Pentacubes: Puzzles & Solutions
| Author: | David Goodger <goodger@python.org> |
|---|---|
| Date: | 2016-11-10 |
| Revision: | 634 |
| Web site: | http://puzzler.sourceforge.net/ |
| Copyright: | © 1998-2016 by David J. Goodger |
| License: | GPL 2 |
Contents
The pentacubes are named as per the pentominoes (planar), with the non-planar pentacubes following Kadon's "Superquints" naming. High-quality hardwood sets of pentacubes are available from Kadon Enterprises as Super Deluxe Quintillions (equivalent to the Pentacubes Plus pieces below). Add the Poly-4 Supplement to get all polycubes of order 1 through 4 as well (see Polycubes: Puzzles & Solutions).
Pentacubes
The 29 pentacubes cannot form a simple box-shaped solid. Other forms are possible however:
5x5x6 Towers
Tower 1: solutions incomplete (X3D model) |
Tower 2: solutions incomplete (X3D model) |
Tower 3: solutions incomplete (X3D model) |
Open Boxes
(closed on the bottom)
5x7x7 Open Box (design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model) |
3x9x9 Open Box: solutions incomplete (X3D model) |
18x3x3 Open Box (design from Kadon's Super Quintillions booklet): solutions incomplete (X3D model) |
2x7x15 Open Box: solutions incomplete (X3D model) |
Highrise Towers
3x3x20 Tower 1 (design by Nick Maeder): solutions incomplete (X3D model) |
3x3x20 Tower 2 (design by Nick Maeder): solutions incomplete (X3D model) |
3x3x17 Tower: solutions incomplete (X3D model) |
3x3x19 Crystal Tower: solutions incomplete (X3D model) |
Cubbyholes
5x7x5 cubbyholes (design from Torsten Sillke's pages [1992]): solutions incomplete (X3D model)
9x9x5 cubbyholes (design from Torsten Sillke's pages [1992]): solutions incomplete (X3D model)
Misc
2x11x11 frame (11x11 square bottom layer with 7x7 square frame on top; design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model)
Corner crystal (one empty cube is hidden in the corner; puzzle design by Nick Maeder): solutions incomplete (X3D model)
Built with Kadon's Super Deluxe Quintillions. Click for the full-size image.
Nine slices (design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model)
Great wall (design from Torsten Sillke's pages [1992]): solutions incomplete (X3D model)
5x9x9 fortress (design by & solutions from Nick Maeder): solutions incomplete (X3D model)
3x9x9 mound (design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model)
9x9x9 octahedral planes: no solutions yet, and probably none exist due to extreme parity imbalances
2x13x13 diamond frame (design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model)
2x13x13 diamond panel: solutions incomplete (X3D model)
2x3x2 chair (suggestion from Nick Maeder): 17 solutions (X3D model). Each solution uses only two pentacubes out of 29, and only 17 combinations (out of 406) produce solutions.
11x11x5 pyramid with base (design from Torsten Sillke's pages [1992]): solutions incomplete (X3D model)
Pyramid with spire (design from Torsten Sillke's pages [1992]): solutions incomplete (X3D model)
7x7x5 block: solutions incomplete (X3D model)
X (designed for G4G10): solutions incomplete
Astroid block: solutions incomplete (X3D model)
Diamond wall (design by Nick Maeder): solutions incomplete (X3D model)
Grand Platform (design from Kadon's Super Quintillions booklet): solutions incomplete (X3D model)
Stepped pyramids:
solutions incomplete (X3D model; four hollows are hidden in the base)
11x7 (3x1x1 hole hidden in the base; design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model)
11x7: solutions incomplete (X3D model)
Castle (design from Andrew Clarke's Poly Pages; center hole is one cube deep): solutions incomplete (X3D model)
Panorama (design from Torsten Sillke's pages [CFF Contest 36]): solutions incomplete (X3D model)
Cooling fins (design from Torsten Sillke's pages [10th Pentacube Contest, 1999]): solutions incomplete (X3D model)
Diamond tower: solutions incomplete (X3D model)
Octagonal frames:
central cube in bottom layer empty; design by Nick Maeder: solutions incomplete (X3D model)
Truncated tetrahedron (design by Michael Reid via Torsten Sillke): solutions incomplete (X3D model)
Hollow tetrahedron (design by Michael Reid via Torsten Sillke): solutions incomplete (X3D model)
4 cubes, suggested by Donald Knuth:
I'm still working on two more exercises about pentacubes. [One of them is an interesting shape that you don't seem to have yet: It consists of a 4x4x4 cube with three 3x3x3 cubes attached --- taking advantage of the remarkable fact that 29 times 5 equals 4^3 + 3^4! I found it in the pentacube book that Sivy Farhi published in the 70s. I still haven't seen the book by Kuenzell; it might well be in there too.]
—private correspondence, 2016-10-29
Pentacubes Plus
Sold under the name "Super Deluxe Quintillions" by Kadon, these 30 pieces are the 29 pentacubes plus a second L3 piece (a.k.a. J3), allowing the construction of box shapes.
Solids
5x5x6: solutions incomplete (X3D model)
3x5x10: solutions incomplete (X3D model)
2x5x15: solutions incomplete (X3D model)
2x3x25: solutions incomplete (X3D model)
Misc
11x11x11 octahedral planes (central cube is empty): no solutions yet, and probably none exist due to extreme parity imbalances
Diamond prism: solutions incomplete (X3D model)
Diagonal walls:
Diagonal block: solutions incomplete (X3D model)
Steps:
5x5x10: solutions incomplete (X3D model)
9x5x6: solutions incomplete (X3D model)
Stepped pyramids:
Non-Convex Pentacubes
There are 28 non-convex pentacubes: all of the pieces that have at least one inside corner. These are just the 29 regular pentacubes less the "I" (straight) pentacube, the only convex piece. The advantage of this set is that, like the Pentacubes Plus set, box shapes can be formed. Also, lacking the 5-cube-long "I" pentacube, more convoluted shapes can be formed.
Solids
2x5x14: solutions incomplete (X3D model)
2x7x10: solutions incomplete (X3D model)
4x5x7: solutions incomplete (X3D model)
Misc
Zigzag 1 (design by Robert Trietsch): solutions incomplete (X3D model)
Zigzag 2 (a symmetrical variation of zigzag 1 above): solutions incomplete (X3D model)
Diagonal walls:
Aztec pyramid (stacked Aztec diamonds): solutions incomplete (X3D model)
Stacked squares: solutions incomplete (X3D model)
Stepped pyramids:
9x8: solutions incomplete (X3D model)
13x6: solutions incomplete (X3D model)
Diamond pyramid (design from Torsten Sillke's pages [1992]): solutions incomplete (X3D model)
5x5x8 crystal tower: solutions incomplete (X3D model)
8x4x7 ring wall: solutions incomplete (X3D model)
Dorian Cubes
These puzzles are constructed from the 25 pentacubes that each fit within a 3×3×3 box (omitting the I, L, N, and Y pentacubes).
Designed by Joseph Dorrie. Referenced on p. 41 of Knotted Doughnuts and Other Mathematical Entertainments, by Martin Garder, 1986.
5×5×5 Dorian Cube solutions incomplete (X3D model)
Dorian Cube 5 Towers: the Dorian Cube subdivided into 5 towers, 4 P-pentomino shaped towers around a central X-pentomino tower (designed by Torsten Sillke). Solutions incomplete (X3D model)
Exploded version (all 5 towers separated; X3D model)
