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Neuer (?) Fund über Sudoku-Geometrie
#21
(02.08.2020, 02:30)cdwg2000 schrieb:
(02.08.2020, 02:28)光碟2000 schrieb:
(11.04.2020, 17:00)hist schrieb: 大家好,

我最近仔细考虑了Sudoku网格的几何形状,并发现了一个有趣的发现。因为我不太有经验,所以这已经是老帽子了,这当然是一件好事。因此,我想您的经验,并询问您是否可能意识到这一点。

我的想法从以下几何开始,可以很容易地表明,暗红色块中的数字总和与蓝色区域中的数字总和相对应:
[Bild: bild.php?data=be82f74e-6191-3030303343332d31]
如果要确定整个红色区域中的数字总体和,则基本上有两个选项:
1.将第1、2、8和9行以及第1、2、8和9列相加,并从中到深色红色的笼A, B,C和D,即
8 * 45-
ABCD。2 . 将所有3x3块加到中间的一个,然后再连接中间的蓝色区域,即
8 * 45蓝色区域。
因此我们得到了:蓝色区域中的,数字的总和= A + B + C +D。

但是,令我感到平静的是,在我查看的每个示例中,,总体而言是相同的,而且深红色块和蓝色区域中的所有数字变量经过深思熟虑,我发现了了这一点的证明(我希望是这样的……):
[Bild: bild.php?data=b2a6bb64-6192-3030303343332d32]
为了显示,请考虑第3行和第7行,第3行和第7列以及四个角的现在,让“ n”为1到9之间的一个数字。然后,“ n”必须在4行和4个块中的每个一个中。恰好出现一次。因此,如果“ n”出现在
-绿色单元格中,则覆盖一行和一个框,
-红色单元格出现,一个框被覆盖,
-蓝色单元格出现,一条线被覆盖覆盖,
-橙色单元格出现两个行一箱覆盖。

因此,,“绿色”必须出现在蓝色或橙色单元格中,而反之亦然。由于此适用于于1到9之间的每个数字,因此红色区域必须包含与蓝色和橙色区域相加的数字相同的数字。

你们中有人知道这样的结果吗?的反馈信息,我将不胜感激,当然也请您提供有关任何错误或有趣的摘要的反馈。


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#22
(02.08.2020, 02:31)cdwg2000 schrieb:
(02.08.2020, 02:30)cdwg2000 schrieb:
(02.08.2020, 02:28)光碟2000 schrieb:
(11.04.2020, 17:00)hist schrieb: 大家好,

我最近仔细考虑了Sudoku网格的几何形状,并发现了一个有趣的发现。因为我不太有经验,所以这已经是老帽子了,这当然是一件好事。因此,我想您的经验,并询问您是否可能意识到这一点。

我的想法从以下几何开始,可以很容易地表明,暗红色块中的数字总和与蓝色区域中的数字总和相对应:
[Bild: bild.php?data=be82f74e-6191-3030303343332d31]
如果要确定整个红色区域中的数字总体和,则基本上有两个选项:
1.将第1、2、8和9行以及第1、2、8和9列相加,并从中到深色红色的笼A, B,C和D,即
8 * 45-
ABCD。2 . 将所有3x3块加到中间的一个,然后再连接中间的蓝色区域,即
8 * 45蓝色区域。
因此我们得到了:蓝色区域中的,数字的总和= A + B + C +D。

但是,令我感到平静的是,在我查看的每个示例中,,总体而言是相同的,而且深红色块和蓝色区域中的所有数字变量经过深思熟虑,我发现了了这一点的证明(我希望是这样的……):
[Bild: bild.php?data=b2a6bb64-6192-3030303343332d32]
为了显示,请考虑第3行和第7行,第3行和第7列以及四个角的现在,让“ n”为1到9之间的一个数字。然后,“ n”必须在4行和4个块中的每个一个中。恰好出现一次。因此,如果“ n”出现在
-绿色单元格中,则覆盖一行和一个框,
-红色单元格出现,一个框被覆盖,
-蓝色单元格出现,一条线被覆盖覆盖,
-橙色单元格出现两个行一箱覆盖。

因此,,“绿色”必须出现在蓝色或橙色单元格中,而反之亦然。由于此适用于于1到9之间的每个数字,因此红色区域必须包含与蓝色和橙色区域相加的数字相同的数字。

你们中有人知道这样的结果吗?的反馈信息,我将不胜感激,当然也请您提供有关任何错误或有趣的摘要的反馈。


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#23
Phistomefel- After reading your post I have found 4 more relationships like yours but I used red rectangles instead of red squares

Would it be worthwhile posting them here???

