The Official Solution to Alexander's Star Puzzle

95 Pages·1982·1.305 MB·English

by  Adam Alexander

P,-Ok NMOAN T9
gj;(cid:1)LIN
Po
L A
1I1
ALEXANDIS
SMAR"
THE OFFICIAL SOLUTION TO
ALEXANDER'S STAW
PUZZLE
N
CONTENTS
L Introduction 1
1. RUBIK'S CUBETM 2
2. The Invention of the Star 4
3. The Making of the Star 7
II. Solution Preliminaries 9
4. The Great Dodecahedron 10
5. ALEXANDER'S STAR@: /
Construction and Coloration 16
6. The Geography of
ALEXANDER'S STAR 19
7. Flush Faces 24
8. Following Leads 26
9. Seeing Stars 28
10. A Menu of Moves 31
11. A Plan of Attack 43
m. The Step-by-Step Solution 45
Step 1. First Two Pieces and Orientation 46
Step 2. Completing the North Star 49
Step 3. Filling in the Northern Ring 53
Step 4. Completing the Equator 57
Step 5. Completing the Southern Ring 62
Step 6. Starting the South Star and
Bind-Finding 64
Step 7. The Bind 71
Step 8. Flipping Out 73
IV. Variations on the Star 75
1. The Non-Opposing Solution 77
2. The Triangular Coloring 79
3. The Totally Flipped Star 80
4. The Star-Center Labeling 82
5. The Triangular Labeling 83
6. The Completion of ALEXANDER'S STAR:
The Inner Labeling 84
V. Mathematics and Puzzles 87
THE OFFICIAL SOLUTION TO
ALEXANDER'S STAW
PUZZLE
Adam Alexander
Ballantine Books * New York
Copyright © 1982 by Adam Alexander
ALEXANDER'S STARE is a registered trademark of
Gabriel Industries. a division of CBS Inc.
RUBIK'S CUBETM is a registered trademark of
Gabriel Industries, a division of CBS Inc.
All rights reserved under International and Pan-American
Copyright Conventions. Published in the United States by
Ballantine Books. a division of Random House, Inc..
New York. and simultaneously in Canada by Random House
of Canada Limited. Toronto. Canada.
Library of Congress Catalog Card Number: 82-90771
ISBN 0-345-30842-5
Manufactured in the United States of America
First Edition: December 1982
Drawingsa nd book design by Gene Siegel
I
Introduction
RUBIK'S CUBET
Sometime in the early seventies, Erno Rubik, pro-
fessor of Architecture and Design at the School for
Commercial Artists in Budapest, Hungary. invented
his famous Cube. The Hungarian patent is dated
1975.
In 1979, the Ideal iby Corporation purchased
the rights to manufacture and market RUBIK'S
CUBETD around the world. It was introduced in early
1980, but 1981 was truly the "year of the Cube:' Be-
tween the Cubes made by Ideal and the counterfeits
made by all sorts of small companies, approxi-
mately eighty million Cubes were sold worldwide.
Professor Rubik's motivation in creating the
Cube was to help his students visualize objects in
three dimensions. Whether or not he achieved his
goal is hard to say; but as a device to illustrate the
complexities of simple processes, It created a revolu-
tion. The complexity of world politics was illustrated
on the cover of Newsweek magazine by a RUBIK'S
CUBET™w ith a map of the world on Its sides.
It is my contention that the appeal of the Cube
2
has to do with its simple mechanical structure in
contrast to its complexity of patterns; it is a bridge
between the old mechanical world and the new elec-
tronic computer world.
Of course, there are more basic answers for the
appeal of the Cube. It seems so obvious at first
glance: you simply have to put everything back
where it's supposed to be. Not so simple, after all:
you suddenly realize that each move affects things
in unexpected ways. This Is the key to the appeal of
the Cube. It looks simple at first, and then it reminds
you that there are more consequences to any action
than you ever thought.
Rubik discovered more than just a puzzle. In a
sense, he discovered a principle of interchangeable,
interlocking parts. It was my attempt to extend and
elucidate this principle that led to the creation of
ALEXANDER'S STARK
As a mathematician working in the toy in-
dustry, I watched the RUBIK'S CUBE phenomenon
with great interest. I wondered whether Ideal Ibys
had something in the works, a sequel to the Cube. So
I watched very carefully, and atIby Falr, in February
1981, the event where major toy companies show
their new products, I found out that.. .there was no
sequel to the Cube!
"This is my departments I thought to myself.
After all, when I worked for one large toy company,
my business card gave me the title "Corporate
Mathematician:' So I started the process to figure out
the range of possibilities of "Rubicoid mechanisms:'
my own term for any system of pieces that per-
mitted repositioning while holding themselves
together, in the manner of the Cube.
3
The
Invention of
the Star
The first thing I did was try to generalize the prin-
ciple of the Cube. It is a set of pieces, with subsets
that move with respect to the rest but without the
whole set falling apart. The subsets, which are
overlapping, rotate. That was the first generaliza-
tion: overlapping subsets of pieces that could each
rotate with respect to the others while the whole
assemblage held together.
Very good, but what does that say about the
pieces? They have to be alike or of only a few types,
and they have to be replaceable. This tells me that
the entire shape must be symmetrical, it's got to look
like itself from many different angles, the way a cube
looks like itself from many different directions.
That's what I needed: a shape with symmetry.
Where do I look for shapes with lots of symmetry?
Solid geometry, especially the study of the regular
figures.
So that was the beginning. From tetrahedron
to octahedron to cube to dodecahedron to ico-
sahedron, I figured out systems with four, six, eight,
4