
 |----------------------+----------------------+----------------------|
 | 24569  7      269    | 2459   2489   2458   | 1      3468   2368   |
 | 8      1469   1269   | 2479   3      124    | 2467   5      26     |
 | 1245   145    3      | 2457   12478  6      | 2478   48     9      |
 *--------------------------------------------------------------------*

*1 . . | . 1 . | . . 1
 . 1 . | .#1 1 | . 1*1
 . . . | . . 1 | . 1 .
 ------+-------+------
 . . . | 1 . . | . . .
 1-1 . | . . . | . . 1
 . 1 1 | . . . | .*1 .
 ------+-------+------
 . . . | . . . | 1 . .
 . 1 1 | . . 1 | . . .
 1 1 . | . 1 . | . . .

If r5c2=1 => r1c1=1 => r6c8=1 => r2c9=1
=> conflict (according to symmetry r2c5=1)
=> r5c2<>1 and r2c5<>1

Next there's a nice symmetrical turbot fish:

Code: Select all

 1 . . | . 1 . | . .-1
 . 1 . | . . 1 | . 1 1
 . . . | . . 1 | . 1 .
 ------+-------+------
 . . . | 1 . . | . . .
*1 . . | . . . | . .*1
 . 1 1 | . . . | . 1 .
 ------+-------+------
 . . . | . . . | 1 . .
 . 1 1 | . . 1 | . . .
-1 1 . | . 1 . | . . .
If r9c1=1 => symmetry gives r1c9=1 => no 1 in r5 => r9c1 and r1c9<>1

Brings ER down to 7.1.

Next time you get stuck there's a similar symmetrical turbot fish on digit 9.

Other one not rated by gsfr...

Very nice puzzle! SE 1.1 on my old XP2400+ gives me the "cannot solve logically" message in only 28 seconds... Never seen it that fast before, probably means that it doesn't need to follow the forcing chains very far before hitting the wall...

RW

[udosuk]

There are several techniques you can develop for yourselves to get expertise in recognising potentially symmetric sudokus. One is this: remember there has to be a certain minimum quota of symmetric boxes, and two boxes can only be symmetric to each other if they contain the same number of BRICKS, because no scrambling can ever change the number of bricks in a box.

I didn't realise the BRICKS have such significance on the symmetry issues... Anyway I'll show my way, which could be a tad bit more solid as for recognising the symmetry... However, it's quite tedious if you do it manually using pen & paper, but is fairly easy if you use a program like Simple Sudoku...

Let's take GC31 as an example:

Code: Select all

7..|96.|8..
..6|2..|.3.
.3.|..1|...
---+---+---
...|3..|4..
.6.|...|.5.
4..|.1.|..8
---+---+---
91.|.8.|..6
..2|..9|...
3..|4..|2..

The key is to recognise which digits are matching which... Let's take a look at the pencilmark grid:

Code: Select all

 7      245    145    | 9      6      345    | 8      124    1245   
 158    4589   6      | 2      457    4578   | 1579   3      14579 
 258    3      4589   | 578    457    1      | 5679   24679  24579 
----------------------+----------------------+---------------------
 1258   25789  15789  | 3      2579   25678  | 4      12679  1279   
 128    6      13789  | 78     2479   2478   | 1379   5      12379 
 4      2579   3579   | 567    1      2567   | 3679   2679   8     
----------------------+----------------------+---------------------
 9      1      457    | 57     8      2357   | 357    47     6     
 568    4578   2      | 1567   357    9      | 1357   1478   13457 
 3      578    578    | 4      57     567    | 2      1789   1579   

Now, we have to realise that for a grid with symmetrical clues, not only the solution would be symmetrical but also the pencilmark grid... For example if 19|28|37|44|55|66 are the matching pairs then a cell with candidate {1358} should have its opposite cell being {9752}...

Therefore, all matching digits in the pencilmark grids should have the isomorphic patterns of distribution over the 9 boxes... And if we count the appearances of each digit in each box it's very likely we can spot the matching pairs... With the "filtering candidate" function it's quite easy to do...

So, let's do it for this grid:

Code: Select all

digit   #times appearing in each box (sorted)
    1   0,0,0,1,2,4,4,5,5
    2   0,0,0,1,2,4,4,5,5
    3   0,0,0,0,1,2,2,3,3
    4   0,0,0,2,2,3,4,4,5
    5   0,4,4,5,5,5,5,6,6
    6   0,0,0,0,1,2,2,3,3
    7   0,4,4,5,5,6,6,7,7
    8   0,0,0,2,2,3,4,4,5
    9   0,0,0,2,2,2,5,5,6

It should be fairly easy to see the pairings: 12|36|48|55|77|99...

(In case if 3 or more box-appearance list being identical, we could do 2 things:
1. Carefully examine the shape of each pattern in each box, e.g. the number of BRICKS would be a good guideline
2. Exhaustively apply a certain type of technique eg. all naked singles, all hidden singles or all locked candidates and see if the situation changes which quite likely would...)

