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In June 1999 Christopher Monckton launched the original Eternity puzzle which took him 14 years to develop. Chris said, "It won't be a computer which solves it and it won't be a mathematician either". A few months later the puzzle was solved by two mathematicians (Alex Selby and Oliver Riordan) using two personal computers.
In 2000 Chris's initial idea for Eternity II was a 1001 piece puzzle forming a rhombic dodecahedron. This time Chris, having learnt from his previous mistake, sought advice from Alex and Oliver. Later Chris's rhombic dodecahedron idea was dropped and by 2005 Alex and Oliver had designed a computer program that generated Eternity II.
Using only the above criteria it is possible to logically and mathematically determine the remaining details of the puzzle.
From my experience the hardest puzzle of a given size requires:
1. Compact final shape
The most compact shape you can make with 256 squares is a square with side length 2561/2 = 16
2. Uniform tileable pieces
Symmetric pieces are harder to tile than asymmetric pieces because they have less unique orientations.
Duplicate pieces are harder to place then the equivalent number of different pieces.
There are differences in the border and interior piece tileability that requires leveling. There are less border pieces than interior pieces. Also the border pieces can only be placed with 1 orientation while the interior pieces have 4 orientations. To level out the piece tileabilities tune the number of border and interior edge types.
The more uniform the edge type distribution the more uniform the piece tileability.
3. One expected solution
Let M = Interior edge types
On average there are 2 joins per interior piece.
On average there is 1 border edge type join per border piece.
©2007 Brendan Owen