Polycubes: Puzzles & Solutions

Author: David Goodger <goodger@python.org>
Date: 2015-02-24
Revision: 600
Web site:http://puzzler.sourceforge.net/
Copyright: © 1998-2015 by David J. Goodger
License:GPL 2
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Contents

High-quality hardwood sets of these polycubes are available from Kadon Enterprises: Super Deluxe Quintillions (pentacubes) and Poly-4 Supplement (polycubes 1-4).

Tetracubes

The 8 tetracubes are composed of 32 unit cubes.

Soma Cubes

It is a beautiful freak of nature that the seven simplest irregular combinations of cubes can form a cube again. Variety growing out of unity returns to unity. It is the world's smallest philosophical system.

—Piet Hein

Soma Cubes include all non-convex polycubes of order 3 (1) and 4 (6). They were invented by Piet Hein in 1933. The 7 Soma cubes are composed of 27 unit cubes. Many of the designs below are from the Thinkfun Block By Block challenge cards. The two castles are from Dennis Nehen's Soma Cube pages.

If we consider the "a" and "b" pieces (a chiral pair) to be interchangeable in reflection, there are exactly half as many solutions as shown for each of the puzzles below. In other words, we can consider two solutions that are mirror reflections of each other to count as only one solution, but only if we allow pieces "a" and "b" to be exchanged (in the counts that follow, we don't allow this).

images/cubes/soma-3x3x3.png

3x3x3 cube: 240 solutions (X3D model)

images/cubes/soma-cornerstone.png

Cornerstone: 10 solutions (X3D model)

images/cubes/soma-curved-wall.png

Curved Wall: 66 solutions (X3D model)

images/cubes/soma-high-wall.png

High Wall: 46 solutions (X3D model)

images/cubes/soma-long-wall.png

Long Wall: 104 solutions (X3D model)

images/cubes/soma-screw.png

Screw (not self-supporting!): 14 solutions (X3D model)

images/cubes/soma-pyramid.png

Pyramid: 14 solutions (X3D model)

images/cubes/soma-castle-1.png

Castle 1: 10 solutions (X3D model)

images/cubes/soma-castle-2.png

Castle 2: 10 solutions (X3D model)

Diabolical Cube

The Diabolical Cube puzzle dates from the 19th century, published in Puzzles Old and New by Professor L. Hoffmann, London 1893. The puzzle contains one piece from each of the solid polyominoes of orders 2 through 7 (i.e one solid domino, ..., one solid heptomino), for a total of 27 unit cubes. More info in The Puzzling World of Polyhedral Dissections.

Polycubes of Order 2 Through 4

These puzzles use the 1 dicube, 2 tricubes, and 8 tetracubes, for a total of 40 unit cubes.

Polycubes of Order 1 Through 4

These puzzles use the 1 monocube, 1 dicube, 2 tricubes, and 8 tetracubes, for a total of 41 unit cubes.

Polycubes of Order 1 Through 5

These puzzles use the 1 monocube, 1 dicube, 2 tricubes, 8 tetracubes, and 29 pentacubes, for a total of 186 unit cubes.

Solid Hexominoes

These puzzles use the 35 solid hexominoes (planar hexacubes), for a total of 210 unit cubes.

Solid Hexominoes Plus

These puzzles use the 35 solid hexominoes (planar hexacubes), plus a second copy of the N06 piece (called S16), for a total of 216 unit cubes (which equals 6³).

6³ Cubes: Polycubes of Order 1 Through 5 Plus Select Hexacubes

These puzzles use the 1 monocube, 1 dicube, 2 tricubes, 8 tetracubes, 29 pentacubes, and a selection of 5 hexacubes, for a total of 216 unit cubes (which equals 6³). The 5 hexacubes chosen (arbitrarily, as "the five most interesting") are as follows. (Names in parentheses are from Kadon's hexacube naming system.):

images/pieces/polycubes/Ba6.png

Ba6 (B)

images/pieces/polycubes/Nt6.png

Nt6 (N2b3)

images/pieces/polycubes/O06.png

O06 (O)

images/pieces/polycubes/Qe6.png

Qe6 (Q4b1)

images/pieces/polycubes/Tp6.png

Tp6 (T4b4)

This set of polycubes was selected because it can be used to illustrate both sides of the first solution to the Diophantine 3.1.3 equation (the only solution consisting of consecutive integers):

3³ + 4³ + 5³ = 6³

For more on this curious equation and dissection puzzle, see: