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22 unique edge colors (plus gray, making 23 in total)
Each tile has its tile number printed on the back, so is not reversible.

There are around 1.2 million unique combinations of 4 tiles arranged in a square. I call this arrangement a ‘Quad’. Of these, 1050 quads fit in the corner; and around 43000 fit along an edge. There are 460 unique edge patterns formed by all possible quads.

This problem really is effectively impossible, and I will be very surprised indeed if anyone solves it, ever — except by trying possible board layouts at random, and being very very very lucky.

I purchased the puzzle a week after it was released in Australia and sat down for 4-5 hours and only got about 7 lines of pieces done. That’s only about 22 pieces done per hour doing it manually. I realised that this puzzle will only ever be solved by a computer, (unless you can pick winning lotto numbers) so I joined eternity2.net and have my computer running 24/7.

Do you have a program set up to work it out on the computer yet? I am very interested in seeing a competitor to eternity2.net.

p.s. I heard that if you join up with the eternity2 yahoo group you are automatically disqualified from the comp. Is this true?

Good luck with the algorythms.

The edge pieces provide you a way to split the problem it two parts and of course to simplify it, but you still have a problem to relate both solutions (1. Solution for the border and 2. Solution from the middle part). In practice you can arrange first all possible borders and after that to arrange 14×14 by all possible ways starting from the piece 139. At the end you still will need to mach solutions form the borders with the solutions of the middle part.

Is it possible to use non-deterministic algorithms (like GA) in solving such a problem? Did some of you try to solve only borders of the puzzle? The corner pieces are 4 and can be arranged in 4! ways. After that we have 4 edges by 14 edge pieces or 56 and they can be arranged in 56! ways. To find all possible borders it should be: 4!*56!

Solving a puzzle like this can be conceptualized as a search problem within a massive decision tree. When placing the first piece on the board, you have 256 different pieces (and four rotations per piece) to choose from. Chances are, the choice is incorrect, since the puzzle has so few correct solutions.

But how long does it take to realize that the choice was incorrect? Often, you won’t know you made a bad placement until you’ve already traversed deeply into the decision tree (empirically, on this puzzle, you can often get 100 or 200 nodes deep into the decision tree before discovering that you’ve hit a dead end). To reduce the amount of wasted time searching through unproductive branches of the tree, it’s essential to discover those dead-ends as soon as possible.

One of the techniques I’m using (and most of the other puzzlers on the Eternity2 Yahoo group) is to find the most constrained positions on the board, and start trying to find valid pieces for those locations first.

As it turns out, the most constrained positions are the corners. Each corner location can support only four possible pieces. On the borders, there are only 56 possible pieces for each location. In the center of the board, there are 196 different candidate pieces for each location.

Constraints are also added to the board each time a new piece is placed, since the occupied grid cells have to have matching edges with the new pieces placed adjacent to them.

Starting in the corners, progressing through the edges, and then working on the center (always placing new pieces adjacent to existing pieces), maximized the amount of constraint on the board throughout the solver.

Eliminating the corners and borders would get rid of a lot of those constraints, so it’d be harder for the solver to know whether it was searching through an unproductive branch of the search tree.

On the other hand…

Eliminating those border constraints would make it easier to tile pieces in the center region. It’d be much easier to place each piece without all those extra constraints, and you’d fill out the puzzle grid more quickly.

Of course, you’d often create false 14×14 solutions (since many of those solutions would be impossible to combine with a legal border solution).

I’ve been working on a supervised learning technique for consuming imperfect puzzle solutions (either with mismatching edges or duplicate pieces) and using their imperfections to learn about the puzzle’s larger constraint structure. It actually could be quite useful, from this perspective.

If I had a few million 14×14 inner solutions, I might be able to study the characteristics of those false solutions, leveraging their statistical properties to better trim the global search tree toward finding a ream 16×16 solution.

I like the way you think, Lyndon :-) Are you working on the puzzle?