Also what format do I need to use for the pictures of my Sudoku puzzles?

If this not appropriate for this forum or thread let me know. this my first post on this forum.
Thanks, Dale E. Kloss
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#24
I think, this is the right place for it. You can attach any common picture format.
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#25
(18.11.2020, 19:53)Dandelo schrieb: I think, this is the right place for it. You can attach any common picture format.

Trying to post but a strange thing happens. Posting from a Window 10 Intel PC.

My post has pictures of various Sudoku puzzles embedded in it but at first when I post it the puzzles are visible but then a second or two later the pictures disappear and are replaced by a big empty space. I have tried cut & pasting a .png file, and copying the image from a Word document but the picture always disappears.

Is there a special procedure for including pictures in the blog ???

An alternative I could do would be to include an attached PDF file if the blog supports attached files.

Thanks in advance. Dale E. Kloss
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#26
 

After reading Phistomefel’s post about his new finding, I have been thinking about using red rectangles in place of Phistomefel’s red squares as the corner elements (I have been retired for a long time so I have lots of time on my hands and in my brains- a small joke).
 
I have found 4 other relationships (although the fourth one is obvious & simple but it illustrates the use of the technique)
 
Remember, for all puzzles, the red cells are counted twice the first sum, but once the second count.
 
SEE THE ATACHED SUDOKUPATTERNS.PDF FILE FOR PICTURES; I COULD NOT GET THE PICTURES TO POST CORRECTLY. BAH , pictures will not attach. I have written up a description of cell placements in each puzzle. Sorry

The donut hole is colored white, green cells are all unmentioned cells.

For all puzzles, R= red cells, G=green cells, B = blue cells, the donut hole is the box in the center of the puzzle 
======================================
The first puzzle uses a 1-cell height by 2-cell width rectangle.
  +----+-----+----+
  | RRG | GGG | GRR |
  | GGB | BBB | BGG |
  | GGB | BBB | BGG |
  +------+-----+------+
  | GGB |       | BGG |
  | GGB | W   | BGG |
  | GGB |       | BGG |
  +------+----+------+
  | GGB | BBB | BGG |
  | GGB | BBB | BGG |
  | RRG | GGG | GRR |
  +-----+------+------+
   1X2 pattern


Red cells = Row 1 & 9 Columns 1 & 2
Blue cells = rows 2,3,8,9  in columns 4 to 6
                 columns 3 & 7 in rows 2 to 8 


 
Counting the cells along rows 1 and 9, then along columns 1, 2, 8, and 9 we get:
 
   2R + G = 6 occurrences of all the numbers 1 to 9
 
Counting the cells in the 8 edge boxes we get:
 
     R + G + Blue = 8 occurrences of all the numbers 1 to 9
 
Taking the difference, we get
 
      R +   2 occurrences of all the numbers 1 to 9 = B
Or
     The contents of the red cells + 2 occurrences of all the numbers 1 to 9 = the contents of blue cells
 
Including the rotations & interchanges, I get 13,122 possible versions of the puzzle as follows:
 
2 rotations
9 positions of the donut hole
3 positions of the line for red cells at the top
3 positions of the blue cell in the leftmost top box
3 positions of the blue cell in the rightmost top box
3 positions of line for red cells at the bottom
3 positions of the blue cell in the leftmost bottom box
3 positions of the blue cell in the rightmost bottom box
 
======================
 
The second puzzle uses just the 4 corner cells.

 +-----+-----+----+
  | RGG | GGG | GGR |
  | GBB | BBB  | BBG |
  | GBB | BBB  | BBG |
  +-----+------+------+
  | GBB |        | BBG |
  | GBB | W    | BBG |
  | GBB |        | BBG |
  +-----+------+------+
  | GBB | BBB | BBG |
  | GBB | BBB | BBG |
  | RGG | GGG| GGR |
  +-----+-----+------+
   1X1 pattern

Red = r1c1, r1c9, r9c1, r1c9
Blue = Columns 2,3,7,8 in rows 2 to 8
           Rows 2,3,7,8 in columns 4,5,6

 Counting the cells along rows 1 and 9, then along columns 1 and 9 we get:
 
