The Official Solution to Alexander's Star Puzzle

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gj;LIN Po L A

1I1 ALEXANDIS

SMAR"

THE OFFICIAL SOLUTION TO

ALEXANDER'S STAW PUZZLE

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CONTENTS

L Introduction 1. RUBIK'S CUBETM 2. The Invention of the Star 3. The Making of the Star II. Solution Preliminaries 4. The Great Dodecahedron 5. ALEXANDER'S STAR@: Construction and Coloration 6. The Geography of ALEXANDER'S STAR 7. Flush Faces 8. Following Leads 9. Seeing Stars 10. A Menu of Moves 11. A Plan of Attack

m. The Step-by-Step Solution Step 1. First Two Pieces and Orientation Step 2. Completing the North Star Step 3. Filling in the Northern Ring Step 4. Completing the Equator

1 2 4 7 9 10 / 16 19 24 26 28 31 43 45 46 49 53 57

Step 5. Completing the Southern Ring Step 6. Starting the South Star and Bind-Finding Step 7. The Bind Step 8. Flipping Out

62 64 71 73

IV. Variations on the Star 75 1. The Non-Opposing Solution 77 2. The Triangular Coloring 79 80 3. The Totally Flipped Star 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 Drawingsand book design by Gene Siegel

I Introduction

RUBIK'S CUBET

Sometime in the early seventies, Erno Rubik, professor 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:' Between the Cubes made by Ideal and the counterfeits made by all sorts of small companies, approximately 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 revolution. The complexity of world politics was illustrated on the cover of Newsweek magazine by a RUBIK'S CUBET™with 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 electronic 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 industry, 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 permitted 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 principle 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 generalization: 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 icosahedron, I figured out systems with four, six, eight, 4

twelve, twenty, and even thirty axes. (The Cube has six.) And an important theorem of solid geometry told me that I could have only a limited number of symmetries in three dimensions and, therefore, only a limited number of shapes. I knew I had a good number of shapes to pick from for symmetry, but could they all be made into Rubicoid mechanisms? Again, I had to generalize on the Cube. What makes it possible for the whole system to hold together and not fall apart? What makes it possible for a given piece to rotate around any of a few different axes? This was the principle I had to describe. Keeping the Cube in mind, I came up with this analysis. If you take the Cube apart by taking out only those eight pieces from the top that surround the top center and then reassembling them, top down, you will see that an approximately circular hole will be left in the remaining part of the Cube and that a corresponding cylinder will seem to come out of the reassembled pieces. The extended cylinder on the reassembled part will have a hole in its center corresponding to the shaft holding the top center piece of the remaining part.The extended cylinder fits into the circular hole and permits the top pieces to rotate around the shaft. But it also prevents the top pieces from sliding around horizontally, and the top center piece prevents the other top pieces from moving upward. Well, that's fine for the top slice. What about the sides and the bottom? They work exactly the same way. In fact, the edges of the top center piece that prevent the top pieces from moving up, are really arcs of the circular holes that hold the side slices in place. These circular holes overlap. Overlapping cylinders That was the secret. 5

There was only one last question to answer: which of all these possible shapes to use? Invention was described by Thomas Edison as 10 percent inspiration and 90 percent perspiration. What I needed was some of that 10 percent. It came in a strange way. Around March 1981, a toy design company called me and asked me if I could come up with an electronic puzzle "like RUJBIK'S CUBE:' "An electronic RUBIK'S CUBE?!" I thought to myself, "Why try to improve this perfect product with electronics? Now, if you'd asked for an extension of the principle, I'd know just what to do:' At that moment the image of the Star came into my mind. I quickly ended the conversation and drew a few sketches. The shape I had thought of is called the "great dodecahedron:' I had known of it for many years and had always liked it, and felt that if I were ever to have the chance to popularize any one shape from solid geometry, it would be this one. The phone call had jogged my memory and brought to completion many intentions, old and new. For a few days, I thought about the procedure to construct a model. Finally. I sat down to make a working model in cardboard.

6

The Making of the Star Paper or cardboard is my medium. I have been making models of geometric figures for years, and I have developed the knack of using cardboard, paper, pencil, ruler, compass, knife, and glue to achieve the desired result. The making of the prototype of the Star, a working model in cardboard, took about sixty hoursthree twelve-hour sessions and one twenty-fourhour session. I had to have a pattern to work from, so I drew one with ruler and compass on tracing paper. Then. using the point of the compass, I poked through the tracing paper into the cardboard, leaving tiny holes. These I then connected with pencil lines. Having thus traced my pattern onto the cardboard, I chose which edges to put tabs on. Using the X-acto knife, I cut and scored the cardboard, and finally glued the various pieces. I really didn't proceed so easily. I would do some poking, connecting, cutting, scoring, and gluing and make one or a few parts. Then I would start 7

the process again, making other parts. There were times when I would get very frustrated; at times I would make mistakes. I got a painful case of "ruler elbow" from holding the ruler down on the cardboard in order to cut it with the knife. Eventually. I made progress. For the last session, instead of working twelve hours as I had planned, I saw the end in sight and continued working another twelve hours as well. I was exhausted, but I had a working model of the Star.

8

Solution Preliminaries

The Great Dodecahedron Just as RUBIK'S CUBETm is in the form of a cube, ALEXANDER'S STARO is in form of a great dodecahedron. The word "great" is used to distinguish this shape from the well-known shape of the regular dodecahedron. The regular dodecahedron (or just "dodecahedron") has been known for thousands of years. while the great dodecahedron has been known for only about 200 years. It was discovered by the French mathematician Poinsot. In order to understand the great dodecahedron, we must start with a regular dodecahedron. If you look at Figure 1, you will see a regular dode-

Figur I 10

cahedron. Each side of the dodecahedron is a regular pentagon, a five-sided figure. There are twelve of these pentagons on the sides of the regular dodecahedron. In fact, that is where it gets its name: Do = two; deca = ten; hedron = side; so "dodeca-

hedron" means a twelve-sided (solid). The pentagons meet three atatime at each comer. So sometimes the dodecahedron is referred to as 59, meaning three five-sided figures at each corner. There are twenty of these corners, and thirty edges in between them. In order to get the great dodecahedron from the regular dodecahedron, we must go through the process known as "stellation:' In stellating, we extend the faces of the dodecahedron until they meet themselves again. We apply this process twice to get to the great dodecahedron. If you look at Figure 2, you will see that the five pentagons surrounding the top pentagon of the dodecahedron have been extended so that they meet in a point and form a five-sided pyramid. We have thus grown a pyramid on the top pentagon.

