What Jessica Fridrich Says About her 20 Years of Speedcubing Experience

May 26th, 2005 by rubikscubesolverssociety

The first time I met the cube
"face-to-face" was when I was 16 years old in March 1981. I was hooked
since the first moment I saw this absolutely unique combination of
simplicity and ingenuity. There was no need to explain what needs to be
done with it - a self-explanatory, remarkably difficult puzzle with a
devilishly mysterious mechanism inside - a fascinating silent
challenge. The owner of the cube was a 14-year old boy who could solve
the cube in about a minute. He lent it to me for a few minutes just
enough to assemble one face.

Although in March 1981 the cube was being
sold by thousands in other countries and despite the fact that the
invention took place in the neighboring country, it was impossible to
buy the cube in Czech Republic. A classical example of how inefficient
and impotent the Eastern Block economies were. I got my hands on a
primitive solving system from a Russian magazine Kvant
long before I actually owned the cube. I would analyze simple moves and
their action on a piece of paper, trying to figure out algorithms based
on the commutator principle.
Then later in the spring, our local astronomy club leader bought the
cube during his trip to Hungary. He was unable to solve it and could
not find anybody who would put the cube back into its original state.
With the help of the commutator principle and those "Russian" moves, I
solved the cube for the first time. It took me several hours.

I desperately wanted to get my own cube
but whoever was lucky enough to own it, would never sell it. So, I had
to wait a little longer and finally got my first cube in July 1981. A
French family was visiting my sister and their two teenage boys brought
the cube with them. When they saw how attached I quickly became to the
piece of plastic, they did not have the heart to take the cube with
them back to France. That meant that I could finally start working on
my system! During the Summer, I persuaded my parents to visit Hungary,
where I bought three more cubes. It was still a challenge to get the
cubes because they were not available in stores. I bought the cubes
from an old lady who was selling magazines and souvenirs in the street.
When I mentioned "Buvos Kocka" to her, she smiled, quickly looked left
and right and handed it to me in a brown lunch bag, put her index
finger across her mouth, and said "Shhh, one hundred and fifty
Forints". I know all this sounds funny now, especially to those from
Western countries where it was a no-brainer to buy the cube. But this
is how I really started.

At first, I was using the layer-by-layer
system that I learned from a Czech magazine. It was actually already
quite advanced. First the first layer, then the four middle edges (just
one algorithm), then flipping the edges, moving the edges, flipping the
corners, moving the corners. These were my basic algorithms that I
started using and I quickly got my average to about 1 minute. It was
September 1981, three months after I started playing with the cube.

My last year of high school was a strange
and exciting time. A kid solving the cube in public was a head-turner.
Cube was a good conversation starter, although later it was termed by
many as a conversation killer . It was normal for two cubers
who met on a bus to start the conversation without looking at each
others eyes, saying: "How do you this?" "Hmmm, could you do it slowly?"
"Thanks". This is how we built our first primitive systems. I had one
schoolmate from high school who had the same disease - an unconditional
love for the Rubik’s cube. His name was Ludek Marek. He was using the
same system as me, but for some reason he was always trailing about 20
seconds behind me. He once noted while I was solving the cube pointing
to my cube: "Oh, I like this "T" pattern, because when you turn the
edges, the whole last layer will actually flip correctly." It was the
shortest 6-move that influences only the last layer - the move that
perhaps all cubers know. And that sentence stuck in my mind. It was the
germ that later blossomed into the current system. I realized that in
the system I was using it was possible to first flip the edges, then
the corners, then position edges and position corners. This is because
the moves commuted. So, what if I had an algorithm for all flipping
patterns and all permutations? Then I could solve the last layer always
in just two algorithms. Also, the number of patterns was not that big
and they were easy to recognize fast. But where to get the algorithms?
I already knew some portion of them and I gradually started adding
more. Whenever I encountered an orientation that I did not know, I did
it the old way - flip edges and then flip corners. And whenever I
encountered a permutation for which I did not have an algorithm, I
would combine the permutation from the algorithms I already knew. I
began improving very steadily as I improved my system and my ability to
recognize positions quickly. In December 1981, 6 months since time
zero, I was averaging about 35. Occasionally, I would read an article
about a student from Great Britan who solved the cube in 28 seconds,
then about a guy from USA who did it in 24, etc. I was always chasing
the world, trying to catch up. I often felt like it was not possible to
squeeze my times anymore, as if I was already at the limits of what I
can do with my system, but nevertheless, with time, I was able to get
to those magic numbers I previously read in the newspaper. At that
time, I was getting ready for my final examination to finish my high
school. I was combining test preparation with cubing. When I was
messing up the cube, I was staring into the text, learnig the subject,
then pausing for a while and solving the cube. And I could go like this
for hours and hours. Surprisingly, during the late Summer and Fall
1981, the cubes became finally available in Czech Republic as well. The
cube crazy has officially began. Local championships popped up at high
schools and universities. I always participated in them often with my
friend Ludek. Both of us always left the competition far behind. There was nobody I knew with whom I could compete. I was thus chasing the clock and the world.

