The New Sudoku Players' Forum

[P.O.]

this puzzle has the IBPU, TDC, IBPA patterns mentioned in this article, whose knowledge makes its resolution trivial, particularly the IBPA pattern which allows you to directly position the 9 templates on the grid

Code: Select all

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

.2.4.............6.5.7.912.3....59.8.7831..4..4..........5...9...7.3....5...9.2.1


basics:

Hidden Text: Show

Code: Select all

( n4r4c5   n5r6c3   n7r4c8   n9r5c1   n9r6c4   n9r1c9   n1r6c8   n7r7c9   n7r9c6
  n9r8c2   n9r2c3   n4r9c3   n7r6c5   n8r6c6 )

intersection:
((((6 0) (8 8 9) (5 6 8)) ((6 0) (9 8 9) (3 6 8))))

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


Code: Select all

1678  2     136   4     568   136   3578  358   9             
47    138   9     128   258   123   47    358   6             
468   5     36    7     68    9     1     2     34             
3     16    126   26    4     5     9     7     8             
9     7     8     3     1     26    56    4     25             
26    4     5     9     7     8     36    1     23             
1268  1368  1236  5     268   1246  348   9     7             
1268  9     7     1268  3     1246  458   568   45             
5     368   4     68    9     7     2     368   1           


this puzzle is more difficult but its resolution is also trivial knowing that it has the IBPA pattern
...4.....78...34....6..912..1....97....3.2...6.5.7.....31.6..9.8......6......72..

[Cenoman]

First puzzle has several symmetries (aka automorphisms). Here are four (not exhaustively):

Code: Select all

 +-----------------------+----------------------+--------------------+
 |  168-7  2      136    |  4      568   136    |  3578   358   9    |
 |  47     18-3   9      |  128    258   123    |  47     358   6    |
 |  468    5      6-3    |  7      68    9      |  1      2     34   |
 +-----------------------+----------------------+--------------------+
 |  3      16     126    |  6-2    4     5      |  9      7     8    |
 |  9      7      8      |  3      1     26     |  56     4     25   |
 |  26     4      5      |  9      7     8      |  36     1     23   |
 +-----------------------+----------------------+--------------------+
 |  1268   1368   1236   |  5      268   1246   |  8-34   9     7    |
 |  1268   9      7      |  1268   3     1246   |  458    68-5  45   |
 |  5      368    4      |  68     9     7      |  2      368   1    |
 +-----------------------+----------------------+--------------------+

Main diagonal symmetry, with digit relabelling (1-1)(2-7)(3-4)(5-9)(6-6)(8-8) =>main diagonal contains only (1,6,8); ste

Code: Select all

 +-----------------------+----------------------+--------------------+
 |  1678   2      136    |  4      568   136    |  3578   358   9    |
 |  47     138    9      |  128    258   123    |  47     5-38  6    |
 |  468    5      36     |  7      68    9      |  1      2     34   |
 +-----------------------+----------------------+--------------------+
 |  3      16     126    |  26     4     5      |  9      7     8    |
 |  9      7      8      |  3      1     26     |  56     4     25   |
 |  26     4      5      |  9      7     8      |  36     1     23   |
 +-----------------------+----------------------+--------------------+
 |  1268   1368   1-236  |  5      268   1246   |  348    9     7    |
 |  1268   9      7      |  1268   3     1246   |  458    568   45   |
 |  5      368    4      |  68     9     7      |  2      368   1    |
 +-----------------------+----------------------+--------------------+

Second diagonal symmetry, with digit relabelling (1-1)(2-4)(3-7)(5-5)(6-8)(9-9) =>second diagonal contains only (1,5,9); ste

Code: Select all

 +-----------------------+----------------------+--------------------+
 |  17-68  2      136    |  4      568   136    |  7-358  358   9    |
 |  47     138    9      |  1-28   258   123    |  47     358   6    |
 |  4-68   5      36     |  7      68    9      |  1      2     34   |
 +-----------------------+----------------------+--------------------+
 |  3      16     126    |  26     4     5      |  9      7     8    |
 |  9      7      8      |  3      1     26     |  56     4     25   |
 |  26     4      5      |  9      7     8      |  36     1     23   |
 +-----------------------+----------------------+--------------------+
 |  1268   1368   1236   |  5      268   1246   |  348    9     7    |
 |  1268   9      7      |  1268   3     1246   |  458    568   45   |
 |  5      368    4      |  68     9     7      |  2      368   1    |
 +-----------------------+----------------------+--------------------+

Column Sticks symmetry: swap bands 2-3, then swap columns 2-3, 5-6, 8-9, with digit relabelling (1-1)(2-3)(4-4)(5-6)(7-7)(8-9)
=> sticks r123c1, r123c4, r123c7 contain only (1,4,7); ste