   2R + G = 4 occurrences of all the numbers 1 to 9
 
Counting the cells in the 8 edge boxes we get:
 
     R + G + Blue = 8 occurrences of all the numbers 1 to 9
 
Taking the difference, we get
 
      R +   4 occurrences of all the numbers 1 to 9 = B
Or
 
     The contents of the red cells + 4 occurrences of all the numbers 1 to 9 = the contents of blue cells
 
Since the rotation produces the same positions of red & blue cells I get (after interchanges) 6,561 different versions of the puzzle as follows:
 
9 positions of the donut hole
3 possible rows for the red cells at the top
3 possible positions of the of the leftmost red cell at the top
3 possible positions of the of the rightmost red cell at the top
3 possible rows for the red cells at the bottom
3 possible positions of the of the leftmost red cell at the bottom
3 possible positions of the of the rightmost red cell at the bottom
 
==============================
 
The third puzzle uses a 2-cell height by 3-cell width rectangle.

 +-----+-----+----+
  | RRR  | GGG | RRR |
  | RRR  | GGG | RRR |
  | GGG  | BBB | GGG |
  +------+------+-----+
  | GGG |        | GGG |
  | GGG | W    | GGG |
  | GGG |        | GGG |
  +-----+ -----+------+
  | GGG | BBB | GGG |
  | RRR | GGG | RRR |
  | RRR | GGG | RRR |
  +-----+------+------+
   2X3 pattern

Red = rows 1, 2, 8, & 9  in columns 1,2,3,7,8,9
Blue = rows 3 & 7 in columns  4, 5, & 6
 
Counting the cells along rows 1, 2, 8, and 9, then along columns 1, 2, 3, 7, 8, and 9 we get:
 
   2R + G = 10 occurrences of all the numbers 1 to 9
 
Counting the cells in the 8 edge boxes we get:
 
     R + G + Blue = 8 occurrences of all the numbers 1 to 9
 
Taking the difference, we get
 
      R = B + 2 occurrences of all the numbers 1 to 9
Or
     The contents of the red cells = the contents of blue cells + 2 occurrences of all the numbers 1 to 9
 
After rotations & interchanges, I get 162 different versions of the puzzle
 
2 rotations
9 positions of the donut hole
3 possible rows for the blue cells at the top
3 possible rows for the blue cells at the bottom
 
=============================================
The fourth puzzle uses a 1-cell height by 3-cell width rectangle
(A simple example of the technique just for completeness)

  +------+------+------+
  | RRR  | GGG | RRR |
  | GGG | BBB  | GGG |
  | GGG | BBB  | GGG |
  +------+------+------+
  | GGG |         | GGG |
  | GGG | W     | GGG |
  | GGG |         | GGG |
  +------+------+-----+
  | GGG  | BBB | GGG |
  | GGG  | BBB | GGG |
  | RRR  | GGG | RRR  |
  +------+------+------+
   1X3 pattern


Red = Row 1 in columns 1, 2, 3, 7, 8, & 9
Blue = Rows 3 & 4 in columns 4, 5, & 6 
Counting the cells along row 1 , we get
 
   R + G = 1 occurrence of all the numbers 1 to 9
 
Counting the cells in the top center box, we get:
 
     G + Blue = 1 occurrence of all the numbers 1 to 9
 
Taking the difference, we get
 
     R = B
 
Or
     The contents of the top red cells = the contents of top blue cells
 
==================================
For Phistomefel’s original post, I get 729 different versions as follows:
No difference between rotations.
 
9 donut hole positions
3 possible lines for the blue horizontal line at top
3 possible positions of the vertical blue line in the leftmost side of the puzzle
3 possible positions of the vertical blue line in the rightmost side of the puzzle
3 possible lines for the blue horizontal line at bottom
 
=================================
Obviously, of mine, the 2x3 is the least complicated to implement, has the least number of versions, and the least number of cells to be considered.  Phistomefel’s puzzle or the 2x3 puzzle could be used in some solver.
 
It would seem to me that if one of these patterns were to be added to a solver, it would fastest to find eliminations by first searching for missing numbers that fell on the corner of a rectangle.
 
That would determine the donut hole position.  That determines the positions of all the elements of the rest of the pattern.
 
Please let me know if I made any mistakes.  First time poster.
 
Dale E. Kloss, Portland, Oregon, USA
 
                                               
                                               
 
 
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