CUBIC CIRCULAR

A quarterly newsletter for Rubik Cube addicts

Issue 1 Autumn 1981

Published by David Singmaster Ltd., London, England. ISSN NO 0261-8362

2

WELCOME, ADVERTISEMENT AND INTRODUCTION

Welcome to the first issue of the CUBIC CIRCULAR. Let me begin by telling you my plans for the Circular. Primarily it will be a newsletter to distribute information on the Cube. However, there have been many new puzzles as a result of the enormous success of Rubik's Cube™, and the circular will discuss these as well.

There has been so much interest in such puzzles and other mathematical puzzles, that I have established a company to distribute them. I will use the Circular to announce interesting new products as they appear, both from ourselves and from other sources. We are always delighted to hear about new items for possible distribution and we maybe able to organise production of them. Samples, drawings, patent specifications, etc. will be gratefully received.

There has been a persistent demand for a newsletter like the Circular ever since the fifth edition of my "Notes of Rubik's 'Magic cube'" appeared in September 1980. As a consequence of the explosion of Cube mania at the beginning of 1981, I have been receiving letters at several times the rate at which I can read and reply to them. Included are manuscripts, books, computer programmes and computer printouts which can run to hundreds of pages and require weeks to study! Consequently my study is piled with unanswered correspondence and I would like to apologise to everyone who has written and hasn't yet had a reply. I am hoping that the Circular will contain the answers to some of the more standard questions and perhaps reduce the number of letters sent in by providing the answer. The Circular may also be usable as a standard response letter.

Because much recent material In my study has not yet been properly read and filed, I find myself sometimes remembering facts that I cannot relocate and so I am sometimes unable to give proper credit to the source of an idea or fact.

I am delighted to receive copies of articles on the cube or related material - but please be sure to put the date, name and place of publication on it. I would prefer an original version rather than a photocopy, if at all possible. I would appreciate it if you can include such details as the page number(s), transliteration and/or translation of the title and/or abstract of the content for articles not in English, French, German or Italian. Information on radio and TV items will also be appreciated - but again, please be as detailed as possible.

At present, I have enough material to fill several Circulars - indeed, I am presently preparing a bibliography which would fill several issues by itself! This issue contains rather less than I had intended - all technical material has been deferred. Nonetheless, I would be happy to receive short articles, news items, announcements, etc. for inclusion.

The Circular can be considered as further addenda to the fifth edition of my Notes which is now being published by Penguin. A Spanish translation of my Notes by Gilberto Torbeck is being published by Altalena Editores, Cochabamba 2, Madrid 16, Spain at 400 pesetas. A German translation by Suzanne Bischoff will soon be published by Springer-Verlag, Berlin & Heidelberg. A Dutch version is under way. We shall stock copies when they are available. References to the Notes will be in the form N5, NS3, meaning page 5 and page S-3 of the Notes. (In the American printing, pages i, ii became v, vi; pages S-1 to S-4 became 61 to 64, pages B-1 and B-2 were extended to 65 to 68 and I-1 to I-5 became 69-73). I will refer to page 2 of Circular 1 as C1-2, etc. I will use the notation of my Notes since it seems to be the most widely accepted of all notation. For typographic convenience, I will abbreviate 3×3×3 as 3³, etc.

I had hoped this circular would appear in early October. Apologies for the delay. I will try to get the next circular done over Christmas.

3

ERNO RUBIK AND HIS CUBE

[Corrections to this article can be found at the end. - J ]

I have recently been informed by Cedric Smith that the symbol above the ő in Rubik's first name is not an umlaut (") but a mark indicating a long o. This symbol is not available on English typewriters and the attempt to produce it gives ö which is a little odd looking, but acceptable.

Ernő Rubik was born in Budapest in 1944 to an eminent Hungarian family. His father, Ernő Rubik Senior, was an aeronautical engineer who designed many gliders and was awarded the Kossuth Prize (Hungary's highest honour). Rubik first studied sculpture, matriculating from the Fine and Applied Arts Gymnasium in 1962 with a distinction in sculpture. He then shifted to architecture, obtaining a diploma from the Budapest Technical University in 1967. He then did a post-graduate diploma in interior architecture or architectural planning (the sources vary - perhaps it was a joint course or the English translations refer to the same subject?) at the Academy of Applied Arts. Upon completing this course in 1971, he was appointed a lecturer at the Academy and began teaching three-dimensional design among other subjects. He is now head of the studio for the study of form.

Rubik found that his students, even the post-graduates, had great difficulty with three-dimensional visualisation. Consequently he was thinking about various exercises for his students. Cutting a cube with various planes is a fairly standard exercise and leads to the well-known 2³ and 3³ arrays. Rubik has a life-long interest in mechanisms, undoubtedly assimilated from his father. While contemplating the 2³ and 3³ arrays about May of 1974, this interest led him to ask, "How can I make the faces move?" Within six weeks, he had a solution, first for the 2³ and then for the 3³ (which is much easier than the 2³). It took him a month to solve the cube when he made his first model. He applied for a patent on 30 January 1975.

At the end of 1976, he approached Trial, the major toy distributor in Hungary, and they sent him to Politechnika Ipari Szövetkezet (i.e. Politechnika Co-operative), which made plastic chess sets and similar games. Gyula Nemcsók, the chairman and co-founder of Politechnika, recalls "When I took the Cube in my hand, excitement immediately seized me. I took it to our chief engineer, who likewise recognised the fantastic possibilities. We decided on the spot to obtain the necessary capital for production". (My translation from A. Havas, Der Zauberwürfel, Wochenpost 28:23 [5 Jun 81].) By late 1977, they could supply Trial. Trial made an initial order for 5,000 Magic Cubes (as they were initially named) and anticipated that advertising would be required to sell them. Instead they found that the problem was getting enough to meet demand. The Cube won a prize at the Budapest International Fair in 1978, which led to a number of foreign orders. In 1978, Hungarians were using the Cubes as hard currency which they could take or send abroad! Hungarian customs officials now ask departing people "How many Cubes have you got?" In 1979, the Cube received a prize from the Hungarian Ministry of Education and Culture. It received the Toy of the Year awards for 1980 in the UK and Germany.