In the Winter 1981/82, the Czech magazine
Mlady Svet called for a national championship and people started
submitting their times. In February 1982, the magazine printed a
preliminary table showing the best ten times submitted. And I could not
believe my eyes to see my name on the first place. I also noticed the
name Mirek Goljan who was literallily "breathing on my back". That gave
me more energy for my practicing and I went into the national
championship on May 11, 1982 averaging about 25 seconds with my
personal best of 18. Ten months from time zero.

I won the semifinals and 5 best advanced
into the finals. Among them, Mirek Goljan and my friend Ludek Marek.
The finals were in front of TV cameras. We were allowed to use our own
cubes. The best-of-three time determined the winner. We all solved the
cube at the same time. I won the first and second rounds and Mirek won
the last third round. My second time of 23.55 got me the first place,
Mirek was the second, trailing about 2 seconds, and Ludek ended on the
third place. The first prize was a plane ticket to Budapest to the
first World Championship.

I became a "celebrity" for a few weeks
receiving a lot of letters all asking for one thing - the description
of my system. The letters actually did not have my proper address, just
the name and city, no street address or zip code. They all were
delivered. I decided to publish my system in Mlady Svet. It contained
all algorithms for permutations and orientations and a few moves for
the F2L. Most people were disappointed to learn that the method is
actually quite "complex" requiring a lot of practicing and
memorization. Most expected a simple trick that one can explain in a
few minutes. What did you say about the free lunch? I remember one
really funny story that happened to me on a train when I commuted to
college from my home town. A guy was sitting next to me playing with
the cube. I asked him about his system. He said: "I am using the
Fridrich method." I asked with a surprise in my voice: "You actually
memorized ALL algorithms?" His answer was: "No, that’s too much. I know
only some of them." I replied with: "Well, you need to memorize all of
them otherwise you are not really utilizing its strength." He looked at
me frawning and said with his mouth half open: "Yeah, so what’s your
system?" I answered with a big smile: "I use the Fridrich method, too,
because I am Fridrich." He did not blink an eye, did not say anything
and handed me his messed-up cube. I solved the cube in about 20 seconds
to prove my words and we both laughed at the coincidence.

I was acepted to college and still kept
on improving. Later in 1982, I changed my F2L system to the current
system. Before, I would do the first layer and then insert two cubies
from the last layer into the middle layer. I developed the algorithms
and also algorithms that moved / flipped the cubies in the middle
layer. When I switched to the current system for the F2L, I instantly
improved by several seconds and got my average to around 20 (15 months
from time zero). By 1983, I was consistently averaging 17 seconds. I
knew three more cubers capable of achieving sub-20 averages
consistently. We practiced together. As the cube rage cooled down, I
stopped working on my system. The second Czech Championship took place
in March 1983. Robert Pergl won all three rounds (if I remember
correctly) with a best of 17.04. He was using basically my system but
he knew more than 600 algorithms (I was actively using about 120-150)
and one could say that he was using a "multi-system". From time to
time, he was able to solve the last layer in just one algorithm,
perhaps due to preparing the LL a little before finishing the F2L. And
he stayed cool and psychologically stable during the whole event.
Psyche is a very important factor in championships. There is little
value in being able to solve the cube in 16 seconds on average if the
nerves slow you down to 20 during the competition. You can’t win a big
event unless you work on the psychological factor as well. And Robert
indeed was consulting with a psychologist, preparing very carefully for
the whole year. What can I say - it paid off.

