Each puzzle piece has its edges on one side marked with different shape/colour combinations (collectively called "colours" here), each of which must match precisely with its neighbouring side on each adjacent piece when the puzzle is complete. The other side of each piece is blank apart from an identifying number, and is not used in the puzzle. Thus, each piece can be used in only 4 orientations. There are 22 colours, not including the gray edges. This puzzle differs from the first Eternity puzzle in that there is a starter piece which must be placed near the center of the board. (See PDF rulebook on official website.)
Two Clue Puzzles were available with the launch of the product, which, if solved, each give a piece position on the main 256-piece puzzle. Clue Puzzle 1 is 6 by 6, with 36 pieces and Clue Puzzle 2 is 12 by 6, with 72 pieces. Two further puzzles were made available in 2008. Clue Puzzle 3 is 6 by 6, with 36 pieces, and Clue Puzzle 4 is 12 by 6, with 72 pieces.
The number of possible configurations for the Eternity II puzzle, assuming all the pieces are distinct, and ignoring the fixed pieces with pre-determined positions, is 256! × 4256, roughly 1.15 × 10661. A tighter upper bound to the possible number of configurations can be achieved by taking into account the fixed piece in the center and the restrictions set on the pieces on the edge: 1 × 4! × 56! × 195! × 4195, roughly 1.115 × 10557.
The Eternity II puzzle was designed by Monckton in 2005, this time in collaboration with Selby and Riordan, who designed a computer program that generated the final Eternity II design. According to the mathematical game enthusiast Brendan Owen, the Eternity II puzzle appears to have been designed to avoid the combinatorial flaws of the previous puzzle, with design parameters which appear to have been chosen to make the puzzle as difficult as possible to solve. In particular, unlike the original Eternity puzzle, there are likely only to be a very small number of possible solutions to the problem. Owen estimates that a brute-force backtracking search might take around 2 steps to complete.
Monckton was quoted by The Times in 2005 as saying:
Although it has been demonstrated that the class of edge-matching puzzles, of which Eternity II is a special case, is in general NP-complete, the same can be said of the general class of polygon packing problems, of which the original Eternity puzzle was a special case, so it is unclear as to whether this has any relevance to the difficulty level of Eternity II.
Like the original Eternity puzzle, it is easy to find large numbers of ways to place substantial numbers of pieces on the board whose edges all match, making it seem that the puzzle is easy. However, given the low expected number of possible solutions, it is presumably astronomically unlikely that any given partial solution will lead to a complete solution.
The $2Million Jigsaw; Viscount Unveils Sequel to the 'Unsolvable' Puzzle That Lost Him [Pounds Sterling]500,000
Jan 25, 2007; Byline: GORDON TAIT WHEN he launched the original eight years ago, Christopher Monckton was so sure it would stump players he...