三维拼图谜题三重挑战

Three 3D Tiling Puzzles

2023-06-24 22:44:33

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Summary

The target to be built is a block of 64 cubes arranged as an array of 4 levels each containing 16 cubes.

A full scale version of the target is provided for teaching and to make the construction of the target clear. The individual cubes are outlined. Note that the target is symmetrical with a center of symmetry and three 4 fold rotation axes.

The pieces that build the target are octacubes, eight cubes connected by their faces. See the wikipedia article on polycubes -https://en.wikipedia.org/wiki/Polycube - Eight octacubes are required - a total of 64 cubes.

The tiling pieces occur in sets of eight identical pieces. If the pieces were all different the puzzles would be difficult to solve. When all the pieces are the same the puzzles are easier to solve, the solutions have symmetry, and are artistically more appealing.

The solutions contain sets of 2 and 4 tiles related by symmetry. If the tiles are printed in sets of four with different colors interesting patterns can be created. The three puzzles have unique tilings.

Trays are given which can serve to store the puzzle pieces. They can be used as aids during the solution process as they limit the placement of the tiling pieces; pieces can only be placed inside the boundaries of the trays.

Make 8 pieces of octa_7.stl, octa_78.stl and octa_128.stl to create the three puzzles. For a single simple puzzle just print 8 copies of octa_7; an octacube where all the cubes lie on a plane. Solutions are given in How Done.

Print Settings

Printer brand:
MakerBot
Printer:
MakerBot Replicator+
Notes:
octa_128.stl has an overhanging cube and needs support.

How Done

There are 6922 octacubes. They were used with the tiling program,tiling-05.tar from http://djm.cc/dmoews.html to search for all sets of 8 identical octacubes that would tile a 4x4x4 array of cubes. 212 octacube sets were found.

The combination of a symmetrical target and 8 identical tiles produced tilings where combinations of 2 or 4 tiles related by symmetry were common. 126 of the sets of eight tiled as 2 identical arrays of 4x4x2 cubes. Most of the remaining sets tiled as two identical sets of four tiles which intermesh to form the 4x4x4 array. 60 unique tilings were found.

Exploded views of the tilings are shown below. A single octacube is shown at the upper left. From left to right the octacubes are added 2 at a time to complete the tiling.

The openSCAD code has more information and can be used to create any of the remaining octacube sets that tile the target.