I would dare to say that nothing important happened in speed cubing and cubing in general over the next decade. Then, in 1992 Herbert Kociemba
developed a computer algorithm with a performance very close to the
God’s Algorithm (the shortest moves from any position). It was, in my
opinion, the biggest event in cubing in general. Suddenly, we could
obtain the shortest moves for any position and any pattern.
Surprisingly, Kociemba’s algorithm always seemed to find a solution
within 20 face moves. The famous cube-in-cube pattern turned out to
have an elegant short solution L F L D’B D L² F² D’F'R U’R'F² D as we
suspected for a long time but never found it. Progress has been made in
identifying the farthest positions on the cube (superflip and
supertwist). To give you an idea how revolutionary Kociemba’s discovery
was, the previous best computer solution was always able to solve the
cube within 38 moves, but could not guarantee better (Thistlewaite’s
algorithm). Even though Kociemba’s algorithm did not provide a proof
that the diameter of the cube group is indeed 20 in face counting, it
has been an impressive piece of work, indeed. As the computer speed and
memory increased, optimal solvers came to life, and suddenly, to my opinion, the cube lost much of its enigma.

I put my system in electronic form on the Internet in January 1997 after I had discussions with Mike Pugh on the Cube Lovers mailing list
(one of the oldest mailing lists ever, established in 1980). He
persuaded me that making my system available in electronic form would
be useful for other cubers. I made copies of my old, now yellowish,
notebook and he made those small pictures you now see on my pages. I
included the patter and set up the site. I never put a counter on my
page, so little did I know how popular the system became. Actually, to
be completely honest, I was convinced that nobody in their right mind
will have the energy and will to learn the system in its entirety. I
thought that speedcubing was inactive and not popular enough for
anybody to have the motivation to go through the pain of memorizing the
algorithms. I know now how wrong I was. One should never underestimate
the power of the cube. I still admire those of you who entered the
speedcubing now. Back in 1981, the cube was mysterious. We did not have
computers powerful enough to develop the shortest moves for us. We did
not know if those algorithms we found by trial and error were the best
or shortest. The unknown and unanswered questions were an important
ingredient for many cubers. They were the engine that powered us
forward. I do not intend to sound as an old lady complaining while
recalling the old good days, but I am trying to convey what most of us,
if not all, felt as we were trying to uncover the curtain of secrecy of
the cube.

At the end of 1996, I sent a postcard to
Mirek Goljan and I typed the 14-move algorithm for cube-in-cube and
nothing else on it. Mirek and I have not seen each other for at least
12 years and I was already pursuing my PhD in the US by that time. We
got in touch again via phone and later via e-mail. In 1997 I visited
Czech Republic and after almost 14 years, we started cubing together,
admiring Kociemba’s algorithm, and sharing our personal stories. Mirek
later joined SUNY Binghamton and pursued his PhD degree in the same
field as me - steganography and digital watermarking. We became
professional colleagues and today we work together on puzzles of data
hiding and discovering them in digital images. After 14 years, our
journeys joined again - two top Czech speed cubers uncovering the
secrets of images.

That’s all folks. Thanks for reading!

The Fridrich Method

May 26th, 2005 by rubikscubesolverssociety

steps: cross/F2L/OLL/PLL

This is the Fridrich method created by Jessica Fridrich. With this
method you can solve the cube on average using about 56 moves and under
20 secods. First you make a cross on the bottom face, this usually
takes about 6 moves. Then you solve the F2L(first 2 layers). Each
corner and edge pair takes about 7 moves, so it usually takes about 28
moves for that. Now you only have 1 layer left. This last layer will be
solved using 2 algorithms (an algorithm is a set of moves that
accomplishes a certain goal on the cube). First you orient the entire
last layer using 1 algorithm (meaning you flip over all the last layer
cubes making the last layer 1 solid color). Then you permute the last
layer with 1 algorithm and the cube is solved. There is a total of 57
orintation algorithms and there is 21 permutation algorithms. Don’t let
this number intimidate you, once you get the hang of memorizing these
algorithms
it gets very easy. When learning these algorithms don’t rush them,
learn a few, then make sure you master them, then learn a few more.