Code: Select all

 +-----------------------+----------------------+--------------------+
 |  1678   2      136    |  4      568   136    |  3578   358   9    |
 |  47     138    9      |  128    258   123    |  47     358   6    |
 |  468    5      36     |  7      68    9      |  1      2     34   |
 +-----------------------+----------------------+--------------------+
 |  3      16     126    |  6-2    4     5      |  9      7     8    |
 |  9      7      8      |  3      1     2-6    |  56     4     25   |
 |  26     4      5      |  9      7     8      |  36     1     23   |
 +-----------------------+----------------------+--------------------+
 |  1268   1368   1236   |  5      268   1246   |  348    9     7    |
 |  1268   9      7      |  1268   3     1246   |  458    568   45   |
 |  5      368    4      |  68     9     7      |  2      368   1    |
 +-----------------------+----------------------+--------------------+

Central symmetry, with digit relabelling (1-1)(2-3)(4-7)(5-9)(6-8) => -2r4c4, -6r5c6 (e.g.); ste



Second puzzle is rated higher by S.E., but it has the same symmetries:

Code: Select all

 +-----------------------+-------------------------+-----------------------+
 |  1     259     239    |  4      258      568    |  67    38     679     |
 |  7     8       29     |  126    12       3      |  4     5      69      |
 |  345   45      6      |  7      58       9      |  1     2      38      |
 +-----------------------+-------------------------+-----------------------+
 |  234   1       2348   |  68-5   458      4568   |  9     7      2458    |
 |  49    479     4789   |  3      18-459   2      |  568   148    14568   |
 |  6     249     5      |  189    7        18-4   |  38    1348   12348   |
 +-----------------------+-------------------------+-----------------------+
 |  245   3       1      |  258    6        458    |  8-57  9      4578    |
 |  8     24579   2479   |  1259   123459   145    |  35    6      1345    |
 |  459   6       49     |  1589   134589   7      |  2     1348   18-345  |
 +-----------------------+-------------------------+-----------------------+

Main diagonal symmetry, with digit relabelling (1-1)(2-7)(3-4)(5-9)(6-6)(8-8) =>main diagonal contain only (1,6,8); lclste
(NP 18b5p59 =>-8r4r4; ste)

Code: Select all

 +-----------------------+-------------------------+-----------------------+
 |  1     259     239    |  4      258      568    |  67    38     9-67    |
 |  7     8       29     |  126    12       3      |  4     5      69      |
 |  345   45      6      |  7      58       9      |  1     2      38      |
 +-----------------------+-------------------------+-----------------------+
 |  234   1       2348   |  568    458      5-468  |  9     7      2458    |
 |  49    479     4789   |  3      159-48   2      |  568   148    14568   |
 |  6     249     5      |  19-8   7        148    |  38    1348   12348   |
 +-----------------------+-------------------------+-----------------------+
 |  245   3       1      |  258    6        458    |  578   9      4578    |
 |  8     59-247  2479   |  1259   123459   145    |  35    6      1345    |
 |  59-4  6       49     |  1589   134589   7      |  2     1348   13458   |
 +-----------------------+-------------------------+-----------------------+

Second diagonal symmetry, with digit relabelling (1-1)(2-4)(3-7)(5-5)(6-8)(9-9) =>main diagonal contain only (1,5,9); ste

Code: Select all

 +-----------------------+-------------------------+-----------------------+
 |  1     259     239    |  4      258      568    |  7-6   38     679     |
 |  7     8       29     |  1-26   12       3      |  4     5      69      |
 |  4-35  45      6      |  7      58       9      |  1     2      38      |
 +-----------------------+-------------------------+-----------------------+
 |  234   1       2348   |  568    458      4568   |  9     7      2458    |
 |  49    479     4789   |  3      14589    2      |  568   148    14568   |
 |  6     249     5      |  189    7        148    |  38    1348   12348   |
 +-----------------------+-------------------------+-----------------------+
 |  245   3       1      |  258    6        458    |  578   9      4578    |
 |  8     24579   2479   |  1259   123459   145    |  35    6      1345    |
 |  459   6       49     |  1589   134589   7      |  2     1348   13458   |
 +-----------------------+-------------------------+-----------------------+

Column Sticks symmetry: swap bands 2-3, then swap columns 2-3, 5-6, 8-9, with digit relabelling (1-1)(2-3)(4-4)(5-6)(7-7)(8-9)
=> sticks r123c1, r123c4, r123c7 contain only (1,4,7); ste