Figure 2

We can grow a pyramid on all the pentagons of the dodecahedron. You will see the result, if you look at Figure 3. This shape is called the "small stellated 11

Figur

a

dodecahedron' Each pentagon of the original dodecahedron has grown into a flat five-pointed star. with its central area covered by one of the pyramids. We can continue the process of stellation by extending the faces and filling in the areas between the pyramids. If you look at Figure 4, you will see that a chunk has been grown between two pyramids. with sides that are extensions of two of the stars of the small stellated dodecahedron. This chunk is technically a tetrahedron because it has four sides. If we grow thirty of these chunks between all the

ruw* 4

12

Figure 5

pyramids, we will get the great dodecahedron in Figure 5. The pentagons that are the sides of the regular dodecahedron grew into the flat flye-pointed stars of the small stellated dodecahedron. These stars have now grown into pentagons again, but much bigger than the original ones. But these big pentagons are a little hard to see because most oftheir area is hidden inside the solid. All that can be seen of each pentagon are five little triangles, as in Figure 5. If we think of the entire great dodecahedron as a surface covered by twelve self-intersecting pentagons, we can describe this shape as a surface tied into a knot! It's an amazing idea, and maybe that's why this shape has been known for only 200 years, not thousands. But a simpler way to understand this shape is to think of it as constructed from 1 regular dodecahedron, 12 five-sided pyramids, one on each side of the dodecahedron, and 30 chunks between the twelve pyramids. (See Figure 6.) These three kinds of pieces are present, slightly modified, in your ALEXANDER'S STAR. 13

Flgure 6 Partially exploded view of a great dodecahedron

The great dodecahedron is a beautiful shape. No matter how you look at it, you see a threedimensional five-pointed star on a flat pentagonal background. Each star shares an arm with another star, so each star-arm (chunk) does double duty, causing an optical illusion. To see It, hold your ALEXANDER'S STAR with an edge facing you, as in Figure 7. You can see a star on the left, and you can see a star on the right. They share an arm, the one facing you directly. But try to see both stars at the same time. You can't! 14

How many stars are there? Well, there is one on each background pentagon, and there are twelve background pentagons, so there are twelve stars. You can locate them by the points at the center of each star.

Pgwe

15

ALEXANDER'S STAR Construction and Coloration ALEXANDER'S STAR is constructed in much the same way as the great dodecahedron. It has a dodecahedron in the center that has been reduced slightly so there is some room between it and the pyramids. The pyramids no longer have simple triangles on their sides.The base of each triangle is now an arc of a circle. Five of these arcs form a full circle, and allow the chunks to rotate. A shaft goes down the center of each pyramid connecting it to the dodecahedron, allowing the pyramid to rotate. (See Figure 8A.) The chunks are no longer tetrahedrons; two of their sides are still intact, but the other two (the sides that are hidden) have been removed, and two little springy clips have been put in to hold the chunks between the arc bases of the pyramids. (See Figure 8B.) The five chunks that touch a given pyramid form a star-not a flat star, but a three-dimensional five-pointed star.These are the stars that give ALEXANDER'S STAR its name, and cause the optical 16

Figure OA

Figure 35

illusion mentioned at the end of the last chapter. But in ALEXANDER'S STAR, they also rotate as a set; since each chunk belongs to two stars, it can rotate around each of two axes. As was noted before, there are twelve of these stars, and they all turn. It is important to turn a star so that its points come to rest on the corners of its background pentagon, otherwise you won't be able to turn any of the stars next to that star. ALEXANDER'S STAR is colored with six colors: red, yellow, blue, green, orange, and white. The Star comes to you packaged in one of its solved states with every background pentagon colored a solid color.There are twelve background pentagons, so there will be two background pentagons of each color: these two pentagons will be on opposite sides of the Star. This pattern-of solid background pentagons with opposite pentagons colored alike-is the object of any solution to ALEXANDER'S STAR. 17

Because of this coloration, each possible combination of two different colors will be found on two pieces in the Star. No piece has the same color on both sides. When the Star is solved, any two opposite pieces-pieces on opposite sides of the Star-will have the same color combination.

18

The Geography of ALEXANDER'S STAR0 ALEXANDER'S STAR is not a simple shape, but with the right description, it can be easily understood. First we must establish a vocabulary. We will refer to the entire thing as ALEXANDER'S STAR, or just the Star; with a capital S. A star on the surface that turns will be referred to as a star, with a small s. The pieces that we referred to earlier as "chunks" or "star-arms" will be called pieces from here on. Each piece has two colored triangular sides called faces. The line separating the two faces on a piece is called the edge. The ends of the edge are called the tips of the piece. (See Figure 9A.) The pentagon that forms the background of a Ed~g

A Face >

A Piece 1 9A 19

~ATIp

Northern plane

Figures9

star will be referred to as a backgroundpentagon or just background. A point that is the center of a star will be called a point. At each point, five pieces meet; that is, their tips meet. These five pieces form a star called "the star of that point:' In the same way. every star has one point at its center called "the point of that star:" A piece that makes up part of a star will be said to "belong" to that star. Every piece belongs to two stars, so the two stars to which the piece belongs must always be specified. Hold your Star so that one point is on top with opposite point on the bottom, as in Figure 9B. The point on the top is called the north pole. The point on the bottom is the south pole. The star of the north pole is called the north star and the star of the south pole is called the south star. The background of the north star is called the northern plane. The background of the south star is called the southern plane. The northern plane is made up of faces from 20

five pieces that form a ring around the north star. These five pieces are called the northern ring. In the same way, the ring of five pieces around the south star is called the southern ring. In between the northern and the southern ring are ten pieces zigzagging around the middle. These ten pieces are called the equator. We can think of the pieces in the equator as forming five V's. A piece in the equator on the left side of a V is called a leftleaning piece. A piece in the equator on the right side of a V is called a right-leantng piece. As we go around the equator, we find five left-leaning pieces, and five right-leaning pieces. We have now classified the pieces. A piece can be in a) the north star b) the northern ring c) in the equator and left-leaning d) in the equator and right-leaning e) In the southern ring fO in the south star. Now that we have some classification of the pieces, we have to be able to refer to the stars. We have already classified two of the twelve stars: the north star and the south star. The ten stars that are left fall into two sets: the northern set of stars and the southern set of stars. We can identify them by their center points. The five points at the tips of the pieces in the northern ring are the center points of the stars of the northern set. The five points at the tips of the pieces in the southern ring are the center points of the stars of the southern set. (See Figure 10.) We will now identify the faces of pieces on the Star. This will be fairly simple for most of the pieces except those in the north and south star. For a piece in the northern ring, we will refer to its upper face as 21

of stars them set

A star of thf northern se (shaded)

of stars them set

Figure 10

its northernface or its "face in the northern plane:' Its lower face will be referred to as its southernface. For a piece In the southern ring, we will refer to Its lower face as its southemface or its "face in the southern plane:' Its upper face will be referred to as its northernface. For a piece in the equator, whether left-leaning

orthem face uthern face

Flou" I

22

or right-leaning, we will refer to its upward face as its northernfaceand its downward face as its southern face. (See Figure 11.) For a piece in the north star, we do not have a sense of north or south so we will use this rule: Point the piece directly at you. The face on your right is its right face; the face on your left is its left face. Left faces in the north star may be said to face clockwise; right faces, counterclockwise. (See Figure 12.) We now have words and a system of orientation for the whole star, and we're ready for the next step.

Looking down on North Star PFgure 12

23

Flush Faces

The word flush means lined up, on the same level, touching. We will use this word to refer to two faces in the same plane or background, with tips thatjust touch. In Figure 13, you can see two sets of flush faces. One set, marked A, is part of a background. The other set, marked B. seems to be part of a star. Two flush faces will always look like a big triangle with its center section partly hidden by another piece. Flush faces are not hard to see, and they are

These two faces are ftush

These two faces are flush

Plour 13

24

the simplest relationship that two faces can have (besides being on the same piece). We want to eventually get all backgrounds to be solid colors-or we couldjust as well say we wantall pairs of flush faces to be the same color. In fact, that is really the way we will go about the solution to ALEXANDER'S STAR: by moving the pieces around so that, one by one, pairs of flush faces will end up the same color. This will be especially Important in the beginning of the solution.

