Polysticks: Puzzles & Solutions
| Author: | David Goodger <goodger@python.org> |
|---|---|
| Date: | 2012-10-07 |
| Revision: | 573 |
| Web site: | http://puzzler.sourceforge.net/ |
| Copyright: | © 1998-2012 by David J. Goodger |
| License: | GPL 2 |
Contents
Polysticks of Order 1 Through 3
These puzzles use the 1 monostick, 2 disticks, and 5 tristicks, for a total of 20 line segments on the square grid.
4x4 grid with clipped corners:
Tetrasticks
One-Sided Welded Tetrasticks
The "welded" tetrasticks are those that contain junction points, or welds, and therefore do not form simple connected paths (in other words, they branch). There are 6 welded tetrasticks, 4 of which are asymmetrical, therefore there are 10 one-sided welded tetrasticks.
5x5 grid: 3 solutions
Triangle: 9 solutions
Trapezoid: 7 solutions
15 of 16 Tetrasticks
Due to an imbalance in horizontal/vertical parity, the 16 tetrasticks cannot be formed into a symmetrical shape. But by omitting one of the five tetrasticks that have an excess of 2 vertical or horizontal line segments (H, J, L, N, or Y), symmetrical shapes can be formed.
6x6 grid: 1795 solutions
In this solution, the "Y" piece is omitted:
Aztec Diamond of order 3: 3 solutions
In this solution, the "J" piece is omitted:
One-Sided Tetrasticks
9 of the 16 tetrasticks are asymmetrical, therefore there are 25 one-sided tetrasticks.
5x5 diamond lattice: 107 solutions
Calculating all 107 non-isomorphic solutions took approximately 11 weeks using one core of a 2.66GHz Intel Core 2 Duo E6750 CPU (Python 2.6.6, Windows XP) on an otherwise idle and always-on machine. The calculated solutions correspond to those listed in "Covering the Aztec Diamond with One-sided Tetrasticks, Extended Version", by Alfred Wassermann, University of Bayreuth. (Wassermann, and Knuth before him, mistakenly called the shape an "aztec diamond", but an aztec diamond is a subtly different shape. This puzzle actually corresponds to a centered square number.)
The solution above is number 9 in Wassermann's paper, and number 58 in the Polyform Puzzler solutions. Both Wassermann's and the Polyform Puzzler solution sets are split into six sub-puzzles by position of the "X" piece, and correspond as follows:
X-Position (sub-puzzle)
Polyform Puzzler
Wassermann
Solutions
A (symmetrical)
6
19
B
5
36
C (symmetrical)
2
11
D
4
19
E
1
15
F
3
7
Note: The Polyform Puzzler order for the X-positions is in increasing distance from the center of the puzzle. Wassermann's order is as follows: with all X-piece possitions in the upper-left quadrant in two horizontal rows (rotate the diagrams in the paper 90° clockwise), ordered from left to right, top to bottom. Symmetrical X-positions (X-piece on the diagonal) have the I-piece limited to horizontal for both Polyform Puzzler and Wassermann (when rotated as noted above).
8x8 grid with center hole: solutions incomplete
8x8 grid with one clipped corner: solutions incomplete
8x8 grid with two clipped corners 1: solutions incomplete
X (designed for G4G10):
10x5 trapezoid: solutions incomplete
Polysticks of Order 1 Through 4
This puzzle uses the 1 monostick, 2 disticks, 5 tristicks, and 16 tetrasticks, for a total of 84 line segments.
7x7 grid: solutions incomplete
10x5 grid (with a hole): solutions incomplete
3x7 diamond lattice: solutions incomplete
Truncated diamond lattices:
6x4: solutions incomplete
8x3: solutions incomplete
12x2: solutions incomplete
Octagons:
Four overlapping squares: solutions incomplete
Three overlapping squares: solutions incomplete
