The New Sudoku Players' Forum

[Leren]

Code: Select all

*-----------*
|...|4..|.5.|
|..1|..9|7..|
|8..|.6.|..2|
|---+---+---|
|..6|8..|4..|
|.5.|.1.|...|
|7..|...|..9|
|---+---+---|
|.4.|.2.|..6|
|...|..7|1..|
|..9|5..|.8.|
*-----------*
...4...5...1..97..8...6...2..68..4...5..1....7.......9.4..2...6.....71....95...8.

[P.O.]

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..1.....2.3.4...5.9...6.8...6...2.....4.3.9.....8...1...2.1...6.5...9.3.8.....4..   
...4...5...1..97..8...6...2..68..4...5..1....7.......9.4..2...6.....71....95...8.


aren’t these 2 puzzles isomorphic? from a template point of view they are similar.

[m_b_metcalf]

Code: Select all

..1.....2.3.4...5.9...6.8...6...2.....4.3.9.....8...1...2.1...6.5...9.3.8.....4..   
...4...5...1..97..8...6...2..68..4...5..1....7.......9.4..2...6.....71....95...8.


aren’t these 2 puzzles isomorphic? from a template point of view they are similar.

My own minlex program tells me they look very alike:

Code: Select all

.....1..2..3....4..5..6.7....7..8..5.1.4.....6...2.3....8..7.6..4.3..1..2...5....
.....1..2..3....4..5..6.7....7..8..5.1.4.....6...2.3....8..7.6..4.3..1..2...5....

Mike

[P.O.]

i don't have a procedure for testing isomorphism so i couldn't be positive but isomorphic puzzles have invariants in common that i can calculate like the template numbers at the start or the clue configuration and these two match
of course, the opposite is not true, two puzzles having the same invariants are not necessarily isomorphic

the resolution path is also very similar:
basics:

Hidden Text: Show

Code: Select all

QUINTE BOX: ((1 1 1) (2 3 6 9)) ((1 2 1) (2 3 6 7 9)) ((1 3 1) (2 3 7)) ((2 2 1) (2 3 6)) ((3 2 1) (3 7 9))
(((2 1 1) (2 3 4 5 6)) ((3 3 1) (3 4 5 7)))

QUINTE BOX: ((4 6 5) (2 3 5)) ((5 6 5) (2 3 4 6)) ((6 4 5) (2 3 6)) ((6 5 5) (3 4 5)) ((6 6 5) (2 3 4 5 6))
(((4 5 5) (3 5 7 9)) ((5 4 5) (2 3 6 7 9)))


Code: Select all

2369   23679  237    4      378    1238   3689   5      138             
45     236    1      23     358    9      7      346    348             
8      379    45     137    6      135    39     1349   2               
1239   1239   6      8      79     235    4      1237   1357           
2349   5      2348   79     1      2346   2368   2367   378             
7      1238   2348   236    345    23456  23568  1236   9               
135    4      3578   139    2      138    359    379    6               
2356   2368   2358   369    3489   7      1      2349   345             
1236   12367  9      5      34     1346   23     8      347             

8r128c5 => r5c4 <> 7
 r1c5=8 - c5n7{r1 r4}
 r2c5=8 - r2n5{c5 c1} - c1n4{r2 r5} - r5n9{c1 c4}
 r8c5=8 - r7n8{c6 c3} - c3n7{r7 r1} - r3n7{c2 c4}


basics:

Hidden Text: Show

Code: Select all

( n9r5c4   n7r4c5   n7r3c4   n9r8c5   n1r7c4   n8r7c6 )

intersections:
((((4 0) (9 5 8) (3 4)) ((4 0) (9 6 8) (3 4 6))))

PAIR ROW: ((3 2 1) (3 9)) ((3 7 3) (3 9)) 
(((3 6 2) (1 3 5)) ((3 8 3) (1 3 4 9)))

( n9r7c8   n7r9c9   n7r5c8   n7r7c3   n7r1c2 )