At the end of 1979, Ideal Toy, a major US and international toy firm best known for introducing the teddy bear some years ago, took over the distribution for most of the western world. They planned to call it 'The Gordian Knot' but someone came up with the name 'Rubik's Cube'. I know no other toy named after its inventor - a unique tribute to a unique invention! Ideal wisely spent time and money in retooling the production lines. The resulting product is lighter, smoother in finish and much easier to turn. The original Hungarian production, like much of the unlicensed Taiwanese production, was so stiff that cubists developed 'Cubist's Thumb' or 'Rubik's Wrist'.)

The new version began to appear in mid-1980 and the resulting runaway success is now well chronicled. Ideal sold about 4½ million Cubes in 1980 and anticipated sales of about 10 million in 1981. Although figures are not yet known for 1981, I estimate they have sold perhaps 20 million so far. The Taiwanese are reported to have sold about three times as many! There are reports of a Munich shop Selling 800 Cubes in an hour and of a Wolverhampton (UK) shop selling 2000 in an hour. Market traders sold 1000 easily on Saturday. During the great Cube famine of early 1981 when Ideal's stocks were completely depleted and the Taiwanese knock-offs had not arrived, I heard of a second hand Cube market and offers of £15 for a Cube.

4

Rubik's name has become a household word in many languages and has been entered into the Oxford English Dictionary and the Brockhaus Lexicon. UK Customs even has a new category: Rubik's Cube and the like.

But what of Rubik himself? Early rumours said he was getting little royalties and he still does not have a telephone. However, Ideal pays him a royalty on the use of his name which has started arriving and recent reports describe him as the richest man in Hungary or the first communist millionaire (though the currency is not specified). He receives between 1 and 2p per cube from Ideal and is said to be receiving about $30,000 per month after taxes! He says it is just as well that he doesn't have a telephone.

The Cube is said to be Hungary's biggest single earner of foreign currency. With all the production sent to the West this Eastern Block is almost unobtainable in the Eastern Block Indeed an East German magazine has an article telling how to make your own Rubik Cube! Recent reports indicate it is available and widespread in Poland

Rubik, his wife Rozsa and his 3 year old daughter Anna live in a hilly suburb of Buda in a house he built with his own hands literally on top of his parent's house. His appearance is more Mediterranean than Central European. He is of medium height, thin, with small neat features. He has straight brown hair and smokes heavily. He reminds me a bit of Alain Delon or Gerard Philipe. He is initially reserved but opens out when the conversation gets into three dimensions. His conversation and manner are serious and intense. He often replies in a way which opens up more questions and thoughts to consider. He is one of the few people that I have met who immediately comes over as a genius.

Rubik consciously applied the fundamental principles of design when devising the Cube. In particular, he aimed to make each piece as simple and compact as possible. He believes form follows function. He is particularly pleased with the spring-loaded screws, which provide the dynamic tension to hold the cube together and to take up wear. We have discovered several earlier attempts to make Magic Cubes as well as several later independent designs. (Details will appear in a later circular). The measure of Rubik's success at simplifying the mechanism is that none of the other designs for the 3³ has yet been produced.

Rubik has succinctly analysed the features of the Cube which made it unique among puzzles when it appeared.

A. The pieces stay together.

B. More than one piece moves at a time.

C. The pieces have orientation as well as position.

I have written a section on the Fascination of Rubik's Cube for Rubik's forthcoming book and have added some analysis of the more obvious points such as the three-dimensionality, the coloration and the complexity. A point which I have recently realised, is that the Cube has many subgoals - one can get pleasure out of simply doing one side or just the corners. Like any masterpiece in any field, everyone can find his own special aspects of the cube to fascinate him.

I have not yet been able to think of any previous puzzle with the three features Rubik has pointed out. Now we have many variations and many new puzzles based on these points. Perhaps we could now call them "Rubik's Rubric" for puzzle inventors.

5

CUBE VARIATIONS

One of the marks of greatness which Rubik's Cube has demonstrated, is the ability to inspire imitations and variations in profusion. The two most obvious variations have been in the face patterns and in size. Most of the patterned cubes require one to solve the supergroup (N 18, 22, 45). Patterns include magic squares, fruit, suits, die patterns, chessmen, balls, O's and X's, and two types of Royal Wedding Cube - surely the ultimate, symbol of 1981! One of these has nine identical pictures of each face: Charles, Diana and flags of England, Scotland, Wales and the UK. The other is made in England and each face is a single nine part picture: Charles, Diana and four Union Jacks. With the latter, you can interchange the couple's hairdos or their ears! There are Cubes with the stickers modified into circles or octagons and with glittery, pastel or grainy colours. Some cubes have been used for advertising. Ideal is about to introduce a Calendar Cube with syllables, letters and numbers allowing you to set the day, date and month on one face This is an obvious executive toy which should keep the executive busy every morning.
[ see also C7/8-4. - J ]

Tamás Varga pointed out that one can modify the colour patterns on the Cube to get a number of simpler Cubes. I have made up a number of these, of which the following are the most interesting.