25

(0

Following Leads

A lead is a set of three pieces in which two faces of one piece are flush with a face on each of two other pieces. (See Figure 14.) The three pieces form a sort of arrowhead or dart shape. This is called a "lead" because the outer faces, A and B, lead into the faces, a and b, of the third piece. This third piece is called thefront of the lead. A is flush with a, and B is flush with b.

A lead (shaded) Mquwr 14

26

If the pieces with faces A and B are correctly positioned at some stage of the solution, then the piece that should go in the position of the piece with faces a and b is completely determined: the piece to go there must have the same color as face A on its face a, and the same color as face B on its face b. Just how to get such a piece in the right position without disturbing the pieces that are correctly positioned is another question, to be answered later. But for now, the lead tells us which piece must go where, and in what orientation.

27

Seeing Stars

It is easy to see that the solid backgrounds are the object of this puzzle, but the question keeps coming up: "I can see that the background should be all one color, but what about the star? Is it just multicolored? Can the stars be solid colors?" No, the stars can't be solid colors. Every piece has two different colors on it; since each star shares its pieces with other stars, if one star were a solid color, then all the stars would be that color and the whole Star would be one color. So the stars are multicolored but when the Star is solved, each star has exactly five colors on it in a special pattern. In order to understand this pattern, we should look at a simple five-pointed star. Most people think of a five-pointed star like the one in Figure 15A; at first glance, it appears to have ten short sides around it. But another way to look at a five-pointed star is the way many people draw it as in Figure 15B: not with ten short lines, but with five long lines going from point to point, skipping a point each time. 28

x

Furly Ise

Thinking of a five-pointed star this way allows a better understanding of the relationship between the ten short lines of the star shown in 15A. The two short lines marked x and y are really part of the same line. Figure 15C shows a star from ALEXANDER'S STAR. You can see that the faces that correspond to the lines of the star in 15B are flush faces. (In fact, itis a good exercise to run your eye around this star, following a face into its flush face, going around the tip of that piece, following that face into its flush face and so on until you come around to the beginning.) Thinking of the stars of ALEXANDER'S STAR this way will help you to see thata star on a solved ALEXANDER'S STAR is not "Just" multicolored, but is multicolored with all five pairs of flush faces the same color. We call a star whose flush faces are the same color a good star: 29

A good star numberss represent colors)

Figure 16

The first step in the solution to ALEXANDER'S STAR will not be to get a solid backgroud, but to get a good star. The numbers on the faces of the star in Figure 16 represent colors, and different numbers represent different colors; this star then is a good star. If we call two flush faces of a star a side of the star, then a star is good when it has solid-colored sides. It is important to remember that each side of a star is also part of a background. Sometimes the image of a star is so strong we forget this.

30

00 A Menu of Moves

A. Talking About Moves All the stars on ALEXANDER'S STAR rotate, so we will need to specify the rotations we want. A star can be rotated to any offive possible positions. The smallest rotation you can make, one-fifth of the way around in either direction, is called a tum. If you rotate a star five turns in one direction, you will bring it back to its original state. The direction can be either clockwise or counterclockwise. If we label a star with a letter X then we can use that letter to tell us how to move the star. X, as a move, represents one turn in the clockwise direction. X' represents two turns in the

clockwise direction. X- I represents one turn in the counterclockwise direction. X-' represents two turns in the counterclockwise direction. (See Figure 17.) Very often you will use moves involving only two stars. In these two-star moves, you'll turn one star a certain way, the other star another way.Then, you will undo what you did to the first star and undo 31

Figus 17

what you did to the second star: this will leave only three pieces affected.The two stars involved in these kinds of moves always share a common piece, so we will refer to these moves in terms of that common piece, which will be held horizontally, facing you directly. The star on its left will be called the left star, or L, and the star on its right will be called the right star or R. Instead of talking about clockwise or counterclockwise in these cases, we will use the terms up and down. You can see in Figure 18 two arrows representing turns of the left and right stars in the down direction. If the right star is rotated one turn

19ure is 32

down, it is the same as rotating it one turn counterclockwise. If the left star is rotated one turn down, it is the same as rotating it one turn clockwise. A move can then be represented like this: L

R

down down up up This means: first turn the right star one turn down (counterclockwise); then the left star one turn down (clockwise); then turn the right star one turn up (clockwise); and finally turn the left star one turn up (counterclockwise). You can remember this move as "down, down, up, up, starting on the right:' (It could also be written as R- 'L R L-1 .) It is a very useful move. It is called the Trn-spin, since it moves a triangle of pieces, each into the next. The effect of this move is diagrammed in Figure 19. The plain arrows at L and R indicate that the piece seems to remain connected at that point and swing in the direction of the arrow.The looped arrow

Pru.I

33

indicates that the piece seems to turn over or "flip" as it swings on point A. Only the shaded pieces are affected by this move. S. The Menu

Move 1: The Trl-spin L

down up

R

down up

The left Tri-spin L

R

down down up up

(Not The left Tri-spin cancels out the Tri-spin.)

34

The lower Tri-spin L

up down

R

up down

I

The left lower Tr-spin L up

R

down

up down

wOre The left lower Trilpin cancels out the lower Tri-spin.)

All the following moves could be described in their four different forms as we have done for the MIspin. That is, reversing right and left and/or reversing up and down. We have done it for the Tri-spin because it is an important move used in many ways. The lower Tri-spin and the left lower Tri-spin are especially important for the solution because they only affect pieces below the LR piece; since the solu35

don works downward, they do not destroy work already accomplished. Below, then, are all the possible four-twist type of moves. By changing right and left, or up and down, each move can represent three more moves. You can get the mirror image of a move by reversing the left and right moves. And you can invert a move, top to bottom, by reversing the "up" and "down" instructions. If you practice these moves, you will begin to feel the rhythm, the feeling of doing something, doing something else, undoing the first, and undoing the second. If you want to undo a move, you can repeat it twice more, or you can undo it from the bottom up substituting "up" for "down" and "down" for "up" in all cases. Move 2S The Ilttl Z L up

R

down

down

The face marked 'A" ends up on the far, or hidden, face of the position Indicated.) (Note

36

Move 3: TheWm* U L

R

down down

down

down up up

up

up

(mmTThe face marked "A"ends up where the face marked 'B" now appears.)

Move 4: The wid Z L up up

N

down down up

down down

(woe The face marked "A"will end up on the far, or hidden, face of the position Indicated.)

37

Move 5: The skew U R

L

down down

up

down

up up

("ale The face marked "A' ends up

where the face marked "B" now appears.)

Move 6: The rivers skew U L

down

up

R

down down up

up

(wOtE The face marked "A" ends up where the face marked "B" now appears.)

38

Move 7: The Hook L

up up down down

R

down

(NOmE The face marked "A"ends up on the far (hidden) face of the position indicated.)

Move 8: The rever

Hook L

up

R

down down up

down

(momT The face marked WA' ends up on the under (hidden) face of the position indicated.)

39

These moves can be combined to generate more useful moves. The following two moves are important for the solution. Move 9: The Fan L

up down down up

R

down up down up

(NOTE This move is really the little Z

followed by left Trl-spin.)