WCA Speedcubing World Records

May 26th, 2005 by rubikscubesolverssociety

WCA Speedcubing World Records

Acknowledged by the World Cube Association

Updated: October 17, 2005.
World record history.


Event

Type

Record holder

Country

Competition

Record

Details

Rubik’s 3×3x3 Cube

Fastest time

Jean Pons

France

Dutch Open 2005

11.75

Rubik’s 3×3x3 Cube

Average of 5

Shotaro Makisumi

Japan

Caltech Fall competition 2004

14.52

13.16 13.77 15.49 14.29 15.60

Rubik’s 2×2x2 Cube

Fastest time

Shotaro Makisumi

Japan

Horace Mann Spring competition 2005

4.13

Rubik’s 2×2x2 Cube

Average of 5

Shotaro Makisumi

Japan

Horace Mann Spring competition 2005

6.29

14.23 5.00 7.59 6.28 4.13

Rubik’s 4×4x4 Cube

Fastest time

Chris Hardwick

USA

Caltech Dallas Summer competition 2005

55.38

Rubik’s 4×4x4 Cube

Mean of 3

Yuki Hayashi

Japan

Kyoto Rubik’s Cube contest 2005

1:04.97

1:00.38 1:04.10 1:10.44

Rubik’s 5×5x5 Cube

Fastest time

Frank Morris

USA

Caltech Dallas Summer competition 2005

1:51.41

Rubik’s 5×5x5 Cube

Mean of 3

Olivier Gaucher

France

Dutch Open 2005

2:06.15

1:53.40 1:59.21 2:25.83

Rubik’s 3×3x3 Cube
Blindfolded

Fastest time

Tyson Mao

USA

Northern California Fall Open 2005

1:58.32

Rubik’s 4×4x4 Cube
Blindfolded

Fastest time

Dror Vomberg

Israel

European Championship 2004

19:54

Rubik’s 5×5x5 Cube
Blindfolded

Fastest time

Stefan Pochmann

Germany

European Championship 2004

2:34:36

Rubik’s 3×3x3 Cube
One-handed

Fastest time

Shotaro Makisumi

Japan

Horace Mann Spring competition 2005

23.76

Rubik’s 3×3x3 Cube
One-handed

Mean of 3

Shotaro Makisumi

Japan

Horace Mann Spring competition 2005

27.56

23.76 29.98 28.95

Rubik’s 3×3x3 Cube

Fewest moves

Mirek Goljan

Czech Republic

Caltech Winter competition 2005

28 moves

Rubik’s 3×3x3 Cube

24 hours

Jess Bonde

Denmark

December 16/17, 2002

2000 cubes

Used 22 hours 16 minutes

Rubik’s 3×3x3 Cube

Youngest solver
in competition

John Ismael Ugelstad

Norway

German Open 2005

5 years
233 days

1:43.63

Rubik’s 3×3x3 Cube

Youngest solver

John Ismael Ugelstad

Norway

December 28, 2004

5 years
117 days

Rubik’s 3×3x3 Cube
With feet

Fastest time

Kåre Krig

Sweden

Swedish Open 2005

2:14.33

Rubik’s Clock

Fastest time

Stefan Pochmann

Germany

Dutch Open 2004

9.05

Rubik’s Clock

Mean of 3

Stefan Pochmann

Germany

German Open 2005

9.74

9.76 10.23 9.22

Rubik’s Magic

Fastest time

Stefan Pochmann

Germany

Dutch Open 2005

1.23

Rubik’s Magic

Average of 5

Alexander Ooms

Netherlands

Dutch Open 2005

1.48

DNF 1.51 1.44 1.50 1.32

Rubik’s Master Magic

Fastest time

Bob Burton

USA

Caltech Dallas Summer competition 2005

3.05

Rubik’s Master Magic

Fastest time

Stefan Pochmann

Germany

Dutch Open 2005

3.36

4.18 3.53 3.06 3.28 3.28

Square-1

Fastest time

Lars Vandenbergh

Belgium

Dutch Open 2004

25.29

Square-1

Mean of 3

Lars Vandenbergh

Belgium

Dutch Open 2004

33.21

40.17 25.29 34.18

Megaminx

Fastest time

Stefan Pochmann

Germany

German Open 2005

1:55.34

Pyraminx

Fastest time

Gunnar Krig

Sweden

Swedish Open 2005

6.55

Pyraminx

Average of 5

Piotr Kózka

Poland

Polish Open 2005

11.22

12.25 10.90 10.83 9.77 11.94

Siamese Cube

Fastest time

Kenneth Brandon

USA

World Championship 2003

70.96

Rainbow Cube

Fastest time

Chris Parlette

USA

Horace Mann Spring competition 2005

10.20

Rainbow Cube

Mean of 3

Chris Parlette

USA

Horace Mann Spring competition 2005

14.32

19.26 10.20 13.50

Source: www.speedcubing.com

Micellaneous Things of What Erno Rubik Thinks of His Cubes

May 26th, 2005 by rubikscubesolverssociety

It took Ernö Rubik 1 month of extensive practice to resolve for the first time its puzzle.
Before, he wasn’t even sure that there was a method to succeed.