1. All faces green - the Irish Cube (suggested by R. G. S. Hankin).
2. 5 black faces, one white face - a Beginner's Cube.
3. 3 yellow faces, 3 blue faces - with the yellows meeting at a corner.
   (This is the pattern that Varga's colleague Julienne Szendrei showed me.)
4. 2 opposite faces of red, white and blue.
5. Remove the centre stickers.
6. Remove the centre and edge stickers. This simulates the 2³.
7. Remove the centre and corner stickers.
8. Remove stickers so the remainder form patterns like on a die or domino (suggested by John Ergatoudis). One could also make such patterns using just white stickers on the black cube.
9. Make an all white cube and trace a continuous circuit through all the 54 facelets. Finding such a circuit is a problem in itself. (Due to DBS based on a related picture in Deledicq & Touchard's book).

Several of these patterns have indistinguishable pieces, which makes them easier at first but can lead to complications later on. They are good tests of one's understanding of the Cube.

[ See also Sheperd's Cube in C7/8-7. - J ]

EXERCISE. How many ways are there to put 1,2,..., 6 colours on the six faces of a cube? The answer depends on whether you consider the actual colours as important or not and whether one allows reflections or not.

EXERCISE. Find all the circuits through the 54 facelets.

Some South African students have just suggested some further patterns: the South African Voters Cube - all white; the Bantustan Cube which starts with 5 black areas and one white area and winds up with one black area and 5 white areas; a six coloured 1³, called the Total Apartheid Cube since the colours never get mixed.

A UK firm has produced a solid version of the Irish Cube called Rube Mick's Cube. A number of other colour patterns are shown in the March 1981 Scientific American and the October-November 1981 Jeux & Stratégie. In the latter, one version has all stickers diagonally split between black and white. One could also use nine colours - one of each on each face.

Stickers are now available to make each face a magic square on a gold background. This is much harder than the previous magic square version which had different coloured background. Dan Feldman sends a set of stickers from Israel but I'm not quite sure what one does with it. I have seen a Do-It-Yourself Cube Kit which you assemble, but it has ordinary stickers.

One can put one's own choice of pictures on a Cube. This requires careful cutting and gluing. The result should be protected with some artist's fixative, but one must be careful not to make the Cube stick together. Joe and Jamie Buhler put six playboy pictures on a Cube for 3-D Jackson. He took hours to do it because it was all pink! Daniel Minoli sent a pictorial cube with rather abstract pictures in only a few colours. He threatens to send one with a short story in small print!

6

A Cambridge student, whose name I have forgotten, made white stickers with the same letter along each edge so the letter could be read from each side as in the diagram. He then found 6 3 by 3 letter patterns so that each row, column and diagonal is an English 3-letter word in one direction or the other. This gives 48 different words, though he says there are 54 of them - perhaps some are words in both directions. Neil J. Rubenking suggests labelling the 26 cubelets (or cubies) with the letters of the alphabet, trying to spell as many 3-letter words as possible.

There have been several inquiries about Cubes for the blind. (I also had an inquiry about Cubes for the colour blind, but this has been solved by many of the patterned Cubes). Bernard Morin, the well-known blind French topologist, used Dymo-Tape to produce tactile labels. Rainer Seitz, the founder of the Rubik's Cube Club in Germany (of which more later), devised the simple technique of using a hot needle to create craters in the plastic, in the form of die patterns. One can do this oneself though it takes some practice to get the right size needle, to get the right heat and to make the holes neat. Seitz demonstrated this technique on German TV and about a thousand inquiries were received. He was subsequently proposed for an award for services to the disabled. Cubes of this type and another type with brass studs are now made in Germany and we can import them. James Dalgety, of Pentangle, says that some may be produced here but so far I have no confirmation of this proposal.
[ see also C7/8-4. - J ]

The variations in size are more straightforward. The standard Rubik's Cube is 57mm on an edge. Ron Leary, of the Mechanical and Production Engineering shops at my Polytechnic, has produced some 114mm cubes in wood for demonstration purposes. I have recently seen that cubes of similar size are being made in Germany, but I have no details. Commercially, one can obtain cubes with edges: 52mm, 44mm, 38mm, 31 mm, 30mm, 27mm, 20mm. The smaller ones come with key chains or neck chains attached to the cube or its package.

A different kind of variation has appeared. 2³ cubes have been produced in Japan. One type has Popeye characters on coloured labels and used Ishige's mechanism (N 37). My example works very well, but Dalgety says his doesn't work so well. I gather this is no longer being produced. Another 2³ has a magnetic mechanism - each piece has a magnet in it which holds it to a central steel sphere. Unfortunately, rapid turning causes the pieces to come off. The outside of this version had all black labels with a silvery path through the facelets, The path crossings are labelled so you know which end is matched with which - leaving the labels off would have made a more interesting cube. Dalgety says that Ideal will be bringing in a 2³ soon. Jerry Slocum tells me he has three other types from Japan. Wim Osterholt, a Dutch enthusiast, has made several types of 2³ by hand and points out that the Rubik and Ishige mechanism are ends of a continuum of mechanisms. Dan Sleator, now at Bell Telephone, made a 2³ while a graduate student at Stanford, which I saw in Summer 1980. His mechanism is like Rubik's, with one additional point not specified in Rubik's patent. The theory of the 2³ is discussed in my Notes, especially N28, 29, 31. Solving it is quite easy, but there is one point where you can go astray if your algorithm for the 3³ does the edges before the corners of the last layer.

