The New Sudoku Players' Forum

[urhegyi]

gurths-theorem.png (14.69 KiB) Viewed 1163 times

In the discussion on this forum about generating a new sudoku from an existing solution grid m_b_metcalf generated a nice example with "crossword symmetry" based on the solution grid I created myself (22 clues and rated ED=9.0/1.2./1.2). Thank you very much Mike for providing this. Because the solution was created with the symmetrical placement theorem it can be used to solve this very extreme sudoku and reduce so the hardness level with a considerable amount. I wonder nobody has detected this or at least posted remarks on it. Now have fun solving it.
Further exploring the batch of generated examples, I found a few very good Gurth theorem examples:

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...........4.35...97.6..8...9....5...6.8.2.4...5....1...2..4.31...57.6...........

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..........24..51..97.6.......8....633.18.29.774....2.......4.31..95..68..........

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..............51.69736.....29......3.6.8.2.4.7......18.....47314.95..............

[ghfick]

Andrew Stuart's solver identifies Gurth's Theorem and gives a solution path with nothing harder than AICs.

[mith]

thunderstorm-t38219.html

[urhegyi]

After doing more study on the theorem. I came to these conclusions:
a) First rotate the grid by 180 degrees to check the theorem is applicable and find the mappings.
b) Looking which candidate maps to itselfs: in this solution grid digit 5 which can be placed at R5C5.
c) Check sudoku solves with singles after placement at center.
d) yes:done
no : finding more eliminations by using other solving methodes.
If you come to an elimination by a method , there's always a second one which eliminates the complementary candidate.

The three examples I posted before don't solve by putting 5 at the center.
I will study them further.

To do: rotation by 90 degrees and mirroring on diagonal axis

[Leren]

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*-----------*
|.2.|6..|..5|
|4..|..1|.3.|
|..9|...|...|
|---+---+---|
|8..|.23|.9.|
|...|4.6|...|
|.1.|78.|..2|
|---+---+---|
|...|...|1..|
|.7.|9..|..6|
|5..|..4|.8.|
*-----------*

.2.6....54....1.3...9......8...23.9....4.6....1.78...2......1...7.9....65....4.8.

Leren

[StrmCkr]

A bit more comicated then a reflection to confirm the semtrics.
Whrn considering how it applies to other puzzkes all transpositions must be considered

Ps there is There is also diffrent symetry techniques

[Cenoman]

I - urhegyi puzzles

For the 1st puzzle, the central symmetry is not useful to place +5r5c5, already placed with basics, but it helps to solve it in one step:
Central symmetry: digit pairs 19, 28, 37, 46, 55

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 +------------------------+--------------------+------------------------+
 |  1256    125    16     |  247  c89    89    |  134-2  267    23467   |
 |  1268    128    4      |  27    3     5     |  19-2   2679   2679    |
 |  9       7      3      |  6     24    1     |  8      25     245     |
 +------------------------+--------------------+------------------------+
 |  23478   9     g78     |  134   146  f67    |  5      2678   23678   |
 |  137     6      17     |  8     5     2     |  39     4      379     |
 |  23478   2348   5      |  34   d469  e679   |  23     1      23678   |
 +------------------------+--------------------+------------------------+
 | a568    a58     2      |  9    b68    4     |  7      3      1       |
 |  1348    1348   19-8   |  5     7     38    |  6      289    2489    |
 |  34678   348    679-8  |  12    12    368   |  49     589    4589    |
 +------------------------+--------------------+------------------------+

(8)r7c12 = r7c5 - (8=9)r1c5 - r6c5 = (9-7)r6c6 = r4c6 - (7=8)r4c3 => -8 r89c3; inferred symmetric eliminations -2 r12c7; ste

Second puzzle, solved with the symmetry placement:

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 +------------------------+------------------------+------------------------+
 |  1568    1568    356   |  2479   24789   789    |  38    24579   24569   |
 |  68      2       4     |  379    3789    5      |  1     79      69      |
 |  9       7       35    |  6      1248    18     |  38    245     245     |
 +------------------------+------------------------+------------------------+
 |  2       9       8     |  147    1457    17     |  45    6       3       |
 |  3       56      1     |  8      46+5    2      |  9     45      7       |
 |  7       4       56    |  39     3569    369    |  2     1       8       |
 +------------------------+------------------------+------------------------+
 |  568     568     27    |  29     2689    4      |  57    3       1       |
 |  14      13      9     |  5      1237    137    |  6     8       24      |
 |  14568   13568   27    |  123    12368   1368   |  457   2459    2459    |
 +------------------------+------------------------+------------------------+

Central symmetry: digit pairs 19, 28, 37, 46, 55 =>+5r5c5; ste

Third puzzle solved in two steps:

