cinelli wrote: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):
.
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