7

Osterholt has built and patented a 4³. He brought it to London and it worked quite nicely once we lubricated it with Holt's rubber and nylon lubricant. Rainer Seitz showed me some German patents or applications for 4³ and 5³. The 4³ has 8 corners, 24 edges and 24 centres. If these are all distinguishable and their orientations are noticeable, then we have 8!·24!·24!·3⁸·2²⁴·4²⁴ = 4.8 × 10⁷⁷ constructible patterns. However, if we have each face a solid colour, then the edges occur in indistinguishable pairs and the centres in quadruples, while the centre orientations are unnoticeable. Thus we get only 8!·24!·24!·3⁸·2²⁴ / 2¹²·24⁶ = 2.2 × 10⁵¹ distinct constructible patterns for the standard colouring. In the positions obtainable by layer turns, I have so far determined that: i) each position must be an even permutation of the corners and centres together; ii) the orientation of a corner is fixed by the orientation of the other seven corners; iii) the orientation of an edge or centre is determined by its location. This gives at most 8!·24!·24!·3⁸ / 2·3 = 1.7 × 10⁵⁵ patterns in the 4³ supergroup. If we use a standard colouring with solid faces, then we get to divide this number by 2¹²·24⁶ as above, to give at most 2.2 × 10⁴³ distinct patterns in the cube group.
[ See corrections at C2-8 and C3/4-15. - J ]
Osterholt showed me a process which appeared to exchange one pair of centres, though it is actually a 3-cycle with two centres indistinguishable. Consequently, it looks like all the above patterns are indeed achievable by first getting the centres correct, then the edges and corners by use of commutators of face layer turns.
[ See also C3/4 - 14-17. - J ]

Shape variations have also appeared. The simplest is a Truncated Cube, i.e. a Cube with the corners trimmed off to leave 8 small triangles which are coloured gold (Figure 1). The main Cube faces are thus octagons. The triangles are irrelevant so they can be coloured all different. If one truncates further so the triangles just meet, one has a Cubo-octahedron with 6 square faces and 8 triangular faces (see Figure 2 and the catalogue). This is sold as a Diamond Puzzler. The corner pieces retain bits of the main cube faces, so the triangles are still irrelevant, but they are coloured with 8 new colours. If one carries the truncation further until none of the Cube faces are left, one has an Octahedron which is dual to the cube. The 8 triangular faces are coloured with 8 colours. A face centre of the Cube is now a corner of the Octahedron, and has four colours on it. By adjusting the relative spacing of the Cube's bisecting planes so the edge pieces are no longer cubical, we can make it so that a corner of the Cube is now a triangular face centre of the Octahedron with only one colour on it. The edges of the two shapes correspond, but the colour pattern behaves differently on them (Figure 3). It is possible to simulate this Octahedron by colouring a Cube with eight colours in a pattern looking like a 2³ superimposed on the 3³ (See Figure 4 or Scientific American, March 1981, p.15, bottom right picture.) I call this a 'Dual' Magic Octahedron because the corners turn, rather than the faces. This has been patented by Josef Trajber of Vienna and shown to me by Rainer Seitz. Its solution is similar to the cube, but requires some new insights. It has 8!·12!·2¹²·4⁶ / 2·2·2 = 4.1 × 10¹⁹ = 40 50301 90700 29824 patterns. I don't know when this may be produced.
[ see C3/4-11. - J ]

Fig. 1	Fig. 2

8

Fig. 3	Fig. 4

Another shape variation, which is still essentially a Cube, is the Magic Ball. This is simply a rounded out Cube, approximately the circumsphere of a standard Cube, with diameter about 95mm. The face centres are nearly square areas, but the edges are long rectangular areas and the comers are large spherical triangular areas. The coloured stickers form large circular areas about each face centre, tangent to one another, leaving some black area on the corner faces - see the figure in the catalogue. Because you don't have the Cube's edges to align the eye or hand as you turn, the Ball is quite difficult to do. A 75mm version exists and miniature key chain versions of 31mm diameter exist. The latter have solidly coloured faces, forming 8 columns of three like coloured pieces, with the North and South Poles differently coloured or made to match one of the other 8 colours. There is also a Magic Globe! An amazing combination of Geography and Geometry! Now you can sort out the world! It comes complete with globe stand. At the time of writing these are not readily available in the UK.

MAGIC BALL	MAGIC GLOBE

More interesting are shape variations which are not quite the same as the Cube. The first of these was the Magic Octagonal Prism, which looks like a Cube with its four vertical edges trimmed (Figure 5 and the catalogue). This gives eight vertical faces of three facelets and two octagonal faces which are coloured with 10 colours altogether. The middle layer edge pieces now have only one outside face, which means that you cannot tell which way round they are. This makes the restoration easier at first, but harder overall since only half the ways of putting in these pieces are possible when the rest of the Prism is restored. (Compare with the comment of some of the patterned cubes). The colours of the vertical faces corresponding to the trimmed edges are not related to the other colours, so they may be rearranged, but again only half the ways are possible. In order to deal with these problems, one must develop two further tricks beyond those used to restore the Cube. (I will give a detailed solution in a later Circular). The Prism has only 1/8 of the distinct patterns of the cube, i.e. 5406 50040 93112 32000 patterns and 12 of these can be considered as START patterns. The Prism rapidly gets into non-convex shapes because it lacks cubical symmetry. There are miniature Prisms with edges 37mm, 34mm, 30mm.
[ or see my octagon page - J ]

9

MAGIC OCTAGONAL PRISM

Fig. 5

An odder shape is a Cube which is octagonalised along two axes (Figure 6). Mine is labelled Million Space Shuttle on the two poles and I have heard it called a cushion or pillow.
[ see also C7/8-4. - J ]
I have also recently obtained a Cube which has had all its edges and corners trimmed to give 6 square centre faces, 12 rectangular edge faces and 8 triangular corner faces (Figure 7). If the rectangles were squares, this would be a Small Rhombi-cubo-octahedron! I will leave the detailed descriptions of these for a later Circular.
[ See also C3/4-11 and C7/8-5. - J ]

During a recent visit to Liverpool, Alexander Wall showed me an object he had created by taking half of an Octagonal Prism and half of a Cube-octahedron and putting them together. I suggest we call this a Chimerical Cube. Obviously there are slot of possibilities here since almost all the standard size cube variations have the same internal dimensions. We could even mix three or more kinds to produce the most amazing shapes!
[ See also C3/4-11. - J ]

I have recently put the catalogue of my cube collection onto a computer file, so it can be kept up to date easily. Copies can be obtained by writing me with a stamped addressed envelope or a mailing label and stamps or International Reply coupons appropriate for a 30gm or 1 oz letter.

