History

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.

Criteria

• Hardest edge-matching puzzle
• 256 one-sided square pieces
• Final shape has a grey border
• One compulsory hint

Design

Using only the above criteria it is possible to logically and mathematically determine the remaining details of the puzzle.

• Final shape is a 16 x 16 square
• Asymmetric pieces
• Unique pieces
• Different number of border and interior edge types
• Flat edge type distribution
• Interior edge types = 17
• Border edge types = 5

Design details

From my experience the hardest puzzle of a given size requires:

1. Compact final shape
2. Uniform tileable pieces
3. One expected solution

1. Compact final shape

The most compact shape you can make with 256 squares is a square with side length 256^1/2 = 16

• Final shape is a 16 x 16 square

2. Uniform tileable pieces

Symmetric pieces are harder to tile than asymmetric pieces because they have less unique orientations.

• Asymmetric pieces

Duplicate pieces are harder to place then the equivalent number of different pieces.

• Unique 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.

• Different number of border and interior edge types

The more uniform the edge type distribution the more uniform the piece tileability.

• Flat edge type distribution

3. One expected solution

Let M = Interior edge types
Let B = Border edge types

On average there are 2 joins per interior piece.
Interior solutions = 195! x 4¹⁹⁵ / M^{2 x 196} = 1
195! x 4¹⁹⁵ / M³⁹² = 1
M = (195! x 4¹⁹⁵)^1/392
M = 16.85

• Interior edge types = 17

On average there is 1 border edge type join per border piece.
On average there is 0.5 interior edge type joins per border piece.
Border solutions = 56! x 4! / (B⁶⁰ x M^{0.5 x 56}) = 1
56! x 4! / (B⁶⁰ x 17²⁸) = 1
B = (56! x 4! / 17²⁸)^1/60
B = 4.97

• Border edge types = 5