Highest Score

Personally I feel this puzzle is far too hard to be solved until more than the currently advertised 5 hints become available. My personal aim is to understand this puzzle and find the highest score possible within the time limit. Solving these types of puzzles is still an important process in understanding Eternity II.

Solving

Solving this puzzle involves placing pieces and checking if they fit. If all the pieces fit you have the solution otherwise keep trying. Some methods of solving are worse than others. I'll start with the worst methods and then move to the better ones.

By hand

Contradicting what Christopher Monckton says, I feel it is very unlikely Eternity II will be solved completely by hand. Although playing with the pieces might give you the inspiration required to solve the puzzle.

Place all then check

Placing all the pieces and then checking would find a solution eventually providing you tried all possible combinations. The problem is that the number of combinations is huge.

4²⁵⁶ x 256! = 2x10⁶⁶¹

Place all in correct areas then check

A slightly better method would place pieces in the correct regions. Always put the interior pieces in the middle, edge pieces on the edges, corner pieces in the corners and hint piece in it's location. There is still, however, a huge number of combinations to be checked.

4²⁵⁶ x 195! x 56! x 4! x 1 = 6x10⁵⁹³

Place all in correct areas and orientations then check

Only the interior pieces have orientation options. The hint, corner and edge pieces all have a fixed orientation. Although this still leaves a large number of combinations to be checked also.

4¹⁹⁵ x 195! x 56! x 4! x 1 = 4x10⁵⁵⁷

Place and check

There is a lot of wasted effort using the above methods. An improved method is to check while you place pieces. If you cannot place anymore pieces you backtrack (remove pieces) and then try other pieces. Calculating the effort required for this method is more complicated as is depends on the order you place pieces. The calculation details can be viewed under Theory and Analysis (eventually) but my best estimate is that there are still 2x10^47 configurations that require testing.