Move 10: The"epin

L up up

down down down down up up

R

down down up up

down down up

Un -r (NOm This move Is really the wide Z followed by a left wide U. The four "downs" in the L column could be replaced by an "up:' but it would destroy the rhythm.)

40

In the next move, more than two stars are turned, but it is still a combination of two of the above moves. Move 11: Tho Flip IFIrst do the In-spin: L

R

down down up up

Now turn the whole Star around point P, so that AL moves Into the position of LR. Then do the left Tri-spin:

A

down up

L

down up

Now turn the whole Star back again around point P so that LR Is horizontal. No pieces have changed position In this move, but the two pieces, AL and ARl have changed orientation, or flipped.

41

When you want to flip two adjacent pieces, just hold the Star so that the two pieces form an inverted V.The piece that completes the triangle with these two pieces is the horizontal piece connecting the L and R stars that are to be turned in the first Th-spin of the Flip. When you have the pieces to be flipped in the correct position, you can think of the move as "start right-hand, down, down, up, up, turn counterclockwise, start left-hand, down, down, up, up, turn back, clockwise:'

42

A Plan of Attack

The solution will work downward as follows: In steps I and 2, the north star will become a good star. STEP ONE: Begin a good star by finding or getting a pair of same-colored flush faces and from these determining the north pole. STEP TWO: The completion of the north star. STEP THREE: Fill In the northern ring and, thereby, get the first solid background. STEP FOUR: Fill in the equator by following leads. STEP FIVE: Fill in the southern ring. At this point we will have seven solid backgrounds. STEP SIX: Invert the Star and start work on the south star. STEP SEVEN: Determine if we are in the "bind:' and If we are, perform the "switch" which will undo

the bind. STEP EIGHT: Complete the south star, and thereby complete the solution. -

43

The Step-By-Step Solution

Now that you have read the Solution Preliminaries part of this book, and understand the vocabulary and references used in this solution, you are ready to begin. We will repeatedly go through the process of a) determining which position is to be filled in correctly, b) figuring out what piece is to fill that position, c) finding that piece, wherever it may be on the Star, and d) getting it, through a series of intermediate positions, to that position. Since you would not want to get a piece to a position if it were correctly filled, you can assume that the position to be filled has an incorrect piece in it. We will call it the "position piece"'The piece that is to correctly fill that position will be referred to as the "desired piece:' So, basically, we will be exchanging the desired piece for the position piece, while trying not to disturb the other, correctly placed pieces.

Step 1: First Two Pieces and Orientation Your Star is in your hand. Look at it. Turn it around a bit. The first thing to look for is a good star. You probably won't find one unless somebody took your ALEXANDER'S STAR in its solved state and only made a few turns. If you do find such a good star, pro46

ceed to step 3, and even then, look carefully, because some or all of the northern ring might be intact. If you can't find a good star your first task will be to create one. This will take the first two steps. Look around your ALEXANDER'S STAR for two same-colored flush faces. Usually you can find a few of these sets of faces. (In fact, it's rather difficult to manipulate your ALEXANDER'S STAR to the point where no samecolored flush faces exist. Try it.) If you can't find such a pair of same-colored flush faces easily, just turn a few stars at random. A pair will eventually turn up. Let's say you have found such a pair of samecolored flush faces. Now, look at the other two faces on the pieces with these same-colored flush faces. They must be of different colors. If they are not, find another pair of same-colored flush faces and check the other two faces on these pieces. Keep this up until you find a pair of pieces with same-colored flush faces and differently colored other faces. (See Figure 20.) This'is the main object of Step 1. But there is more to be done in this step. Not only do you now have two pieces in place, you also i point

mee the th pole

These tw face coloi

bes" two Ke colored lffeiently

the same

lubt to 47

have an orientation to the whole Star. The point where the two same-colored flush faces come together is the north pole, as in Figure 20. The star around it containing the two pieces with the samecolored flush faces is the north star. (It is important to keep track of the north pole throughout this solution, especially during the next two steps, when you can easily lose your place. After the northern ring is complete, you will be able to refer to the solidcolored northern plane and not lose your place so easily. But for now, it is best to keep a finger on the north pole so as not to get lost.) In addition to keeping track of the orientation of the Star it is also necessary to keep track of the colors used in this and the next step. We need a way to refer to these colors without specifically naming them, since each time the star is solved, a new set of colors will come up in a new order. For this reason, the colors just placed and to be placed in the next step will be referred to by numbers. The color on

the two same-colored flush faces will be called color 2. Turn your ALEXANDER'S STAR, with the north pole up, so that the two same-colored flush faces, with the color now referred to as color 2, are

igure 21

48

facing to the left. a in Fixgur 21. Cal the coor on the other face of the piece closer to you color 1 and the color on the other face of the piece farther away from you color S. If you followed the instructions correctly, color 1 and color 3 should be different. Now go-on to step 2.

Step 2: Completing the North Star The north star now has two pieces in it, correctly placed. Three more must now be placed. Hold your Star so that the north pole is up and the flush faces with color 2 of the last step are facing to the left, as in Figure 22. Referring to the position occupied by the piece with colors 1 and 2 as A, and the position occupied by the piece with colors 2 and 3 as B, continue around the north star along flush faces referring to the last three positions as C, D and E. as in Figure 22. We will also continue the numbering of the colors as indicated in Figure 22.

A

49

Colors 1, 2, and 3 have already been determined in the last step. You will determine colors 4 and 5 as you proceed, color 4 when you choose the piece to go into position C and color 5 when you choose the piece to go into position D. There will be no more color choices for the rest of this solution. Basically, you are matching the "old" new color on one face and adding the "new" new color on the other face, until the piece for position E is in place. Three pieces will now be placed correctly in positions C, D, and E. First we will find a piece for position C that will have color 3 on one face and any color other than 1 or 2 on the other face. This color will be called color 4. The color 3 face is to end up flush with the color 3 face of the piece in position B. (See Figure 22.) Once you have found such a piece, call it the destredpiece and go to the next step, Procedure 2 (below) to place It correctly in position C. Next we will find a piece for position D that will have color 4 on one face and any color other than 1, 2, or 3 on the other face.This color will be called color 5. The color 4 face is to end up flush with the color 4 face of the piece in position C. (See Figure 22.) Once you have found such a piece, call it the desired piece and go to Procedure 2 to place it correctly in position D. Finally we will find a piece for position E that will have colors 5 and 1 on its faces, the color face to end up flush with the color 5 face of the piece in position D, and the color 1 face to end up flush with the color 1 face of the piece in position A. (See Figure 22.) When you have found this piece, call it the desired piece and go on to Procedure 2 to place it correctly in position E. 50

Procedure 2 The following procedure is a method of getting any desired piece into its specified location in the north star. It has two parts: 2A, getting the desired piece into the northern ring; and 2B, getting the desired piece from the northern ring into the north star.

