Polytwigs (Hexagonal-Grid Polysticks): 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 |
Contents
- One-Sided Polytwigs of Order 1 Through 3
- Tetratwigs
- One-Sided Tetratwigs
- Polytwigs of Order 1 Through 4
- One-Sided Polytwigs of Order 1 Through 4
- Pentatwigs
- One-Sided Pentatwigs
- Polytwigs of Order 4 & 5
- Polytwigs of Order 1 Through 5
- One-Sided Polytwigs of Order 1 Through 5
- Hexatwigs
- One-Sided Hexatwigs
- Polytwigs of Order 1 Through 6
- Quasi-Tritwigs
- Quasi-Polytwigs of Order 1 Through 3
One-Sided Polytwigs of Order 1 Through 3
This puzzle uses the 1 monotwig, 1 ditwig, and 4 one-sided tritwigs,, for a total of 15 line segments on the hexagonal grid.
Triangle: 3 solutions
Tetratwigs
This puzzle uses the 4 tetratwigs for a total of 16 line segments on the hexagonal grid.
Arch: 2 solutions
One-Sided Tetratwigs
These puzzles use the 6 one-sided tetratwigs for a total of 24 line segments on the hexagonal grid.
Triangles with 3 line segments omitted in various positions:
Arches with 2 line segments omitted in various positions:
Polytwigs of Order 1 Through 4
These puzzles use the 1 monotwig, 1 ditwig, 3 tritwigs, and 4 tetratwigs, for a total of 28 line segments on the hexagonal grid.
Hexagons with 2 line segments omitted in various positions:
4x2 inset rectangle: 5,755 solutions
One-Sided Polytwigs of Order 1 Through 4
These puzzles use the 1 monotwig, 1 ditwig, 4 one-sided tritwigs, and 6 one-sided tetratwigs, for a total of 39 line segments on the hexagonal grid.
5x2 trapezoid: solutions incomplete
3x2 elongated hexagons with 2 segments omitted in various positions:
Triangles:
Pentatwigs
These puzzles use the 12 pentatwigs (a.k.a. pentacules), for a total of 60 line segments on the hexagonal grid.
Triangle: 56 solutions
Triangle variations (puzzle designs by Colin F. Brown):
5x3 parallelogram: 194 solutions
4x4 parallelograms:
1 solution (design by Colin F. Brown)
6x3 trapezoid: 184 solutions
5x3 staggered rectangle: 145 solutions
5x3 wave-staggered rectangle (puzzle design by Colin F. Brown): 202 solutions
3x3 chevron: 115 solutions
5x2 chevron: 241 solutions
Butterfly: 8 solutions
Trefoil (puzzle design by Colin F. Brown): 4 solutions
If you look carefully, you can see that the solution above consists of 3 congruent shapes (in fact, all 4 solutions share this property). This is illustrated explicitly in the following exploded form (same 4 solutions, just rearranged):
Möbius strip (puzzle design by Colin F. Brown): 6 solutions?
Take the puzzle strip above, give one end a half-twist, and join it to the other end, resulting in a Möbius strip, a surface with only one side and one edge.
This puzzle was implemented without any coordinate "wrap-around": a piece cannot begin at the right edge and continue (through the join) to the left edge. There may be many more solutions with such a wrap-around configuration.
Elongated rounded rectangle (design by Colin F. Brown): 21 solutions
Rosette clusters (design by Colin F. Brown):
X (designed for G4G10): 32 solutions
5x2 elongated hexagon (with holes; puzzle design by Colin F. Brown): 2 solutions
One-Sided Pentatwigs
These puzzles use the 19 one-sided pentatwigs, for a total of 95 line segments on the hexagonal grid.
12x2 trapezoid: solutions incomplete
7x4 inset rectangle with 1 segment omitted in various positions:
8x2 elongated hexagons (with holes):
3x4 elongated hexagons (with holes):
String of rosettes (design by Peter F. Esser): solutions incomplete
Cross (design by Peter F. Esser): solutions incomplete
Peanuts:
solutions incomplete (design by Peter F. Esser)
Polytwigs of Order 4 & 5
These puzzles use the 4 tetratwigs and 12 pentatwigs, for a total of 76 line segments on the hexagonal grid.