Anyhow, after the pairings it'd be quite obvious to identify the matching boxes in the original grid:

Code: Select all

7..|96.|8..
..6|2..|.3.
.3.|..1|...
---+---+---
...|3..|4..
.6.|...|.5.
4..|.1.|..8
---+---+---
91.|.8.|..6
..2|..9|...
3..|4..|2..

12|36|48|55|77|99
b2<->b7, b3<->b4, b5<->b9 and b1,b6,b8 should be on a diagonal

Next we try to align the correct box on a diagonal... Exchanging the 2nd & 3rd band seems the easiest way:

Code: Select all

7..|96.|8..
..6|2..|.3.
.3.|..1|...
---+---+---
91.|.8.|..6
..2|..9|...
3..|4..|2..
---+---+---
...|3..|4..
.6.|...|.5.
4..|.1.|..8

The next step is to put the correct digit in each box on the diagonal... In b5 the 9 should be moved to r6c6 while in b9 the 4 & 8 shouldn't be in r7c7 & r9c9... Exchanging r5 with r6 and r7 with r9 should do:

Code: Select all

7..|96.|8..
..6|2..|.3.
.3.|..1|...
---+---+---
91.|.8.|..6
3..|4..|2..
..2|..9|...
---+---+---
4..|.1.|..8
.6.|...|.5.
...|3..|4..

Now the grid is perfectly symmetrical about the "\" diagonal... Placing {59} in r2c2+r3c3, {57} in r4c4+r5c5, {79} in r7c7+r9c9, and the puzzle is solved easily

Code: Select all

724|963|815
156|278|934
839|541|672
---+---+---
915|782|346
378|456|291
642|139|587
---+---+---
493|615|728
267|894|153
581|327|469

r6597 -> r5679

724|963|815
156|278|934
839|541|672
---+---+---
915|782|346
642|139|587
378|456|291
---+---+---
581|327|469
267|894|153
493|615|728

r456 <-> r789

724|963|815
156|278|934
839|541|672
---+---+---
581|327|469
267|894|153
493|615|728
---+---+---
915|782|346
642|139|587
378|456|291

But that is not the innotative part... Here it is:

Once you've confirmed the symmetry (diagonal or rotational) and identified the pairing digits/matching boxes, you can just operate the symmetry moves on the original grid!

For this one:

Code: Select all

 7      245    145    | 9      6      345    | 8      124    1245   
 158    4589   6      | 2      457    4578   | 1579   3      14579 
 258    3      4589   | 578    457    1      | 5679   24679  24579 
----------------------+----------------------+---------------------
 1258   25789  15789  | 3      2579   25678  | 4      12679  1279   
 128    6      13789  | 78     2479   2478   | 1379   5      12379 
 4      2579   3579   | 567    1      2567   | 3679   2679   8     
----------------------+----------------------+---------------------
 9      1      457    | 57     8      2357   | 357    47     6     
 568    4578   2      | 1567   357    9      | 1357   1478   13457 
 3      578    578    | 4      57     567    | 2      1789   1579   

We already knew that b1,b6,b8 are the "diagonal boxes"... We can set out to find the "diagonal cells" in these boxes... And they must be along one of the "broken diagonals" (a total of 6 in each 3x3 box)...

In b1, 7 is already in r1c1, so the other 2 "diagonal cells" are either r2c2+r3c3 or r2c3+r3c2...
- Which must be in the former case, so r2c2+r3c3={59}

In b6, 5 is already in r5c8, so the other 2 "diagonal cells" are either r4c7+r6c9 or r4c9+r6c7...
- The latter case is true, therefore r4c9+r6c7={79}

In b8, 9 is already in r8c6, so the other 2 "diagonal cells" are either r7c4+r9c5 or r7c5+r9c4...
- Obviously the first case... And {57} are already there...

So all the morphing is skipped and we can directly progress to the solution without redrawing extra grids!

With this in mind, GC32 & GC33 are quite easily solved:

Code: Select all

GC32

8.4|...|.92
3.7|...|51.
...|...|3.4
---+---+---
.46|..8|9..
9.2|...|.7.
13.|...|...
---+---+---
...|31.|8..
...|5.2|...
...|.64|...

 8       156     4       | 167     357     13567   | 67      9       2       
 3       269     7       | 24689   2489    69      | 5       1       68     
 256     12569   159     | 126789  25789   15679   | 3       68      4       
-------------------------+-------------------------+------------------------
 57      4       6       | 127     2357    8       | 9       235     135     
 9       58      2       | 146     345     1356    | 146     7       13568   
 1       3       58      | 24679   24579   5679    | 246     24568   568     
-------------------------+-------------------------+------------------------
 24567   25679   59      | 3       1       79      | 8       2456    5679   
 467     16789   1389    | 5       789     2       | 1467    346     13679   
 257     125789  13589   | 789     6       4       | 127     235     13579   

Just from the distribution of pencilmarks, it's quite easily to match b2 with b7, b3 with b4 and b5 with b9... Which enable us to spot the pairings: 12|34|56|77|88|99...

Then we spot the "diagonal cells" in the "diagonal boxes" (b1,b6,b8): r3c2=9 and r6c9=8... Afterwards the puzzle is solved with naked singles!