Part 2A: To get the desired piece into the northern ring: 1) If the desired piece is in the north star and is correctlyplaced, go on to the next piece or to Step 3 if it is the last piece to be placed in the north star. 2) If the desired piece is in the north star but not correctly placed, just turn the star of the northern set to which it belongs one turn either way, so that the desired piece moves into the northern ring. Go on to 2B. 3) If the desired piece is in the northern ring already, go on to 2B. 4) If the desired piece is in the equator, turn the north star so that the position piece points to its northern tip, and then turn the star of the northern set to which it belongs either way to bring the desired piece into the northern ring. Go on to 2B. 5) If the desired piece is in the southern ring or on the south star. turn one of the stars of the southern set to which it belongs so that it ends up in the equator. Then go back to move 4 above. 51

Part 2B: Once the desired piece Is In the northern ring, It must be correctly placed In the north star. 1) If the face of the desired piece in the northern plane is to end up on the rightfaceof a piece in the north star, turn the north star so that the position piece is touching it on its right side, and turn the star of the northern ring to which the position piece now belongs one turn clockwise. (See Figure 23.) 2) If the face of the desired piece in the northern plane is to end up on the leftface of a piece in the north star, turn the north star so that the position piece is touching it on its left side, and turn the star of the northern set to which the position piece now belongs one turn counterclockwise. (See Figure 23.) Once a piece has been placed in the north star, go on to the next piece in the north star until it is completed. Go on to Step 3.

FPaw

52

23

Step 3:

Filling In the Northern Ring The north star is now a good star with all its flush faces the same color, and each pair of flush faces a different color, five in all. Since there are six colors on the star. one color has not been used. We will call It

color 6. Color 6 is to be the color of the northern plane, the background of the north star. Thus, each piece in the northern ring must have a color 6 face on its northern plane side. But what about Its southern side? To determine the other color of a desired piece for a position in the northern ring, hold your Star in front of you with a position piece in the northern ring facing you, as in Figure 24A. Hold it so that you can see both its northern and southern faces. Then tip the north pole toward you and you will see two formerly hidden, same-colored flush faces come into view, as in Figure 24B. Their color is the color that the southemface of the piece of the northern ring facing you is to be. Having determined the desired piece for a position in the northern ring, go on to Procedure 3.

Procedure 3: To got a desired piece Into the northern ring by way of the south star: Part 3A To got the desired piece Into the south Stan 1) If the desired piece is in the northern ring and correctlyplaced, leave It alone and go on to the next position of the northern ring, or, if it

53

a MP 4

is the last piece to be placed in the northern ring, to Step 4. 2) Ifthe desired piece is in the northern ring and not correctlyplaced,then call the ends of the desired piece R (for right) and L (for left) and do the lower iT-Spin move: L up down 54

R up down (See Figure 24C.)

MRe 24C

The desired piece will now be in the equator. Go on to 3, below. 3) If the desired piece is in the equator, turn the star of the southern ring to which it belongs either way until the desired piece is in the south star, then go on to Part 3B. 4) If the desired piece is in the southern ring. turn a star of the southern set to which it belongs one turn so that it ends up in the south star, then go on to 3B.

Part 3B: To got the desired pilee from thO south star Into the equator directly under the position plee with oolor 6 on Its northern fao.: With the desired piece in the south star, turn your ALEXANDER'S STAR so that the position piece in the northern ring is facing you. Then turn the south star so that the desired piece is pointing toward you as in FIgure 25. 55

ce

tigus 25

Then turn the star of the southern set to which it belongs two turns clockwise. The desired piece will now be in the equator in a left-leaning position. 1) If the northern face of the desired piece is color 6, go on to 3C. 2) If the northern face of the desired piece is not color 6, turn the star of the southern set to which it belongs one turn clockwise. The desired piece should now be in the equator in a right-leaning position with its color 6 side up. Go on to 3C.

Part 3C: The desired piece should now be In the equator, right-leaning or left-leaning, with Its color 6 face up and adjacent to the position piece above It. 1) If the desired piece is right-leaning,call the tips of the position piece R (for right) and L 56

(for left) and perform the lowerTri-Spin: L

up

R up

down

down 2) If the desired piece is left-leaning, call the ends of the position piece R (for right) and L (for left) and perform the left lowerTfi-Spin: L up

R

down

u down

The desired piece should now be correctly placed with its color 6 face in the northern plane and its southern color matching the opposite faces in the north star. Go on to the next position in the northern ring, or, if the northern ring is completed, to Step 4.

Step 4:

Completing the Equator The pieces to be placed in the equator are now in the equator, the southern ring, or the south star. The procedure will be to determine which piece is to go into a given position in the equator, find that desired piece, get it down into the south star, and then place it in Its position in the equator. The order of placement will start with a left-leaning position, and continue to the right (east!), alternating left-leantng and then right-leaning until the equator is complete. 57

Procedure 4 Part 4A: To determine the desired pieces for positions In the equator by using leads: You can see from Figure 26 that the colors of the faces of a piece for an equator position are determined by colors of faces of pieces in the north star and of southern faces of pieces in the northern ring, pieces that have already been fixed. Eachpiecein the equatoris thefront of a lead. 1) Left-leaning pieces are determined by the right face of a piece above it in the north star and the southern face of a piece to its left in the northern ring. 2) Right-leaning pieces are determined by the left face of a piece above it in the north star and the southern face of a piece to its right in northern ring. NP

Figure 20

58

Part 43: Here we must be cautious, because pieces with the same color combinations appear twice in the equator. We don't want to pick as a desired piece a piece that has already been placed; we want the other piece of that color combination. 1) If the desired (and unplaced) piece is in the equator. we turn the star of the southern set to which it belongs either two turns clockwise, if it is right-leaning, or two turns counterclockwise, if it is left-leaning. so that it is now in the south star. We then turn the south star one turn either way and reverse the turn wejust gave the star ofthe southern set. (This reverse turn is not always necessary but it can prevent destroying already completed parts of the equator.) Go on to 4C. 2) If the desired piece is in the southern ring, we can turn either star of the southern set to which it belongs to bring it into the south star. Then turn the south star one turn either way. Then reverse the turn we Just gave the star of the southern set. Go on to 4C. 3) If the desired piece Is in the southern star. go on to 4C. PkAr 4Cs

With the desired piece in the south star, turn your ALEXANDER'S STAR around its vertical axis so that the star of the southern set to which the position piece belongs is pointing at you. (See Figure 27.) Call it X. and call the south star 8.You should be able to see ll five triangular sides of X's background pen59

Figure 27

tagon. Look at the piece whose face is the lower right-hand triangle of X's background pentagon (a. in Figure 27). Thrn the south star until the desired piece is in that position, and call the star of the southern set to which it belongs Y If the position you are filling is left-leaning, go to Cl: if right-leaning, gotoC2. C 1) To fill a left- leaning position. check the color that is to be the northern face of the piece when placed. If it matches the face you can see of the desired piece. go on to la; otherwise, go to lb. la) Perform the move, X-2SI'X2S (a Reverse Hook).

lb) Perform the move, X2YX-2 Y (a Reverse Hook). C2) To fill a right-leaning position, check the color that is to be the northern face of the piece when placed. If it matches the face you 60

can see of the desired piece. go on to 2a: otherwise, go to 2b. 2a) Perform the move, XYX-'Y 1 (a Little Z). 2b) Perform the move, X2S1X2S (a Skew U). Having placed the desired piece in its location in the equator, go on to the next position to the right. It will lean in the opposite direction from the piece you have just placed. If you have completed thie equator, you may wish to check your work so far. To do this, hold your ALEXANDER'S STAR at the north and south poles, and slowly rotate it around its vertical axis. The background pentagons of the northern set of stars should-have all but their southern ring faces solid colors. (See Figure 28.) Then, tipping the north pole back slightly, rotate again. The background pentagons of the southern set of stars should have all but Backgrounds of -ndh

w-

Background of

nd.e

nars Id for

should b except pei southern i

as

SP Figure 28

61

the two lowest (south star) faces the same color. If it checks out, go on to Step 5.