I am delighted to receive samples of cubes, particularly those which I do not have. Several people have sent examples in the past which I have not had time to acknowledge. My thanks especially to Shiyusuke Nakamuraya who sent several samples from Japan which are not available in Europe. I would be grateful for examples of any ancillary material as well, especially the packaging, any enclosed leaflets and any advertising, posters, etc. which can be sent.

Fig. 6	Fig. 7

10

PYRAMINX - THE MAGIC TETRAHEDRON
[ see also C5/6-18/20, C7/8-18 and my pyraminx page. - J ]

All of the above described objects, except the 2³, 4³ and 5³, are essentially variations on Rubik's basic idea. The first of the long awaited other shapes has just arrived as a Magic Tetrahedron called Pyraminx (or Pyramix). It is a tetrahedron trisected by planes parallel to each face (Figure 8). This gives 4 tetrahedral vertex pieces (V), 4 octahedral corner pieces (C) and 6 tetrahedral edge pieces (E). The Pyraminx rotates on each of the trisecting planes. It has a ball bearing action with detents to produce a smooth action and definite stopping. However, Pyraminx has no face centres, so it is not easy to see just how it works. Basically, it is a 'Dual' Magic Tetrahedron in that it is easier, although equivalent, to imagine the corners as turning rather than the faces. I denote the faces by F, R, L, D and the opposite corners by B, L, R, U. Part of the mystery of this pyramid is understanding how to orient oneself, so I will say no more about it now. The Pyraminx is much easier than the Cube and it has only 75582720 = 75,582,720 patterns, which can be reduced to 11,520 or even 960 patterns of interest. EXERCISE. Derive these numbers. Consequently, it will bring great joy to those who haven't been able to do the Cube. At the same time, it shows the basic techniques which apply to all such puzzles.

Fig. 8

Pyraminx was invented about ten years ago by Uwe Mèffert, a Franco-German inventor of scientific instruments. At the time he was interested in pyramids and the mysterious powers they are reported to have. He found that running the point of a pyramid along the palm relaxed him. Hence he wanted to make some kind of pyramid for the hand and the Pyraminx was the result. It took him and his brother, a talented engineer, about two years to develop the mechanism. When they completed their wooden model, they put it away and thought no more about it. Mèffert only realised the potential of his idea after the great success of Rubik's Cube last year. He showed it to several people in the puzzle field, notably To m Werneck, and they were all enthusiastic, so he began to organise production this year.

Having seen the chaos in the Cube field, Mèffert has taken steps to protect Pyraminx by means of patent, registered designs, trademarks and copyright in every relevant country, especially in Taiwan and Korea which are not signatories of the international patent conventions. He is already successfully enforcing his rights in Taiwan, although a number of pirate versions of the pyramid have appeared in Europe, Australia and Canada, where he has also been taking legal action successfully. An interesting further factor in his favour is that he has used fluorescent orange PVC stickers for one face. This orange is a difficult dye to use and only one firm in Europe can produce the material without using lead salts. The Taiwanese versions are required to use Taiwanese produced material which contains about 5,300 parts per million of lead. The maximum amount of lead permitted under the UK Toy Safety Regulations in a paint film is 2,500 parts per million and other countries have similar or lower permitted amounts. The Regulations definitely refer to other types of coating besides paint, so we believe the local authorities may be able to seize infringing copies.
[ see also C2-4 - J ]
Incidentally, the Taiwanese 'knock-offs' are based on 21 examples stolen from Mèffert's display stand at the Tokyo Toy Fair in June. By August, the Taiwanese were advertising them for sale, complete with a defect that Mèffert had already corrected. The Taiwanese versions are made from much inferior plastic and poorly finished, but are not all that much cheaper than the original.

11

Mèffert has worked with an institution of the blind to produce a Pyraminx for the blind with textured surfaces which are readily distinguished. These have been specially coloured for the benefit of the partially sighted as well. Mèffert has developed two simpler versions and two more complex versions of the Pyraminx, the most complex being comparable to the Cube.

Kersten Meier, a computer scientist from Itzehoe, Germany, independently invented the Magic Tetrahedron in 1980. He told me about it by telephone at Christmas, 1980, but I imagine it looked like Figure 9 (I have recently learned that Peter Ruffhead, of Cambridge, applied for a registered design for a tetrahedral puzzle of this form on 31 March 1981, and Ian Roger James, of Potters Bar, applied for a patent which includes this design on 2 July 1981). But Meier said that he could turn one corner and this did not agree with the results of Andrew Taylor and of Uldis Celmins (N 29, 60). Subsequently, I extended and completed their results, obtaining the number of patterns on any magic polyhedron. However, I then learned about other face patterns from Rubik and later from a picture that Meier sent I did some of these, but didn't get back to the Magic Tetrahedron until I heard reports of it from Jan van de Craats and Tom Werneck in August and September.

