Dominoes Puzzle #3

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

Detailed solution to Dominoes Puzzle #3

 . . . . . 2 1 1 . . . . . . 4 4 . . . 2 . . . . a* . 1 1* 2 . . . b* c 1 2 . . . . e z d . . . . . f g 2 1 . . . .

The (1-4) near the 2 in upper right cannot be a domino -- there would be a second (1-4) domino just to its right. So we draw a line (domino boundary) between the 1 and 4. That 1 must be in either a (1-2) or a (1-1) domino.

There is a 1* near the center of the board which must also be in either a (1-2) or a (1-1) domino. Thus both the (1-2) and (1-1) dominoes are accounted for between these two 1's. Lines (domino boundaries) can therefore be drawn to break up all other (1-1) and (1-2) possibilities.

I have erased the other numbers from this board to show that they are unneeded now, in deducing our first domino (though some of these cells are labeled with letters so we can refer to them). We will prove that (a*,b*) must be a domino!

If (b*,c) is a domino then so is (1,d) and there are five cells left isolated in the lower left. This is impossible -- all regions with whole dominoes must have an even number of cells.

If (b*,e) is a domino, then so is (f,g), so also (2,d), so also (1,c) and a single cell is left isolated at z -- again an impossibility.

Therefore (a*,b*) must be a domino -- this is the only place left for b to connect.

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

Here is the puzzle redrawn, with our new deduction. Ths unusual deduction may be available in many positions, but this puzzle is special -- that deduction leads to our first complete domino, and without this deduction no other domino can be deduced except by trial and error.

We show some new deductions in red. The (3-3) next to our (0-0) cannot be a domino -- again we'd end up with an isolated odd-sized region. The nearby (6-5) cannot then be a domino for the same reason. And the (6-2) cannot be a domino or there'd be another (6-2) right next to it.

Another possible (0-0) is impossible -- there'd be a duplicate domino -- so we draw a boundary line through it.

We've split up another impossible (2-1) as explained above. Finally, we note that (0-6) in the top left is impossible -- it would lead to either an isolated (6) or duplicate (5-5)'s.

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

With the (5-0) forced to be somewhere near upper left, lines are drawn to eliminate remaining (5-0) possibilities.

We've also eliminated the (4-2) near lower right. If it were (4-2), there'd be (4-3) next to it; left corner 4 would be (4-6) and the 4 in the top row couldn't connect anywhere without duplicating a domino. This leaves only one place for the (4-2) so we place it in the next diagram.

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

There is only one site for the (4-0) so we locate it there. The (1-1) follows automatically.

The (4-1) must involve the 4 at middle right. There's nowhere for (4-4) except involving 4*, so the (5,4*) is impossible.

Finally (1*-6) near middle right is impossible -- proceeding from the 1 we'd find two different (5-4) forced dominoes.

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

The four 6's marked as 6* each connect only to 3, 4, 5, or 6, so among them they account for all four of the (6-3), (6-4), (6-5) and (6-6) dominoes. This means other possibilities for these dominoes can be eliminated, as shown with red lines.

This will confirm the location of the (5-4) domino near the upper right, and several other dominoes as shown next.

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

With the (5-4), (6-0), and (5-1) dominoes located, other sites for them can be eliminated (shown with red lines).

This locates more dominoes, and leads to more elimnations, as shown in next diagram.

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

Deductions are coming quite quickly now, but we proceed slowly for clarity.

Several more dominoes have been located here; we show them, and associated eliminations, in the next diagram.

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

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

With (3-3) and (6-5) located (but not (6-3) or (5-3)) the remainder of the puzzle is readily solved.

Dominoes Puzzle #3 -- Final solution

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