Step 5:

Completing the Southern Ring The pieces to be placed in the southern ring are either in the southern ring or in the south star. The order of filling the southern ring is not very important. Finding the pieces that go in the lower ring is easy: they all have a color 6 face. It is best to start with the pieces that are in the south star. Determination of which piece goes in a given position is obvious: it will complete a background pentagon of a star of the northern set and have that color on its northern face, while the southern face will be color 6.

Part A: Find a piece in the south star that has a color 6 face. If no such piece exists, go on to 5B. If you do find such a piece, call it the desired piece and find its position. Hold your ALEXANDER'S STAR with the position piece facing you and with its edge horizontal. lirn the south star until the desired piece is touching the right-hand tip of the position piece. If the face of the desired piece facing you is color 6, go on to 1; otherwise, go on to 2. 1) Ibrn the south star one turn counterclockwtse so that the desired piece is touching the left-hand tip of the position piece.You should now see the other color of the desired piece, which should match the color of the almost completed background pentagon above it. 62

Call the tips of the position piece R (for right) and L (for left) and perform the Left Lower TH-Spin: L up

R

downu

down

Position

(See Figure 28A.)

I piece. If face or 6 go to 5A1

FIgure 28A

go to 5A2.

Then go back to 5A. 2) You should now see the desired piece touching the right-hand tip of the position piece with its other color facing you, which should match the color of the almost completed background pentagon above It. Call the tips of the position piece R (for right) and L (for left) and perform the lowerfli-Spin: L

R

up

down down The go back to 5A. 63

Part B: If you find no color 6-faced pieces in the south star, and not all the pieces of the southern ring have been placed, you must get a piece with a color 6 face that is in an incorrect position out of the southern ring and into the south star. Call this incorrectly placed piece the desired piece. Hold your ALEXANDER'S STAR so that the desired piece is facing you. Label its ends R (for right) and L (for left) and perform the lowerlYi-Spin: L

R up

up down down The desired piece is now in the south star and may be placed by 5A. Continue in this way until all pieces of the southern ring are correctly placed. Then go on to Step 6.

Step 6: Starting the South Star and Bind-Finding If what you have done to this point is right, you should now find that all but the five pieces of the south star have been correctly placed. The five pieces in the south star should be the same five color-pairs as in the north star, although out of order. The task in this step is to position three or more of these five pieces and determine the need for Step 7. If Step 7 turns out to be unnecessary, you will proceed to Step 8, the final step. 64

At this point the entire ALEXANDER'S STAR should be inverted-that is, you should hold the south star pointing upward, even though it will still be called the south. (All terminology will remain the same, even if it seems contrary.) In order to solve the south star, we must discuss the idea of matched pieces. A piece in the south star is called "matched" if it is in the right position even if it is in the wrong orientation. You can determine If it is in the right position using leads. The southern faces (now facing up) of two adjacent pieces In the equator form a lead with a piece in the south star (Piece A in Figure 29). If the faces of the piece in the south star are the same colors as the faces of the equator pieces to which they are flush (Piece B in Figure 29). it is called a good match. If the faces of a piece in the south star are not the same as faces of the equator pieces to which they are flush but if they are such that if the piece were flipped, they would be (Piece C in Figure 29), it is called aflipped match. (A flipped match means that the piece is in the right place but in the wrong orientation.) Because of the

A flipped m

65

flip move, we can take care of incorrectly oriented pieces later. Right now, we will only be interested in matches, good or flipped. Following leads will tell you if a piece is matched. With the idea of match in mind, we will now consider the south star, which has five pieces each of which has a position where It could form a match, but which don't all match at the same time. (Even turning the south star won't match an the pieces at once.) We will turn the south star through all five turns and examine the matchmaking situation. This will be the first part of Step 6.

Part GA: Maximizing Matches and Finding the Bnd Turn the south star and count matches at each position. You are looking for the position with the maximum number of matches. It can't be zero, since you can always pick a piece and find the position it matches. It might only be one; it is more likely to be two or three. It can't be four (Iffourare matched, the last one has only one place to go so it would be matched, too, and we would really have five matches). 1) Ifthe maximum number of matches you can find is five, you're in luck. Go on to Step 8, for any final orientation correction. 2) If the maximum number of matches you can find is 3, check to see that if the two unmatched pieces were switched, they would be matched. If this is true, you are in the bind and will have to go to Step 7. 3) Ifthe maximum number of matches you can 66

find is two, youarenot in thebind. GotoStep 6B. 4) If the maximum number ofmatches you can find Is only one, hold the Star and turn it so that the one matched piece is pointing to the right. Call the tip of the matched piece that is not the south pole, X. Call the south pole S. Holding the Star so that you are looking directly at the matched piece with S to the left and X to the right. perform the Fan move: s

X

down

up down down down up

Up

Now, turning the south star around and examinlng matches again, find the maximum. It should be two or three. Go back to 6A.

Part O: Positonng the South Star Pleo

All the pieces in your star are now correct with the exception of three that are not matched in the south star and the two matched pieces in the south star that may or may not be flipped. There are two possibilities: the two matched pieces may be next to each other, in which case go to 1 below, or the two matched pieces may be separated by an umnatched piece in which case, go to 2 below. 1) Since the two matched pieces are next to each other, the other three pieces, which are mismatched, are together. Point the piece in 67

the middle to the right, and call the mismatched pieces P 9 and R. as in Figure 30. Now, one of two situations is true: Either a) P would be matched if it were where R is, or b) R would be matched if it were where P is. Calling the south star 8, and the other tip of piece Q, X, hold your Star so that piece Q faces you directly with S to the left and X to the right, and perform one of the two Fan moves depending on which situation, a or b, above, is true.

If a is true, perform the following Fan move: s

x

down

up down

up up down

up down

If b is true, perform the following Fan move:

68

$

X

up down down up

down up down

up

All five pieces in the south star should now be matched; that is. in the correct portions but possibly flipped. Go on to Step 8. 2) Since the two matched pieces are separated by a mismatched piece, the other two mismatched pieces are together and the three mismatched pieces form a lead, a dartshaped set of three pieces. Point the dart to the right as in Figure 31, and call the mismatched pieces P. 9, and R as in the figure. Now, one of two situations is true: Either a) P would be matched if it were where R is, or b) R would be matched if it were where P is. Calling the south star 8. and the other tip of piece Q, X. hold your star so that piece Q faces you directly with S to the left and X to

Two matched pieces IX

Thm

MIS-MI pieces

Mow

69

*

the right and perform one of the two A-spin moves depending on which situation, a or b, above, is true. If a is true, perform the following A-spin move: S

x up up

down down

down down

up up up up

up up

down down

down down

If b Is true, perform the following A-spin move: S

up up down down down down up

x down down up up

down down

up

up up-

All five pieces in the south star should now be matched, that is, in the correct positions but possibly flipped. Go on to Step 8. 70

Step 7: The Bind If three pieces are matched in the south star, and the other two pieces are switched, you are in the bind. For it is Impossible to switch just two pieces on an ALEXANDER'S STAR by moves. It is possible, however, to exchange two pairs of pieces. The answer to this problem lies in the fact that there are twins (pieces that are colored alike) all over the star. Every piece has its twin. So the trick is: exchange a pair of twins and exchange the two switched pieces.