Fig. 9

I have recently found a letter from Meier giving a solution algorithm which takes at most 33 moves and has an average of 20.2 moves. My own method takes 50 moves, which can be reduced to 40 moves, at most. Mèffert reports a method that takes at most 38 moves and a newer method that takes at most 30. Nicolas Hammond has used a computer to study the 960 essential patterns and reports that any position is at most 21 moves from START. Further improvement will require studying a larger set of essential patterns.
[ See also C7/8-18. - J ]

Two books on the Pyraminx have already appeared in Germany - one by Tom Werneck (now published in English by Evans Publishers *) and one by Michael Mrowka and Wolfgang J. Weber (which may appear in English as well). Werneck's book is more popular and has lovely colour pictures of Mèffert, the production lines, diagrams, etc. But the Mrowka and Weber book is more interesting for mathematicians, etc., giving a considerably easier algorithm and more mathematical information. Mèffert is working with Professor R. F. Turner-Smith of the Chinese University of Hong Kong on a "Book of Pyramids" which will cover all the different Pyraminxes on both the popular and mathematical levels. This should be appearing early in 1982.

Mèffert has already adapted his principle to an octahedron. Rainer Seitz has shown me a number of German patents for other magic shapes and I think almost every variation has now been considered. Some of these have even reached prototype stage and I have seen some of these, but I don't know when any will become available. Rubik's Magic Domino (N34, 35) has been redesigned and should appear in 1982.

* Mastering the Magic Pyramid by Tom Werneck. Evans, £1.50 (please add 15p for postage if ordered from David Singmaster Ltd.)

12

CLUBS

Three national clubs have been formed.

Rubik's Cube Clube (Deutschland), Amselweg 12, D-6458 Rodenbach 2, Germany. (Founded by Rainer Seitz).

Rubik's Cube Club (Schweiz), CH-4600 Olten, Switzerland. (Apparently an offshoot of the previous club).

Japan Cubist Club, 3-D Daiichi Yazawa Building, 6-4-11 Roppongi, Minato-ku, Tokyo 106, Japan. (Established by Tsukuda, the Japanese agents of Ideal).

Two regional clubs have been established.

Rubik's Cube Club Hamburg, c/o Peter Kellerman, Stormarnerstrasse 1, D-2000 Hamburg 70, Germany.

Hasseltse Kubisten Federatie, do D. Durlinger, Genkersteenweg 179, B-3500 Hasselt, Belgium.

[ See also C5/6-8. - J ]

Three university-based clubs have been formed, at Bristol and Warwick in the UK and Stanford in the USA, but I do not have current addresses for them. Many schools have set up clubs, but I think there may be too many for us to record them in the Newsletter. Nonetheless, I would be pleased to receive details of any clubs at all and I will certainly list any national or regional clubs in future issues of the Circular.

SHORTEST TIMES AND COMPETITIONS

Single times for restoring a cube are not a good measure of ability, but it is amusing to record them as an indication of the fantastic speeds which have been achieved. In my Notes, the shortest known time was due to Nicolas Hammond, namely 36 seconds, Since then Hammond has achieved it in 28 seconds (and perhaps less by now) and did 37 seconds on BBC TV on 24 January 81. Nicolas is an upper sixth former (i.e. about 17 years old) at Nottingham High School (UK). In Harburg, Hamburg, Germany, a 16 year old boy, Ronald Brinkmann, solved his maths teachers cube overnight when the teacher finally gave up. Within a few months, he did 29 seconds on German TV and later reached 24 seconds. When asked for advice, Brinkmann replied, "Turn, turn, turn".

In Paris, Jérôme Jean-Charles, 25, has developed efficient ways of carrying out processes, achieving an average of 36 seconds, then 32 seconds, within a minimum of 21. He can do up to 180 moves per minute and can do (LF')⁶³=I in 37 seconds. In Bootle, Liverpool, Michael Musker, 16, a student at Hillside High School, has been timed at 20 seconds and says he once did 14 seconds when no one was watching. In Australia, Geoff Harris, a 21 year old student at Griffith University claims he did a cube in 7 seconds! At this point, the problem of deciding whether the cube was really in a random position becomes apparent. Although any five or six moves effectively randomises the cube, some random positions are much closer to START than others and this varies depending on the algorithm used by the solver. Hence it is quite unreasonable to take single times. One should have several gos and the cubes ought to be identical patterns for each person. Even then, as in all sporting events, luck will still play some role.

Nicolas Hammond recently appeared on the Paul Daniels Magic Show and reports that Daniels does the world's fastest cube - he throws it in the air and it comes down restored!

After the first competitions in Hungary (N38), the next major competition was on 31 January 81 at the Imperial Hotel in Tokyo, sponsored by the Japan Cubist Club. There were 400 entries and 73 passed the qualifying round. The contest consisted of restoring three new cubes. The winner was a 16 year old schoolboy, Hideki Kitajima, with times of 62, 46, 49 seconds (average: 52 1/3 seconds). He won a new Datsun! Second prize was a Sony video tape recorder, and there were three further substantial prizes. Kitajima said the new cubes slowed him down - he usually does 25 to 30 seconds in his morning practice of 100 cubes.

13

Rainer Seitz organised a Guiness Book of Records bash in Munich on 6-13 March 81 at the Olympia Shopping Centre. The Centre is connected with the German publisher of the Guiness Book). The best times were 38 seconds, again due to new cubes, achieved by Ronald Brinkmann and Jury Fröschl, an 18 year old Munich boy.

Ideal has begun a world championship, though it hasn't yet received much attention in the UK, where the Daily Mirror is co-sponsoring it. Mal Davies (of whom more later) has kindly passed on the first four (of eight) regional results.

The rules allow a 15 second look at the Cube on a table. An automatic timer records the time from when it is picked up to when it is set down. Only one trial is allowed in the regionals, but there will be three trials at the final which will be on 12 December in London. New Cubes are used which are described as randomised by a computer and contestants pick one from a bin full. There will be a world championship in Monte Carlo next year.

The French national finals were held recently and Jérôme Jean-Charles won with a time of 25.6 seconds. The New England regional was won by a 9 year old boy named Jonathan Cheyer in 48.31 seconds.

