Solution to Problem 13: "Dem Bones"

(The crucial frames of this solution, marked [LB], were provided by Larry Baum.)

domino1.bmp (116678 bytes) The only place for the 5-5 domino is at G1-G2.  The 2-2 domino must include F2, so E2-F2 cannot be a domino.  The 3-3 domino must include E3, so E3-F3 and E3-E4 cannot be dominos.  The 6-4 domino must include C1, so C1-C2 cannot be a domino.  The domino that includes A1 must be a 2-4 domino, so F5-F6 is not a domino.
domino2A.bmp (116678 bytes) [LB] Imagine that the 2-6 domino were at C2-C3.  Then the domino that included G3 would have to be the 6-6 domino, so we can set up red lines as shown at the left.  But the three dominos that include B4, C4, and B6 can only be 6-1 and 6-5, so our assumption that C2-C3 was a domino was in error.

The domino that includes C3 must be the 2-1 domino, so E5-F5 cannot be a domino.  Similarly, the domino that includes F5 must be the 2-3 domino, so C6-D6 cannot be a domino.
domino3.bmp (116678 bytes) [LB] The 4-3 domino can either be at E4-F4 or at G5-G6.  Either one of these will force the location of the 2-3 domino.  In either case, the domino that passes through G4 must be G3-G4.

Since we have found the 6-6 domino, C3-C4 and B4-C4 are not dominos.  The two dominos that pass through B4 and C4 must be 6-1 and 6-5 in some order, so the domino that passes through C3 must be C3-D3 and the domino that passes through B6 must be B6-C6.

The only location for the 5-2 domino is E1-F1.
domino4.bmp (116678 bytes) A5-A6, E2-E3, and F2-F3 are dominos. Since we have found the 3-5 domino, D5-D6 must be a domino.

If C4-C5 were a domino, the two large unsolved segments remaining would have an odd number of squares in them, which is impossible. Therefore, B5-C5 is a domino. Since that places 1-5, we know that E5-E6 is not a domino, so we can place dominos at E6-F6, G5-G6, F4-F5, E4-E5, and C4-D4. Finally, since we've placed 4-1, we know that B1-B2, A2-B2, A3-B3, and D1-D2 are not dominos, so the final dominos go at C1-D1, A1-B1, A2-A3, A4-B4, B2-B3, and C2-D2.
domino5.bmp (116678 bytes) The horizontal dominos (marked here in green) are 24 64 52 21 63 56 61 15 62 54.

Comments: I spent a couple of minutes trying this one, but only got up through the first two paragraphs.  In retrospect, I wish I had put my work up to that point into the copier and tried wild guesses.  It's quite a change of perspective that finding any solution is the good enough and who cares if it isn't a pretty process.   (Larry tells me that he did more trial-and-error in the test as well, and only came up with this solution afterward.)