7A: Adjusting the Switohed Pley Every piece on the Star is in its place (although a few in the south star may be flipped) except that two are switched. These two pieces are both in the south star. Any two pieces in a star are either next to each other or separated by another piece. If the two switched pieces are separatedby apiece, go to 7B. If the two switched pieces are next to each other; hold the two pieces so that one is pointing to the right and the other is pointing away from you like the pieces marked P and Qin Figure 30. Calling the south star, S. and the other tip of piece Q, X, perform the Fan move: S

up down down up

X

down up down up

71

Two pieces (not the same two as before the move) are still switched but are separated by a matched piece. Go on to 7B.

7B. The switch Every piece is in its correct position except for two pieces in the south star that are switched. Some of the pieces correctly positioned in the south star may be flipped. The two switched pieces are separated by a matched piece in the south star. The following move is designed to switch two pairs. If one pair are twins, and the other pair are switched pieces, this move will seem to switch only the two switched pieces because there is no way to tell that the twins have been moved. Hold your ALEXANDER'S STAR in the position in Figure 32. You will be looking directly at one of the switched pieces. In the figure the switched pieces are shown as shaded. The pair of twins M and N, to be switched are shown as striped. Calling the stars indicated in the figure S (the south star), A, B, and X, perform the following move: ABX-2S2B-1SX2A-lS2.

Flgu 72

S2

ff you have performed the move correctly, every piece in the south star should be matched. Go on to step 8.

Step 8: Flipping Out All the pieces on your star are now in their correct positions even though some of the pieces in the south star may be flipped.

OA: Unflipping Only an even number of flipped pieces are possible, and only the pieces in the south star could be flipped, so we are limited to either zero, two, or four flipped pieces. If two pieces are flipped, they may be next to each other or separated by a correctly oriented piece. Iffour pieces are flipped, go to 1, below. If two pieces are flipped and separated by an unfltpped piece, go to 2, below. If only two pieces are flipped and next to each other, go to 3, below. If no pieces are flipped, your Star is solved. 1) If four pieces are flipped, then the fifth piece in the south star is not flipped. Call It piece U. Point It to the right, as in Figure 33. Now, using the Flip move as described at the end of the Menu ofMoves chapter, unfllp the two pieces referred to as A and B in Figure 33. Now only two adjacent pieces remain flipped; go on to 3, below. 2) The two flipped pieces are separated by an unflipped piece. Call this separation piece U and point it to the right, as in Figure 33. A 73

Figure 33

and C will be the flipped pieces. Now, using the Flip move described at the end of the Menu of Moves chapterfltp pieces A and U. It is true that U is unflipped but by flipping it. we will end up with two adjacent flipped pieces. Go on to 3. below. 3) Only two pieces are flipped and they are adjacent; to unflip them, just apply the Flip move as described at the end of the Menu of Moves chapter. Once you've done this, your Star will be solved. Congratulations!

74

Variations on the Star

The solution in the previous chapter results in the standard form of ALEXANDER'S STAR in which all the backgrounds are solid colors and opposite backgrounds are colored alike. It is possible. however, to achieve variations on this standard form that, in effect, make your Star into more than one puzzle. Every solution to the Star Is in one sense a variation since there are twelve different solutions with all backgrounds solid and opposite backgrounds alike. If you compare two solved Stars, you can see this. You can always get two sets of samecolored backgrounds to correspond, and the other four colors can be arranged in twenty-four different ways. But if you turn the Star upside down you will find a second arrangement of those four colors, so we get only twelve solutions. None of these solutions is particularly distinguished from the others. If we regard any combination, messy or otherwise, as a variation, we have 3.621 x 1O04 possible variations on the Star, a thirty-five digit number. It is approximately the number of combinations of RUBIK'S CUBET™squared! Another type of variation is the result of applying new labels to the Star, different than the ones it comes with. You might want to try your hand at relabeling your Star yourself by using colored paper and rubber cement or colored tapes available in art supply stores. Or you might just enjoy considering the possibilities and the light they shed on your understanding of ALEXANDER'S STAR. 76

1. The Nonwopposing Solution In the standard form of ALEXANDER'S STAR., 1) every background is solidly colored, and 2) opposite backgrounds are colored alike. But it is possible to drop the second condition to get what we will call a non-opposing solution. You might think that by dropping this second condition almost any distribution of colors to backgrounds could be achieved. As it happens, dropping condition 2 allows only one other pattern. This Is because the backgrounds must meet in such a way that there are two and only two pieces with a given color combination. Even when this condition is met, some patterns force a single flipped piece that can't be corrected. The single non-opposing solution requires that two colors, called the opposing colors, each be on opposite backgrounds as in the standard form, and that the other four colors, called the non-opposing colors, be paired off so that each background of one color has its paired color on the opposite background. Thus, If red and green are the opposing colors, a red background will be opposite a red background, and a green background will be opposite a green background. If the other four nonopposing colors are paired off blue to orange and yellow to white, then opposite each blue background will be an orange background and vice versa, and opposite each yellow background will be a white background and vice versa.. You can choose any colors to be the opposing colors and the non-opposing pairs of colors, but they will always form the same type of pattern. - 77

'lb get a non-opposing solution, follow the solution in the previous chapter but with these changes: We will call 1 and 2 the opposing colors, 3 and 4 a pair of non-opposing colors, and 5 and 6 the other pair of non-opposing colors. 1) The north star will no longer have five colors; it will have four colors with color 3 occurring twice, as in Figure 34. Notice that color 5 is a non-opposing color from the other pair and that it appears only on the pieces with color 3. Coloring pattern for north star and northern plane In onopposing solution

Pru 84

2) The northern plane has as its color the same color 5 as in the north star. The southern faces of the northern ring pieces no longer match the opposite sides of the north star in all cases. Accordingto the rule, color 1 will be opposite color 1, color 2 will be opposite color 2, but color 4 will be on the southern faces of the pieces opposite the color 3 sides of the north star, and color 6 will be on the southern face of the piece opposite the color 5 side of the north star. 3) The equator is to be filled injust as before by following leads. 78

4) The northern faces of the southern ring will complete the backgrounds just as before, but the southern plane will be color 6. since it is opposite the color 5 northern plane. 5) The south star is determined by leadsjust as before, but because there are two pieces with the same color combination (colors 4 and 6), there Is no problem with the bind since it can be eliminated by switching these two pieces.

2.

The Triangular Coloring ALEXANDER'S STAR seems to have stars all over its surface; but it can also be seen as having triangular depressions all over its surface, twenty altogether. Some people ask if it Is possible to move the pieces so that all these triangular depressions (or Just "triangles") are solidly colored, as in Figure 35. The answer is: no....but almost.