The first Inter-school Rubik Cube Match was held on 15 October 81 between St Thomas Aquinas RC School and The Holy Child Jesus School in Birmingham. There were 16 students on each side. The fastest times were Anthony Shally (34 seconds) and John Bartlett (37 seconds). The organiser was Mal Davies, a teacher at the first school, who was filmed giving a lesson to his students and the TV station began the story that this was part of the regular curriculum! Davies wishes to make it clear that such lessons are extracurricular. He has started to teach an Adult Education class on the Cube! This takes three evenings and is held at the Brasshouse Centre, Birmingham.

The youngest solver of the cube appears to be Lars-Erik Andersen, 7, of Lørenskog, Norway. He does the cube regularly but cannot explain how. James Dalgety says an 8 year old in Hampshire has solved the cube. Tanya Shelton, 6, in Oxford, has learned how to do the cube from her father and is now much faster than he is.

Rainer Seitz reports that a boy in Germany does two cubes at once - one in each hand. Mal Davies reports that Richard Hodson, of Queen Mary's Grammar School, Walsall, can do a Cube one-handed in 89 seconds. John White, 19, a second year mathematics student at the University of Warwick can do the Cube behind his back with one look. He demonstrated this for Mal Davies, taking 10 minutes to study the Cube first. Unfortunately it came out with two pieces wrong, which he then corrected behind his back, in a total time of 154 seconds.

14

FUNNY MOVES

Jérôme Jean-Charles has devised the following technique for implementing (F²R²)³. Place your forefingers on the U sides of the UF and UR pieces with your thumbs on the D sides of the DF and DR pieces. Figure 10 shows the placing of the forefingers on the U face. One can now repeat F²R² indefinitely without letting go! Jean-Charles also makes moves just flipping a corner piece with a finger, e.g. B is achieved by flicking the D face of the DRB piece with the forefinger.

Rainer Seitz makes a similar hold on the Cube, as shown in Figure 11. With this hold, one can repeat L²B⁴L⁴B⁴L⁴... continuously. Seitz calls this the Buddhist Prayer Wheel. Perhaps one should chant "Om Rubik om..." or "Hare Rubik, hare Rubik, hare Rubik, hare, hare" while doing it.

Chris Strain-Clark holds her Cube similarly at the points shown in Figure 12. By skilful slipping, one can carry out slice moves without appearing to have done anything at all. Pete Strain-Clark points out that the group of patterns achievable is not a subgroup of the Cube group because the orientation of the Cube is now important. Rather we have a subgroup of the 'oriented Cube group', which has 24N = 1.04 × 10²¹ elements.

Fig. 10	Fig. 11	Fig. 12

ERRATA TO MY NOTES

The following are errata beyond the list of 23 Feb 1981, which were incorporated into the American edition. The items on pages 37, 43, 44, 45, 49 are given on the list of 1 Jul 1981 which is included in the Penguin edition. The items on pages 46 and 51 are corrections to previous errata.

Thanks to Frank Bernhart, Marston Conder, Sandy Frey, Jesper Gervaed & Torben Bisgaard, Nicolas Hammond, Colin Merton and Anneke Treep for finding some of these.

Page 21, Line 21. In the antislice notation, the second process is (FR)³U².
Page 37, Line 5; Page 38, Line -2; Page 51, Line -14; Page I-3, Left, Line -6. Change Nicholas to Nicolas.
Page 43, Line -6. Change to: L'U'B'U²BLUFU²F'.
Page 44, Line -16. Change to: (MBT&KO - 9)
Page 45, Line -7. Delete: (See also .
Page 45, Line -3. Change to: (MBT - 14).
Page 45, Line -1. Change to:R_aF²_aR'_a·U·R_aF²_aR'_a·D'
Page 46, Line 30. ... only moves (A, C-)D- in ...
Page 49, Line -6. Insert:(UF, DL).
Page 50, Lines -15-14. The first non-obvious case is 88 which ... have 8 and 11 cycles...
Page 51, Line 26. Change the total to 355995.
Page 51, Line 31. Change to: 3389 45406 22394 = 3.39 × 10¹³.
Page 51, Line 32. ... about 7.84 10^-7 ...

15

GENERAL ANECDOTES

Fred Warshofsky relates that a football game in Connecticut was delayed due to a missing player who was discovered playing with his Cube in the locker room. Someone told me of a UK court being disrupted by a Cube. Several people have told me that the Cube reduced the productivity of their department to zero. One in London and one in Washington pointed out that because they worked for government departments, one couldn't notice the difference. Paul Samet, head of the University College London Computer Centre, bought a copy of my Notes so he could get his staff back to work.

An Italian engineer in Nigeria, Vittorio Burgatto, won a Chinese dinner for two when the proprietor saw his Cube and said it looked easy! Robert Cole told me that two people followed him out of a waiting room into his train compartment when he was playing with his Cube. Fifteen minutes later one of them realised he was on the wrong train.

Ideals promotion of the cube in the US started with a party given by Zsa-Zsa Gabor in Hollywood on 5 May 80 (N37), followed by a New York do on 8 May 1980 with three generations of Gabors. In Hollywood, Solomon W. Golomb, the inventor (or at least the popularizer) of Polyominoes™, appeared as the mathematical half of the sponsorship. David Sibley was the mathematician on the East Coast. Miss Gabor managed to forget the name of the product at the Hollywood press conference.