Code: Select all

 +-----------------------+-------------------------+-----------------------+
 |  1     259     239    |  4      258      568    |  67    38     679     |
 |  7     8       29     |  126    12       3      |  4     5      69      |
 |  345   45      6      |  7      58       9      |  1     2      38      |
 +-----------------------+-------------------------+-----------------------+
 |  234   1       2348   |  568    458      4568   |  9     7      8-245   |
 |  49    479     4789   |  3      1-4589   2      |  568   148    14568   |
 |  6     249     5      |  189    7        148    |  38    1348   12348   |
 +-----------------------+-------------------------+-----------------------+
 |  245   3       1      |  258    6        458    |  578   9      4578    |
 |  8     24579   2479   |  1259   123459   145    |  35    6      1345    |
 |  459   6       49     |  1589   134589   7      |  2     1348   13458   |
 +-----------------------+-------------------------+-----------------------+

Central symmetry, with digit relabelling (1-1)(2-3)(4-7)(5-9)(6-8) =>+1r5c5

Once solved, it turns out that puzzle 2 has the same solution grid as puzzle 1. So, both have the same set of symmetries.

[champagne]

I knew the use of symmetry of given, and the proof of the validity.
Here, I don't see the property, so I am not sure that cenoman's path is valid.

And reading the article to see how it can be applied requires time, but I see some properties having some potential to optimize my virtual catalog.
The repetitive triplet is part of the code

[Leren]

Like Champagne I don't see Main Diagonal Symmetry of Givens for the first puzzle, although in the solution the Main Diagonal is limited to 168.

Is there some other rule being applied here ?

Leren

[champagne]

My bet is that in the first puzzle the key is the "9" fully assigned showing the described property,
but how to use it in the proper way is till unclear for me

[Cenoman]

Hi champagne and Leren,

Let's say that I have used oriented T&E !
As a reminder, automorphism is an intrinsic property of the solution grid. For a valid puzzle (unique solution), whichever set of givens is made, none can destroy the property.
First puzzle: if it has MDS (Main Diagonal Symmetry), then looking at PMs (after basics), it easy to see that 1,6,8 are the invariant digits in digit relabelling (look at mini diagonals, boxes 1,5,9), and that (2-7)(3-4)(5-9) are the other digit exchanges (look at cell pairs r2c3-r3c2, r4c5-r5c4, r7c9-r9c7). The inferred eliminations on MD solve the puzzle, without contradiction of the MDS assumption (i.e. actual MDS of the solution grid 81 cells). Additional Uniqueness assumption validates such a solution (similarly to a clearly visible symmetry of givens).
Not yet checked the other automorphisms for puzzle 1, nor puzzle 2. Will do later.

[champagne]

Hi champagne and Leren,
As a reminder, automorphism is an intrinsic property of the solution grid. For a valid puzzle (unique solution), whichever set of givens is made, none can destroy the property.

Right, but you know this property if and only if you solved the puzzle.
It seems difficult to me to have a path using it.

Uniqueness of the solution is a property of any valid sudoku, so it can be used.

[P.O.]

the IBPA pattern states that in each band and in each stack the relative position of a value in a box is the same

Code: Select all

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


in band one the 2 is in second position in box 1 and 3 and so r2c5=2
in band three the 2 is in first position in box 3 and so must be in first position in box 1 and 2, in box 2 the only possibility is r8c4, it follows in box 1 the 2 must be in r7c1
after this is done
in stack one the 2 is in first position in box 1 and 3 and so r4c3=2
in stack two the 2 is in second position in box 1 and 3 and so r5c6=2
in stack three the 2 is in third position in box 1 and 3 and so r6c9=2

the same reasoning applies to all values

[champagne]

the IBPA pattern states that in each band and in each stack the relative position of a value in a box is the same

for a naive reader, is missing here why this puzzle has an ibpa pattern.
But I did not have enough time to go through your article.

The only point that I saw in the PM is the fact that the "9" have one pattern that you describe

[P.O.]

all values from 1 to 9 have the same pattern

Code: Select all

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


the ibpa pattern is very restrictive, i only found 36 templates that met it
and from these, i was only able to construct 28 solution grids
all these solutions also have the ibpu pattern as ibpa is a subset of it, and 4 solutions also have the tdc pattern
since i don't have a way to test for isomorphism, i don't know if these solutions are all ED