Triangle: solutions incomplete
Diamond ring: solutions incomplete
4x3 elongated hexagon ring: solutions incomplete
5x3 butterfly ring: solutions incomplete
5x5 inset rectangle ring: solutions incomplete
Four congruent groups:
Colin F. Brown posed this problem: divide the twelve pentatwigs into four groups of three, add one of the tetratwigs to each group. Now find a 19-line region that each (equally divided) group will tile.
Mr. Brown provided one solution:
Example 1 of four congruent groups from the tetratwigs & pentatwigs: 19 solutions.
Others are possible:
Example 2 of four congruent groups from the tetratwigs & pentatwigs: 38 solutions.
Open Problem: Combined Symmetrical Shape
Find a solution to the four congruent groups problem such that the four groups can be combined into a symmetrical shape. The more symmetries the better.
The best case I have been able to come up with so far:
Example 3 of four congruent groups from the tetratwigs & pentatwigs: 2 solutions.
Example 3, combined: one symmetry (180° rotational); 24 solutions when restricted to combinations of the congruent shapes above; many solutions if unrestricted.
Polytwigs of Order 1 Through 5
These puzzles use the 1 monotwig, 1 ditwig, 3 tritwigs, 4 tetratwigs, and 12 pentatwigs, for a total of 88 line segments on the hexagonal grid.
5x5 inset rectangle: solutions incomplete
6x4 parallelograms with 3 segments omitted in various positions:
9x3 trapezoid ring: solutions incomplete
4x3 elongated hexagon with 1 segment omitted: solutions incomplete
8x2 butterfly with 2 segments omitted in various positions:
One-Sided Polytwigs of Order 1 Through 5
These puzzles use the 1 monotwig, 1 ditwig, 4 one-sided tritwigs, 6 one-sided tetratwigs, and 19 one-sided pentatwigs, for a total of 134 line segments on the hexagonal grid.
12x2 butterfly: solutions incomplete
12x3 parallelogram with 3 segments omitted in various positions:
13x3 trapezoid with 3 segments omitted in various positions:
Hexatwigs
These puzzles use the 27 hexatwigs, for a total of 162 line segments on the hexagonal grid.
Triangle: solutions incomplete
Triangle rings:
solutions incomplete (design by Peter F. Esser)
solutions incomplete (design by Peter F. Esser)
14x2 elongated hexagon: solutions incomplete
Hexagon rings:
solutions incomplete (design by Peter F. Esser)
Elongated hexagon ring (design by Peter F. Esser): solutions incomplete
6x3 semiregular hexagon: solutions incomplete
This solution was first found by Peter F. Esser.
Knobbed hexagon: solutions incomplete
X (designed for G4G10):
Trefoils:
solutions incomplete (design by Peter F. Esser)
solutions incomplete (design by Peter F. Esser)
solutions incomplete (design by Peter F. Esser)
"Hexatwigs trefoil 3" built from pieces laser-cut from 5mm acrylic (using this file). Puzzle pieces, solution, and photograph by Peter F. Esser. Click for the full-size image.
One-Sided Hexatwigs
These puzzles use the 49 one-sided hexatwigs, for a total of 294 line segments on the hexagonal grid.
Hexagon ring: solutions incomplete
This solution was first found by Peter F. Esser.
26x2 elongated hexagon (design by Peter F. Esser): solutions incomplete
Polytwigs of Order 1 Through 6
These puzzles use the 1 monotwig, 1 ditwig, 3 tritwigs, 4 tetratwigs, 12 pentatwigs, and 27 hexatwigs, for a total of 250 line segments on the hexagonal grid.
22x2 elongated hexagon (design by Peter F. Esser): solutions incomplete
Quasi-Tritwigs
These puzzles use the 17 quasi-tritwigs, for a total of 51 line segments on the hexagonal grid. See Quasi-Polyforms in An Introduction to Polytwigs.
10x1 parallelogram (design by Colin F. Brown): solutions incomplete
6x2 parallelogram: solutions incomplete
Triangles:
Snowflakes:
2x3 elongated hexagons:
Trefoils:
Quasi-Polytwigs of Order 1 Through 3
These puzzles use the monotwig, the 3 quasi-ditwigs, 17 quasi-tritwigs, for a total of 58 line segments on the hexagonal grid. See Quasi-Polyforms in An Introduction to Polytwigs.
9x2 rounded rectangle: solutions incomplete
Hexagon rings:
5x3 parallelogram ring: solutions incomplete
6x3 trapezoid ring: solutions incomplete