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 +---------------------+------------------------+---------------------+
 |  156   15     156   |  2479   24789   789    |  3     2789   479   |
 |  8     24     24    |  379    379     5      |  1     79     6     |
 |  9     7      3     |  6      1248    18     |  458   258    45    |
 +---------------------+------------------------+---------------------+
 |  2     9      8     |  147    1457    17     |  56    567    3     |
 |  135   6      15    |  8      37+5    2      |  59    4      579   |
 |  7     345    45    |  39     3569    369    |  2     1      8     |
 +---------------------+------------------------+---------------------+
 |  56    258    256   |  29     2689    4      |  7     3      1     |
 |  4     13     9     |  5      137     137    |  68    68     2     |
 |  136   1238   7     |  123    12368   1368   |  459   59     459   |
 +---------------------+------------------------+---------------------+

1. Central symmetry: digit pairs 19, 28, 37, 46, 55 =>+5r5c5;

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 +---------------------+------------------------+---------------------+
 |  156   15     56    | b2479   24789  g789    |  3    h2789  h49    |
 |  8     24     24    | a379    379     5      |  1    i9-7    6     |
 |  9     7      3     |  6      1248    18     |  458   258    45    |
 +---------------------+------------------------+---------------------+
 |  2     9      8     | b147    147     17     |  56    56     3     |
 |  3     6      1     |  8      5       2      |  9     4      7     |
 |  7     45     45    |  39     369    f369    |  2     1      8     |
 +---------------------+------------------------+---------------------+
 |  56    258    256   |  29     2689    4      |  7     3      1     |
 |  4     13     9     |  5      137     137    |  68    68     2     |
 | d16    1238   7     | c123    12368  e1368   |  45    59     459   |
 +---------------------+------------------------+---------------------+

2. (7)r2c4 = (7-41)r14c4 = r9c4 - (1=6)r9c1 - r9c6 = (6-9)r6c6 = r1c6 - r1c89 = (9)r2c8 =>-7r2c8; ste

II - Leren's puzzle

Double diagonal symmetry:

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 +-------------------------+-----------------------+-------------------------+
 |  37+1   2       1378    |  6      3479   789    |  4789    147     5      |
 |  4      68+5    5678    |  258    579    1      |  26789   3       789    |
 |  1367   3568    9       |  2358   3457   2578   |  2468+7  12467   1478   |
 +-------------------------+-----------------------+-------------------------+
 |  8      456     4567    |  5+1    2      3      |  4567    9       147    |
 |  2379   359     2357    |  4      19+5   6      |  3578    157     1378   |
 |  369    1       3456    |  7      8      5+9    |  3456    456     2      |
 +-------------------------+-----------------------+-------------------------+
 |  2369   34689   2468+3  |  2358   3567   2578   |  1       2457    3479   |
 |  123    7       12348   |  9      135    258    |  2345    24+5    6      |
 |  5      369     1236    |  123    1367   4      |  2379    8       37+9   |
 +-------------------------+-----------------------+-------------------------+

First diagonal symmetry: digit pairs 11, 24, 37, 55, 68, 99
Second diagonal symmetry: digit pairs 19, 26, 33, 48, 55, 77

=> +5r5c5 (common invariant digit), +1r1c1, +5r2c2, +1r4c4, +9r6c6, +5r8c8, +9r9c9, +3r7c3, +7r3c7; ste

[Mauriès Robert]

Hi all,
Where can I find demonstrations of Gurth's theorems ?
Robert

[Cenoman]

Hi Robert,
Maybe could you start wit this Help Page in Andrew Stuart's solver. Follow the links inside. Furthermore, search in the forum "Advanced solving techniques" with the keywords "symmetry" "automorphism"

[Mauriès Robert]

Hi Cenoman,
Thank you for these indications.
I understood well that it was about automorphism, but I have not yet found the rigorous demonstration of the statement (attributed to Gurth !!): "if, for a single solution puzzle, there is an automorphism of the (revealed) clues of a puzzle then there is this same automorphism of the solution of this puzzle".
Cordialy
Robert

[eleven]

Note that the puzzle must be kown to be unique.
Then it is easily proven:

If you have such a symmetric puzzle, where for all givens also the symmetric number (according to the automorphism mapping) is given in the symmetric cell (in this case symmetric to the center), you can apply the automorphism to a solution grid to get the givens at the same place. If not all other solution numbers have the symmetry property too, you therefore would have a second solution for the given puzzle.

[Cenoman]

Hi Robert,
Here, you find a rationale in one sentence.

This rationale was exact Gurth's wording, in this thread
Note also in the same thread, Ravel's proposal, to name this technique GSP: Gurth's Symmetrical Placement.

Added: Gurth's rationale is in line with eleven's