Hidden Text: Show

Code: Select all

123456789456789123789123456231564897564897231897231564312645978645978312978312645   tdc
123456789456789123789123456231564978564978231978231564312645897645897312897312645
123456789456789123789123456231645897645897231897231645312564978564978312978312564
123456789456789123789123456231645978645978231978231645312564897564897312897312564
123456789459783126786129453231564897594837261867291534312645978945378612678912345
123456789486729153759183426231564897864297531597831264312645978648972315975318642
123456789489723156756189423231564897894237561567891234312645978948372615675918342
123456789456789123789123456312564897564897312897312564231645978645978231978231645
123456789456789123789123456312564978564978312978312564231645897645897231897231645
123456789456789123789123456312645897645897312897312645231564978564978231978231564
123456789456789123789123456312645978645978312978312645231564897564897231897231564   tdc
123456789459783126786129453312645978945378612678912345231564897594837261867291534
123456789486729153759183426312645978648972315975318642231564897864297531597831264
123456789489723156756189423312645978948372615675918342231564897894237561567891234
123456789756189423489723156231564897567891234894237561312645978675918342948372615
123456789759183426486729153231564897597831264864297531312645978975318642648972315
123456789786129453459783126231564897867291534594837261312645978678912345945378612
123456789789123456456789123231564897897231564564897231312645978978312645645978312   tdc
123456789789123456456789123231564978978231564564978231312645897897312645645897312
123456789789123456456789123231645897897231645645897231312564978978312564564978312
123456789789123456456789123231645978978231645645978231312564897897312564564897312
123456789756189423489723156312645978675918342948372615231564897567891234894237561
123456789759183426486729153312645978975318642648972315231564897597831264864297531
123456789786129453459783126312645978678912345945378612231564897867291534594837261
123456789789123456456789123312564897897312564564897312231645978978231645645978231
123456789789123456456789123312564978978312564564978312231645897897231645645897231
123456789789123456456789123312645897897312645645897312231564978978231564564978231
123456789789123456456789123312645978978312645645978312231564897897231564564897231   tdc


[Cenoman]

Right, but you know this property if and only if you solved the puzzle.
It seems difficult to me to have a path using it.

Nothing is "difficult" in any of my eight paths above. You should have written "It seems not acceptable to me to have a path using it".
And the reason is that such a path uses T&E.

I accept to be reproached to have used T&E. But you know well the French expression "ça me glisse dessus comme l'eau sur les plumes du canard" (It's like water off a duck's back)

Now, these are my last words in that thread. I'd be happy to read experts' methods to use hidden automorphisms, not readable easily in givens configuration.

[champagne]

all values from 1 to 9 have the same pattern

Code: Select all

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


the ibpa pattern is very restrictive, i only found 36 templates that met it
and from these, i was only able to construct 28 solution grids
all these solutions also have the ibpu pattern as ibpa is a subset of it, and 4 solutions also have the tdc pattern
since i don't have a way to test for isomorphism, i don't know if these solutions are all ED

If the solution grid is necessary to see the pattern, I would do the same remark as for cenoma's path, you can not use it to solve the puzzle.

regarding automorphisms in solution grids, you have plenty of results in this thread


http://forum.enjoysudoku.com/grid-quick-minlex-and-grids-auto-morphs-t44302-15.html

As far as I can see, your grid is the one with the highest number of auto morphs

EDIT: and here is the min lexical solution grid morph of your 2 puzzles

Code: Select all

........9.5.7......8.1.345...1.6.......8...3.8...3.5.4.1264..7.6....83.2.7.......;964550;0K1gZ28AgUAi20
...........6.8.1.378..2...6.3...4..75....72...9..3.5....26..9.8...9...1..7...2...;964550;0WgJKaXHAC5HY0


[champagne]

Right, but you know this property if and only if you solved the puzzle.
It seems difficult to me to have a path using it.

Nothing is "difficult" in any of my eight paths above. You should have written "It seems not acceptable to me to have a path using it".
And the reason is that such a path uses T&E.

I accept to be reproached to have used T&E. But you know well the French expression "ça me glisse dessus comme l'eau sur les plumes du canard" (It's like water off a duck's back)

Now, these are my last words in that thread. I'd be happy to read experts' methods to use hidden automorphisms, not readable easily in givens configuration.

If I push your statement to maximum, to solve a grid you just apply T&E and you have the result. This is what I do in my brute force solver.

You made better things.

[P.O.]

If the solution grid is necessary to see the pattern, I would do the same remark as for cenoma's path, you can not use it to solve the puzzle.
regarding automorphisms in solution grids, you have plenty of results in this thread

if you're given a puzzle and told it has the IBPA pattern, you can solve it trivially
i only have Android as my operating system

[champagne]

If the solution grid is necessary to see the pattern, I would do the same remark as for cenoma's path, you can not use it to solve the puzzle.
regarding automorphisms in solution grids, you have plenty of results in this thread

if you're given a puzzle and told it has the IBPA pattern, you can solve it trivially
i only have Android as my operating system

Sure, but this is another game.
I edited my last post giving the ED morph of your puzzles.