DominoBrane - Chained Domino Fitting

Question

So I've been working on a set of domino puzzles, I call DominoBrane recently, it's sort of similar to Dominosa - except that, unlike Dominosa, the rules of the normal domino game must be followed.

• Each domino must connect, as in the normal game, to it's neighbour [0:3][3:3][3:5] etc and make a full circuit of all the 6-dominoes.

• At each square, there must be no ambiguity as to which is the connected domino (eg, for a 4 there will be only one adjacent domino with a 4)

• The dominoes must be constrained to a 7x8 grid.

Here is an example.

╔═══╦═══════╦═══╦═══╦═══════╗
║ 5 ║ 5   4 ║ 4 ║ 3 ║ 3   1 ║
║   ╠═══════╣   ║   ╠═══════╣
║ 6 ║ 0   2 ║ 2 ║ 3 ║ 1   1 ║
╠═══╬═══════╬═══╩═══╬═══════╣
║ 6 ║ 0   0 ║ 0   3 ║ 1   2 ║
║   ╠═══════╬═══════╬═══╦═══╣
║ 4 ║ 1   6 ║ 6   6 ║ 6 ║ 2 ║
╠═══╬═══╦═══╬═══════╣   ║   ║
║ 4 ║ 1 ║ 4 ║ 4   4 ║ 2 ║ 5 ║
║   ║   ║   ╠═══╦═══╬═══╬═══╣
║ 1 ║ 0 ║ 3 ║ 3 ║ 4 ║ 2 ║ 5 ║
╠═══╬═══╩═══╣   ║   ║   ║   ║
║ 1 ║ 0   6 ║ 6 ║ 0 ║ 2 ║ 3 ║
║   ╠═══════╬═══╩═══╬═══╩═══╣
║ 5 ║ 5   5 ║ 5   0 ║ 2   3 ║
╚═══╩═══════╩═══════╩═══════╝

And here it is again, showing the path of the chain.

╔═══╦═══════╦═══╦═══╦═══════╗
║ 5 - 5   4 - 4 ║ 3 - 3   1 ║
║   ╠═══════╣   ║   ╠═════|═╣
║ 6 ║ 0   2 - 2 ║ 3 ║ 1   1 ║
╠═|═╬═|═════╬═══╩═|═╬═|═════╣
║ 6 ║ 0   0 - 0   3 ║ 1   2 ║
║   ╠═══════╬═══════╬═══╦═|═╣
║ 4 ║ 1   6 - 6   6 - 6 ║ 2 ║
╠═|═╬═|═╦═══╬═══════╣   ║   ║
║ 4 ║ 1 ║ 4 - 4   4 ║ 2 ║ 5 ║
║   ║   ║   ╠═══╦═|═╬═|═╬═|═╣
║ 1 ║ 0 ║ 3 - 3 ║ 4 ║ 2 ║ 5 ║
╠═|═╬═|═╩═══╣   ║   ║   ║   ║
║ 1 ║ 0   6 - 6 ║ 0 ║ 2 ║ 3 ║
║   ╠═══════╬═══╩═|═╬═|═╩═|═╣
║ 5 - 5   5 - 5   0 ║ 2   3 ║
╚═══╩═══════╩═══════╩═══════╝

So here's the puzzle: 6-DominoBrane #1, first published here

Following the rules above, find the solution where the numbers match the grid as follows (a blank means 'any number':

╔═══╦═══╦═══╦═══╦═══╦═══╦═══╗
║ 0 ║   ║   ║ 0 ║   ║   ║ 0 ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║   ║ 1 ║   ║ 1 ║   ║ 1 ║   ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║ 3 ║   ║ 2 ║ 2 ║ 2 ║   ║ 4 ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║   ║ 6 ║   ║ 3 ║   ║ 3 ║   ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║ 3 ║   ║ 4 ║ 3 ║ 4 ║   ║ 5 ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║   ║ 5 ║   ║ 2 ║   ║ 5 ║   ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║ 6 ║   ║ 4 ║ 1 ║ 1 ║   ║ 6 ║
╠═══╬═══╬═══╬═══╬═══╬═══╬═══╣
║   ║ 6 ║   ║ 0 ║   ║ 2 ║   ║
╚═══╩═══╩═══╩═══╩═══╩═══╩═══╝

If you solve it, I would really love to know how!
You will be the first person to solve any 6-DominoBrane.

There is exactly one solution to this puzzle.

Be careful to make sure that each square is unambiguous as to where it would go next - the problem is much easier without that constraint.

(Kudos to @bobble for telling me I need to provide single solution puzzles)

Answer 1

Step 1:

Each digit must appear 8 times and connect to itself. Use this to fill in every location where each digit can occur.

Step 2:

Select an arbitrary direction for the dominoes to connect in. We will say the white end connects to the gray end.

Step 3:

Some simple deductions.
The 4 can be filled in because it is necessary for the chain of length 4.

Step 4:

To ensure the 02, 03, 04, and 06 dominoes exist, the domino chain must run against the edge.

Step 5:

The chain must tun clockwise around the outer edge.

Step 6:

The 00 and 33 dominoes can now be filled in.

Step 7:

This allows some more deductions.

Step 8:

The 25 domino can only be in one location.

Step 9:

This allows a long chain of deductions.

Step 10:

Fill in the missing dominoes.