Code: Select all

GC33

2..|5..|...
.3.|.1.|...
...|..8|9.7
---+---+---
.5.|.3.|.6.
..9|...|32.
1..|...|...
---+---+---
...|.72|..1
..8|..3|...
..6|...|.54

 2       146789  147     | 5       469     4679    | 1468    1348    368     
 456789  3       457     | 24679   1       4679    | 24568   48      2568   
 456     146     145     | 2346    246     8       | 9       134     7       
-------------------------+-------------------------+------------------------
 478     5       247     | 124789  3       1479    | 1478    6       89     
 4678    4678    9       | 14678   4568    14567   | 3       2       58     
 1       24678   2347    | 246789  245689  45679   | 4578    4789    589     
-------------------------+-------------------------+------------------------
 3459    49      345     | 4689    7       2       | 68      389     1       
 4579    12479   8       | 1469    4569    3       | 267     79      269     
 379     1279    6       | 189     89      19      | 278     5       4       

Quite so obviously, b2<->b4, b3<->b7, b6<->b8 and the pairings are 15|67|89|22|33|44...
"Diagonal cells" in b1: r3c3=4
"Diagonal cells" in b5: r6c4=2, r5c6=4
"Diagonal cells" in b9: r7c8=3, r8c7=2

And all singles afterwards...

[ronk]

Anybody know the percentage of solutions that can be permuted (morphed) into an Emerald solution?

TIA, Ron

[Mauricio]

One difficult sudoku rotationally symmetric, can we apply gurth's symmetry technique here?

Code: Select all

3 . .|2 . .|. 5 .
. 6 .|. 3 .|. . 9
. . 8|. . 6|3 . .
-----+-----+-----
7 . .|1 . .|5 . .
. 4 .|. 5 .|. 6 .
. . 5|. . 9|. . 3
-----+-----+-----
. . 7|4 . .|2 . .
1 . .|. 7 .|. 4 .
. 5 .|. . 8|. . 7 ER=9.8

Take this rotationally symmetric sudoku, we can apply the symmetry technique

Code: Select all

. . .|7 . .|3 . .
. 2 .|. . 6|. 5 .
. . 8|. 2 .|. . 9
-----+-----+-----
. 3 .|1 . .|. . 4
. . 9|. . .|1 . .
6 . .|. . 9|. 7 .
-----+-----+-----
1 . .|. 8 .|2 . .
. 5 .|4 . .|. 8 .
. . 7|. . 3|. . . ER=10.5 ,gsfr-X=99975

With the symmetry technique we get r5c5=5:

Code: Select all

. . .|7 . .|3 . .
. 2 .|. . 6|. 5 .
. . 8|. 2 .|. . 9
-----+-----+-----
. 3 .|1 . .|. . 4
. . 9|. 5 .|1 . .
6 . .|. . 9|. 7 .
-----+-----+-----
1 . .|. 8 .|2 . .
. 5 .|4 . .|. 8 .
. . 7|. . 3|. . .ER=9.3

//////////////////////////////////////////////////////////////////////////
To show that the symmetries need not be twofold, I present this example with 90 degrees symmetry

Code: Select all

. 7 .|. 4 .|. 8 .
9 . .|. . 1|. . 6
. . 8|6 . .|7 . .
-----+-----+-----
. 2 .|. 1 .|9 . .
1 . .|2 . 4|. . 3
. . 7|. 3 .|. 4 .
-----+-----+-----
. . 9|. . 8|6 . .
8 . .|3 . .|. . 7
. 6 .|. 2 .|. 9 . ER 9.1

With the symmetry technique we get that r5c5=5:

Code: Select all

. 7 .|. 4 .|. 8 .
9 . .|. . 1|. . 6
. . 8|6 . .|7 . .
-----+-----+-----
. 2 .|. 1 .|9 . .
1 . .|2 5 4|. . 3
. . 7|. 3 .|. 4 .
-----+-----+-----
. . 9|. . 8|6 . .
8 . .|3 . .|. . 7
. 6 .|. 2 .|. 9 . ER=7.1

A sudoku with 180 degrees symmetry and almost 90 degrees symmetry

Code: Select all

. 2 .|9 . .|. 8 .
9 . .|. . 6|. . 1
. . 6|. 3 .|9 . .
-----+-----+-----
. 7 .|1 . .|. . 8
. . 4|. . .|2 . .
6 . .|. . 3|. 9 .
-----+-----+-----
. . 7|. 1 .|8 . .
3 . .|8 . .|. . 7
. 6 .|. . 7|. 4 . ER=9.3


With the symmetry technique we get then that r5c5=5, so

Code: Select all

. 2 .|9 . .|. 8 .
9 . .|. . 6|. . 1
. . 6|. 3 .|9 . .
-----+-----+-----
. 7 .|1 . .|. . 8
. . 4|. 5 .|2 . .
6 . .|. . 3|. 9 .
-----+-----+-----
. . 7|. 1 .|8 . .
3 . .|8 . .|. . 7
. 6 .|. . 7|. 4 . ER=8.3

When we solve the cells r6c4=2 and r4c6=4 we could use the 90 degrees symmetry technique.