QUAD ROW: ((1 3 1) (2 3)) ((1 5 2) (3 8)) ((1 6 2) (1 2 3)) ((1 9 3) (1 3 8))
(((1 1 1) (2 3 6 9)) ((1 7 3) (3 6 8 9)))

intersection:
((((8 0) (5 7 6) (2 3 6 8)) ((8 0) (6 7 6) (2 3 5 6 8))) ( n3r5c9 ))

TRIPLET BOX: ((1 9 3) (1 8)) ((2 9 3) (4 8)) ((3 8 3) (1 4))
(((2 8 3) (3 4 6)))

TRIPLET ROW: ((2 2 1) (2 3 6)) ((2 4 2) (2 3)) ((2 8 3) (3 6))
(((2 5 2) (3 5 8)))


Code: Select all

69     7      23     4      38     123    69     5      18             
45     236    1      23     58     9      7      36     48             
8      39     45     7      6      15     39     14     2               
1239   1239   6      8      7      235    4      12     15             
24     5      248    9      1      246    268    7      3               
7      1238   2348   236    345    23456  2568   126    9               
35     4      7      1      2      8      35     9      6               
2356   2368   2358   36     9      7      1      234    45             
1236   1236   9      5      34     346    23     8      7       

2r1c36 => r4c2 <> 3
 r1c3=2 - b1n3{r1c3 r23c2}
 r1c6=2 - r1n1{c6 c9} - r4c9{n1 n5} - r4c6{n25 n3}
 
r1n1{c9 c6} - r3c6{n1 n5} - r4n5{c6 c9} => r4c9 <> 1

ste.


[Cenoman]

Sequentially:
swap stacks 1-2, swap rows 2-3, swap columns 1-3, 5-6, 8-9, swap rows 4-5, circularly swap rows 789-978, rotate half turn, then run digit permutation 123456789 -> 157483296, and surprise :

Code: Select all

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

[eleven]

Just to show, how it can be done manually:

With the information (or suspicion), that there is a digital symmetry we can try to bring it into a normal form:

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

If it is a 2-folded symmetry, the 4 boxes with 2 digits, must either be in b1379 or b2468, the first not possible here.
So switch stacks 12:

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

No chance for a central symmetry, so try to bring them to diagonal symmetry (first: main diaginal):
257 maps to 257 (b37), so (b5) 6 maps to itself, 57 to one another and 2 to itself. Then 18 and 49 must be pairs (b24), the missing 3 maps to itself.
The 6 in b159 must be in the main diagonal. Let's move 6 in b5 to the center (all positions in the main diagonal are possible), swap r45c56:

Code: Select all

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

Now let b5 be fixed and try to adjust the 49's in b68, eg by switching c78 and rotating r789 to 978.
(Simpler is switching r79, but all adjustments will lead to a symmetric puzzle in normal form - if it has the symmetry)

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

This also fixes b9 (if it doesn't, it does not have this symmetry).
Now switch r13 to bring 5 in b3 symmetric to 7r7c3, and switch c12 for digits 52 in b7

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

Voila, also b124 fit.
Diagonal symmetry, (18)(2)(3)(49)(57)(6)

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+----------------------+----------------------+----------------------+
| 6      137    135    | 8      45     379    | 1349   39     2      |
| 358    23     9      | 45     1      236    | 346    7      348    |
| 378    4      23-18  | 2369   237    23679  | 5      3689   138    |
+----------------------+----------------------+----------------------+
| 1      79     2346   | 23-49  2348   5      | 2367   2368   378    |
| 79     8      235    | 1239   6      1239   | 1237   4      1357   |
| 345    236    23456  | 7      2348   23-18  | 1236   23568  9      |
+----------------------+----------------------+----------------------+
| 3489   369    7      | 2356   2358   2368   | 23-49  1      345    |
| 34     5      1346   | 1236   9      12367  | 8      23     347    |
| 2      139    138    | 135    3578   4      | 379    359    6      |
+----------------------+----------------------+----------------------+

Remove all but 236 from the main diagonal, stte

[Leren]

Another predictably brilliant post by eleven. The second time he's done this, on a topic that most of us mortals would think could only be achievable by a computer.