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The reason it is impossible Is that there are ten faces of each color (five in each of the two backgrounds on a solved Star), and ten Is not evenly divisible by three, so these ten faces couldn't color a whole number of triangles. If we tried, we would be able to color three triangles and then have one face left over for each color. Since there are six colors, there would be six left-over faces. And that is the "almost" We can solidly color eighteen triangles, three each of the six colors, if we allow the two remaining triangles to share the six left-over faces. This coloring is called the triangular coloring. The two multicolored triangles can be in any relationship on the Star, although it is easier if they are adjacent, and prettiest if they are on opposite sides of the Star. This variation is not an easy one. There is no simple rule to figure which triangle should be what color. And half the time you will have almost solved the puzzle, only to find that one piece is flipped. It is for this problem that positioning the two multicolored triangles adjacent to each other is easier: the piece with a face in each triangle can be flipped either way.

3. The Totally Flipped Star Let's say you have solved your Star. What would happen if you applied the Flip move all over, so that every piece, still in its correct position, were flipped? We will call this pattern the Tbtally Flipped Star. Figure 36 gives an indication of the pattern. The solid colors, instead of appearing on back80

Same colored faces wheel

tally star

PFgwr 30

grounds, now form wheels. The southern faces of the northern ring and the northern faces of the southern ring form a wheel. The faces of this wheel frame the equator. On the 'Ibtally Flipped Star, every wheel is solidly colored. If you give the Star a little spin, holding it by the north and south poles, or any pair of opposite points, you will see the wheel. It is not simple to flip every piece. It is best to have an order of approach or else you can easily get lost with some pieces flipped and some not. Since the Flip move is performed on an inverted V,we have to divide the pieces of the Star into fifteen V's. as in Figure 36. Five of the V's will consist ofa piece in the north star and a piece in the northern ring to its right. Another five V's will consist of a left-leaning and a right-leaning piece in the equator. And the last V's will consist ofa piece in the south star and a piece in the southern ring to its right. If you perform all flf\teen Flip moves on these V's. you will get the Totally Flipped Star. 81

4. The Star-Center Labeling People ask if it is possible to color ALEXANDER'S STAR in such a way that the stars are solidly colored rather than the backgrounds. As was stated earlier, this is not possible, because each star shares its pieces with five other stars, so all the stars would end up the same color. This analysis not only explains the impossibility of solidly colored stars, but it also points the way to a new coloring that gives part of each star a solid color. Since each piece belongs to two stars, if we divide the piece in half by a slice halfway between the tips, we can assign each half of the piece to the star whose center it touches. If we color all these halves the same color (the color of the half wraps around the edge of the piece), the central portion of each star will be solidly colored. The Star with this coloring is shown in Figure 37. As a puzzle, ALEXANDER'S STAR with the The star-center labeling

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star-center coloring requires almost exactly the same solution as with the solid background coloring. although the piece for each position is determined by matching halves rather than matching flush faces. It is easier to understand the positioning of pieces with this coloring, although it destroys the starry appearance of the whole Star.

5. The Triangular Labeling The triangular coloring has been mentioned in this chapter as a pattern that can be achieved on your standard ALEXANDER'S STAR. But it requires that two triangles be multicolored. A triangular labeling, on the other hand, relabels the faces of the pieces in such a way that every triangle can be a solid color. There are a number of ways to do this. An obvious way would be to solve the triangular coloring problem with the standard Star and then relabel the two multicolored triangles with a seventh color. Unfortunately. this coloring is not symmetrical, forcing more pieces with some color combinations than others. If symmetry and equality of colors is the goal, then five colors create the best pattern.There will be

four triangles of each color, and three pieces of each of the ten color combinations. When solved, all five different color triangles will appear around each star. The pattern for labeling the Star in this fivecolor triangular coloring is shown in Figure 38.This same pattern applies to all five colors. The major problem in solving the five-color 83

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triangular Star is that half the time you will end up with one piece flipped.There is no way to correct this other than exchanging two colors, wherever they occur, all around the Star.

6. The Completion of ALEXANDER'S STAR: the Inner Labeling Let's say your Star is solved. If you turn a star part way, you will reveal the five inner triangles that separate the five same-colored faces of that star's background. These inner triangles, which are the sides of five pyramids, have no labels: they are the color of the plastic from which the star is made. But they could be labeled. The obvious way would be to label them the same color as the rest of the background. We will call this labeling the "inner" 84

lbe Inne W*beling

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labeling, or the completion of the Star. (See Figure 39.) It is a pleasant effect to see the color of a background continue under the pieces of a star. And a certain difficulty is added to the puzzle, since we are trying to get two different sets of faces, the regular and the inner, into alignment. Furthermore, this labeling creates the possibility for many pat-terns. It is possible to solve the Star with the inner labeling so that all the background faces of a star are one color and all the inner faces of that star are another color. There are moves that will exchange background colors without exchanging the inner face colors. There are moves that allow the background faces to remain the same while rotating the colors on one or two pyramids. This inner labeling is the most significant change that can be made to the Star. It permits you to solve the Star just like a standard Star, since the regular faces are not changed, and it permits you to solve the two sets of faces together, creating a more;intriguing puzzle. 85

Ai

Mathematics and Puzzles

Imagine a little boy, seated at the dinner table with his brothers and sisters. The meal is almost over, and it's time for dessert. Ice cream? And then the little boy speaks up: "I don't want any ice cream. I'd rather have spinach?" "Ooh, ugh:' go the others, "spinach!" 'lb love something that's good for you, but most people hate, that's called a spinach lust. Mathematics has been a spinach lust for me, and I have tried never to forget it. Rather than inflicting my compulsion on others, I have always tried to refrain from pushing it except on those who share my compulsion. Therefore, I don't want to use this opportunity to say something like "if you like puzzles, you'll love mathematics:' because you won't, necessarily. In fact, I have found that mathematics may be the most disliked subject in the school curriculum. I would rather have people relieved of some of the burden of learning difficult parts of mathematics so that they wouldn't have such a negative attitude toward something I love. At the same time, you may find that the logical, insightful part of puzzles that you like does extend to mathematics, in which case you may find that you have a spinach lust for mathematics yourself. Don't let me discourage you. From my own experience, I know that mathematics is not "fun' as so many educators say. If you have the mind for it, it can be compelling; certainly, 88

if you like it, parts can be very exciting. But mathematics can be learned only by practice, problem after problem, until what once seemed difficult seems easy and becomes merely another technique for solving a more difficult problem. Puzzles are puzzles and mathematics is mathematics. Puzzles are designed to be compelling to anyone with a brain: it's just between you and the puzzle. Mathematics is more like a language, a standardized system that is part of the language of science and technology. But here and there, out of mathematics come little gems ofa much broader appeal than mathematics itself. Because of my acute awareness of the oddity ofmy skill, I have tried not to promote mathematics as a field but to show that out of mathematics come certain ideas, designs, devices, puzzles, games, and who-knows-what, that non-mathematicians would like and can appreciate. In one sense, mathematics can be compared to music. Its results are enjoyed by many although it is practiced by far fewer. But the melody of mathematics is indirectly heard through science and technology; although, every now and then, it can be heard directly, through a puzzle or an idea or a design that shows the fascination of this abstract field of human endeavor.

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About the Inventor and Author ADAM ALEXANDER is a serious mathematician who developed ALEXANDER'S STARM PUZZLE by analyzing many geometric forms In an effort to extend Rubik's principles to a new and striking shape. He is the second toy inventor in history to have his creation named after himself. He lives in New York City.

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