English Cube-mania (Cubanoia?) began when Rubik appeared on Swap-Shop, a Saturday morning BBC TV children's programme on 24 January 81, along with Nicolas Hammond, who did the Cube in 37 seconds. They had several other bits on the cube in the preceding and succeeding weeks, and at later times. The competing show, in ITV, Tiswas, has just launched a 'Stamp out Cubes' campaign. Songs have been written about the Cube! "Mr. Rubik" by the Barron Knights is available in the UK on Epic EPC Al596. (See the catalogue). [ see also C3/4-2. - J ]. There is an Australian song and a German one by a character called Krazy Kuno.

Rubik's Cube has become a household word - I have seen it used as a metaphor in political, educational and economic contexts as well as in numerous political and humorous cartoons. The first political cartoon was by Felix Mussil in Frankfurter Rundschau on 5 February 81, reprinted in Stuttgarter Zeitung and Der Spiegel. It showed a bemused Chancellor Schmidt with a Cube labelled Berlin, unemployment, etc. On 7 July, Garland did a cartoon in The Daily Telegraph showing Willie Whitelaw and Maggie Thatcher struggling with a Cube labelled racial violence, etc. On 6 September, I saw Reagan's Cube by Paul Conrad of the Los Angeles Times reprinted in The New York Times Weekly Review. It had 27 problems showing! A Mahood cartoon In Punch for 9 September shows The Russian Cube - an enormous cube being parachuted onto troops - "A variation of the Rubik Cube, this game is guaranteed to drive any enemy to distraction within minutes". Numerous other cartoons and even a poem have appeared.

Hamlet Cigars have used the Cube for two commercials. One shows my hands getting a Cube all correct, then it is turned to reveal a blue centre in the white face, whereupon It is dropped in disgust Another uses another pair of hands just to show increasing frustration with the wretched thing, then dropping it.

Several people have discovered their Cubes have moved when they left the house, the only people about were children or workmen who swore they hadn't touched it. Hence we have concluded that either there are poltergeists (which explanation I cannot accept) or the Cube must be able to move by itself. This led one correspondent to wonder if one could force theCube to surrender, he thought of starving it, but he couldn't work out what it ate. He then tried sensory deprivation - keeping it in its box in a dark drawer - but to no avail.

There is a definite mysteriousness about the Cube. There is a hotly denied rumour that one of John Conway's graduate students got his beard caught in a Cube and was slowly pulled into it despite his friends' efforts. It is thought that he will reappear when the Cube is returned to the same position it was in when he was first trapped, but no one noticed what it was. CCC (N 30), which is better denoted as C³, has commandeered all the computers in Cambridge to run through Cube patterns in hopes that the desired pattern will be recognisable, perhaps by the appearance of the missing cubist on the video screen. The German advertising for the Cube says 'Do not take hold of it - it will not let you go'. How prophetically true this is - or is it a warning subsequent to a similar Germany tragedy?

16

There has even been a Cubic Communion. The Reverend Richard Ames-Lewis conducted this at Edenbridge Parish Church on 25 October 81. When I first heard about this, I supposed that his parishioners needed holy help to restore their Cubes, but he says he used the Cube as an image of disorganised life which one is presenting for God's grace. Still, I'm sure some people were secretly hoping their Cubes would come back solved.

The mixture of religion and cubism may be an interesting development.
[ see also C5/6-22 - J ]
Solomon Golomb has devised an analogy between Cube corners and quarks, which he has extended into a complete Cubic cosmology. (More on this and on naked beauty or bare bottom in a later Circular).

A correspondent has recently written that the Cube has allowed him to come out of the closet, the mathematical closet, that is. In the past, when he told people he was a mathematician, they treated him like a social outcast. Now they ask how to do the Cube. Beautiful blondes come up to him in airports asking him to relieve their misery, etc.

So many books have come out on the Cube (details in a larger Circular) that Pan Books is about to publish 'Not Another Cube Book'.

"And he's at it all night too!"

MARITAL ANECDOTES

Rainer Seitz told me of a Cube-caused divorce case last March. He has kindly sent the report from Bild Zeitung, which is datelined Bottrop, Ruhr region, 27 Feb (1981). A 21 year old salesgirl was complaining that her husband no longer spoke to her, fled from visitors, never watched TV and no longer did anything in bed. Seitz volunteered to act as marriage counselor by teaching him how to do the Cube, but nothing came of it.

Seitz also reported that a woman with a large black eye came to a demonstration. Her husband had got two faces correct and she had disturbed it while he was at work She had been given two days to get it right again!

A later report comes from Kassel, "She dreams of sex, he only dreams of six sides". 'I have gone nearly mad, being woken up at six by that stupid klick-klack' complained Ilse K. (33) about her technician husband Otto K. (36). He replied that he bought the Cube as a present for her, because she didn't like flowers.

One a more positive note, Eric Pfisterer (the same one), a schoolteacher in Toronto, says that a girl he had known as a child, now grown up, came to Vienna with a Cube and promised to marry him if he solved it He did and she did! And they are living happily ever after.

In England, Pete Strain, a lecturer at the Open University and a world-class racing cubist with a best time of 21 seconds, met a girl, Chris Clark, at summer school and they found the Cube a mutual fascination. He promised to marry her when she got down to 60 seconds. She did and he did! (Actually she was already down to 70 seconds, so this wasn't so great a challenge as it seems). They are now Pete and Chris Strain-Clark Her two daughters also do the Cube and they are claiming a family record! I gave their name to some media people and they volunteered to do the cube underwater! This was filmed for Anglia TV but I gather all you could see was bubbles. They had facemasks but were holding their breath. Chris' face mask got flooded so she had to complete her cube by memory.

I mentioned the divorces and marriages in a talk I gave to the British Association on 4 September 1981. This got headlined as 'Cube Route to Divorce', etc. Next week Private Eye did a pseudo-Sun front page with "Exclusive - How 'That Cube' can improve your love life - Harvard Prof. Tells All".

© David Singmaster