He left out the final step, which was to identify the locations of the "diagonal" cells in the puzzle. Here is the solution.

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*--------------------------------------------------------------*
| 2369   23679  237   | 4     78-3 *23-18 | 3689   5      138  |
| 45     236    1     |*23    58-3  9     | 7      346    348  |
| 8      379    45    | 17-3 *6     15-3  | 39     1349   2    |
|---------------------+-------------------+--------------------|
| 19-23  19-23 *6     | 8     79    235   | 4      1237   1357 |
|*23-49  5      48-23 | 79    1     2346  | 2368   2367   378  |
| 7     *23-18  48-23 | 236   345   23456 | 23568  1236   9    |
|---------------------+-------------------+--------------------|
| 135    4      3578  | 139   2     138   | 59-3   79-3  *6    |
| 2356   2368   2358  | 369   3489  7     | 1     *23-49  45-3 |
| 1236   12367  9     | 5     34    1346  |*23     8      47-3 |
*--------------------------------------------------------------*

Scrambled Main Diagonal (TLBR) Symmetry [18] [49] [57] + [2] [3] [6]. Elimination Cells r1c6, r2c4, r3c5, r4c3, r5c1, r6c2, r7c9, r8c8, r9c7. Eliminations as marked; stte

I wrote the code for this a long time ago as a joke, thinking that a human couldn't reasonably be expected to reproduce it manually. Eleven 2, Leren 0

A corollary of this that I was thinking about today was that the majority of GSP symmetry puzzles are scrambled. The ones that have been made up are the minory of unscrambled ones. I'll stop there.

Leren

[m_b_metcalf]

Another predictably brilliant post by eleven. The second time he's done this, on a topic that most of us mortals would think could only be achievable by a computer.

Yes. I wrote a program to do this (to look among the Hardest Sudokus for suitable entries for the now defunct Patterns Game). Here's what it came up with:

Code: Select all

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

.5..1....7.......9..68..4....1..97..8...6...2...4...5...95...8......71...4..2...6

A corollary of this that I was thinking about today was that the majority of GSP symmetry puzzles are scrambled. The ones that have been made up are the minory of unscrambled ones.

I produce these using yet another program I wrote.

Mike

[m_b_metcalf]

I produce these using yet another program I wrote.

This was a nice one I found years ago.

[Cenoman]

Code: Select all

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

.5..1....7.......9..68..4....1..97..8...6...2...4...5...95...8......71...4..2...6

Code: Select all

 +------------------------+-----------------------+------------------------+
 |  23-49  5       2348   |  79    1      2346    |  2368    2367   378    |
 |  7      23-18   2348   |  236   345    23456   |  23568   1236   9      |
 |  1239   1239    6      |  8     79     235     |  4       1237   1357   |
 +------------------------+-----------------------+------------------------+
 |  45     236     1      |  23    358    9       |  7       346    348    |
 |  8      379     45     |  137   6      135     |  39      1349   2      |
 |  2369   23679   237    |  4     378    23-18   |  3689    5      138    |
 +------------------------+-----------------------+------------------------+
 |  1236   12367   9      |  5     34     1346    |  23      8      347    |
 |  2356   2368    2358   |  369   3489   7       |  1       23-49  345    |
 |  135    4       3578   |  139   2      138     |  359     379    6      |
 +------------------------+-----------------------+------------------------+

Main diagonal automorphism with digit permutation [18] [2] [3] [49] [57] [6] => main diagonal solutions all in {2,3,6} => -49 r1c1, r8c8, -18 r2c2, r6c6; ste

Thank you, Mike, for the series of symmetric puzzles