Dominoes

[cinelli]

I have laid out all 28 pieces of a standard set of dominoes in an 8 by 7 array as follows (blanks are shown as zeros):

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  2   3   3   1   6   6   0   4  
                            
  5   2   3   0   4   6   1   1  
                                 
  1   4   6   1   3   3   0   1  
                                
  1   0   2   5   6   6   3   2  
                                
  5   5   2   0   5   4   4   5  
                            
  5   5   1   3   2   0   0   3  
                                
  4   4   4   0   2   2   6   6

This puzzle is to indicate how all the pieces are oriented. For instance at the top left corner, is the 2-3 piece horizontal or is the 2-5 piece vertical?

Cinelli

[Gengulphus]

I have laid out all 28 pieces of a standard set of dominoes in an 8 by 7 array as follows (blanks are shown as zeros):

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  2   3   3   1   6   6   0   4  
                            
  5   2   3   0   4   6   1   1  
                                 
  1   4   6   1   3   3   0   1  
                                
  1   0   2   5   6   6   3   2  
                                
  5   5   2   0   5   4   4   5  
                            
  5   5   1   3   2   0   0   3  
                                
  4   4   4   0   2   2   6   6

This puzzle is to indicate how all the pieces are oriented. For instance at the top left corner, is the 2-3 piece horizontal or is the 2-5 piece vertical?

Spoiler...

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OK, first look for dominoes that can only be in one place. They are the 0-0 and 2-4, giving us:

+-------------------------------+
| 2   3   3   1   6   6   0   4 |
|   +---+                       |
| 5 | 2 | 3   0   4   6   1   1 |
|   |   |                       |
| 1 | 4 | 6   1   3   3   0   1 |
|   +---+                       |
| 1   0   2   5   6   6   3   2 |
|                               |
| 5   5   2   0   5   4   4   5 |
|                   +-------+   |
| 5   5   1   3   2 | 0   0 | 3 |
|                   +-------+   |
| 4   4   4   0   2   2   6   6 |
+-------------------------------+

There were originally only two possible places for the 0-6, and that gets rid of one of them, so it must be in the other possible place. That forces the top right 4 to be part of the 1-4, and so allows us to 'wall off' the remaining possible place for the 1-4:

+-------------------+-------+---+
| 2   3   3   1   6 | 6   0 | 4 |
|   +---+           +-------+   |
| 5 | 2 | 3   0   4   6   1 | 1 |
|   |   |                   +---+
| 1 | 4 | 6   1   3   3   0   1 |
|   +---+                       |
| 1   0   2   5   6   6   3   2 |
|                               |
| 5   5   2   0   5   4   4   5 |
|                   +-------+   |
| 5   5   1   3   2 | 0   0 | 3 |
|        ---        +-------+   |
| 4   4   4   0   2   2   6   6 |
+-------------------------------+

That leaves only one place for the 1-1, which then forces a 2-5 and a 3-3 in the top left corner, and only one place for the 1-4. Knowing where the 2-5 and 3-3 are also allows us to wall off the other possible places for them. Also, the 4 and 5 at the bottom of the second column cannot be the 4-5 domino, since then there would be a second 4-5 domino in the bottom left corner. So we can wall them off from each other:

+---+-------+-------+-------+---+
| 2 | 3   3 | 1   6 | 6   0 | 4 |
|   +---+---+       +-------+   |
| 5 | 2 | 3   0   4   6   1 | 1 |
+---+   |                   +---+
| 1 | 4 | 6   1   3 | 3   0   1 |
|   +---+                       |
| 1 | 0   2 | 5   6   6   3   2 |
+---+                        ---+
| 5*  5 | 2   0   5   4   4   5 |
|                ---+-------+   |
| 5   5   1   3   2 | 0   0 | 3 |
|    -------        +-------+   |
| 4   4*  4   0   2   2   6   6 |
+-------------------------------+

The asterisked 5 must be part of the 5-5 and the asterisked 4 part of the 4-4, so we can wall off the other possible positions for the 5-5 and 4-4. That forces the lower 5 in the second column to be part of the 1-5 and allows us to wall off the other possible position for it:

+---+-------+-------+-------+---+
| 2 | 3   3 | 1   6 | 6   0 | 4 |
|   +---+---+       +-------+   |
| 5 | 2 | 3   0   4   6   1 | 1 |
+---+   |                   +---+
| 1 | 4 | 6   1   3 | 3   0   1 |
|   +---+   +---                |
| 1 | 0   2 | 5   6   6   3   2 |
+---+                        ---+
| 5   5 | 2   0   5   4 | 4   5 |
|   +---+---+    ---+---+---+   |
| 5 | 5   1 | 3   2 | 0   0 | 3 |
|   +-------+       +-------+   |
| 4   4   4   0   2*  2   6   6 |
+-------------------------------+

If the asterisked 2 combines with the 0 to its left to form the 0-2, that implies a chain of 4-4, 4-5, 5-5 and a second 0-2 around the bottom left corner. So it doesn't, which implies that it must be part of the 2-2, and thus that the other possible place for the 2-2 can be walled off. That in turn implies the actual position of the 0-2:

+---+-------+-------+-------+---+
| 2 | 3   3 | 1   6 | 6   0 | 4 |
|   +---+---+       +-------+   |
| 5 | 2 | 3   0   4   6   1 | 1 |
+---+   |                   +---+
| 1 | 4 | 6   1   3 | 3   0   1 |
|   +---+   +---                |
| 1 | 0   2 | 5   6   6   3   2 |
+---+   +---+---+            ---+
| 5   5 | 2   0 | 5   4 | 4   5 |
|   +---+---+---+---+---+---+   |
| 5 | 5   1 | 3   2 | 0   0 | 3 |
|   +-------+       +-------+   |
| 4   4   4   0 | 2   2   6   6 |
+---------------+---------------+

Now we can wall off the other possible position for the 0-2, and the implications of that cascade to complete the solution:

+---+-------+-------+-------+---+
| 2 | 3   3 | 1   6 | 6   0 | 4 |
|   +---+---+---+---+---+---+   |
| 5 | 2 | 3   0 | 4   6 | 1 | 1 |
+---+   +---+---+---+---+   +---+
| 1 | 4 | 6 | 1   3 | 3 | 0 | 1 |
|   +---+   +-------+   +---+   |
| 1 | 0 | 2 | 5   6 | 6 | 3 | 2 |
+---+   +---+---+---+---+   +---+
| 5 | 5 | 2   0 | 5   4 | 4 | 5 |
|   +---+---+---+---+---+---+   |
| 5 | 5   1 | 3   2 | 0   0 | 3 |
+---+---+---+---+---+---+---+---+
| 4   4 | 4   0 | 2   2 | 6   6 |
+-------+-------+-------+-------+

Gengulphus

[ReformedCharacter]

Thank you, nice puzzle.

RC

[cinelli]

Congratulations to Gengulphus for a masterly solution, which involved a great deal of typing.

Cinelli

[HBEN]

8 is the answer!