"This object is a wonderful example of the rigorous beauty, the big wealth of the natural laws :
it is a perfect example of the human mind possibilities to test their scientific rigour and to
dominate them. It represents the unity of real and beautiful, which means for me the same thing."
[Ernö Rubik]

The Cube can seem alive as it heats up in your hand. The fact that each face of the Cube is made of three
layers of three blocks has an important meaning.
The number three seems to have a particular significance, relevant in some strange ways to the relation
between man and nature.
"mother-child-father,
heaven-earth-hell,
creation-preservation-destruction,
birth-life-death."
[Ernö Rubik]

The Cube is an imitation of life itself –or even an improvement on life.
The problems of puzzles are very near the problems of life, our whole life is solving puzzles.
If you are hungry, you have to find something to eat. But everyday problems are very mixed–they’re not clear.
The Cube’s problem depends just on you. You can solve it independently. But to find happiness in life,
you’re not independent. That’s the only big difference.
[Ernö Rubik]

What Erno Rubik Says About His History

May 26th, 2005 by rubikscubesolverssociety

Ernorubik2I was born in 1944 in Budapest, Hungary. My father was a mechanic-engineer, glider builder, a renowned specialist, creator of more than twenty six types of gliders. My mother was woman of letters, poetess and artist. The combined presence of these impulses, the technical one and the arts one, haw been for me, I’m sure of it, a determining factor. At first, I tended towards the visual arts : I drew a lot and I have painted a lot. I made my secondary education in a school which was dependent upon the "Beaux- Arts", as a sculptor. From that time, my taste for technical applications awakened. So the next step in my studies naturally became the University of Technical Education of Budapest, and, in 1967, I obtained there my architect diploma.
I still have a passion for architecture, as one have the more complex activities which combines the more characteristic features of science, technical science and arts. With my diploma, I didn’t feel yet like a completely well-trained man and I continued my studies at the Decorative Arts High School in the interior architecture section. My second diploma gave me the title of designer; it was given to me in 1970. These studies made me sensitive to the artistic factors. Since 1970 I have stayed constantly in the High School, teaching plans and constructions, interior architecture drawings, furniture plans and projects, study of form and descriptive geometry. Teaching is the best way to learn, I’m still convinced of that; by passing on our knowledge we continue to discover and learn. Moreover, this activity force us each time to a new formulation of what we want to express, force us to new tries, constant search of new methods. The constant links with youth help us to always have a youthful outlook, make us able to surprise ourselves constantly.(…) I got married in 1977; my wife is an interior architect. Our little daughter, born in 1978, is called Anne.

(…)

Space always intrigued me, with its incredibly rich possibilities, space alteration by (architectural) objects, objects transformation in space (sculpture, design), movement in space and in time, their correlation, their repercussion on mankind, the relation between man and space, the object and time. I think the CUBE arose from this interest, from this search for expression and for this always more increased acuteness of these thoughts.(…) I love playing, I admit it, I particularly games where the partner, the real opponent is nature itself, with its really particular but decipherable mysteries. The most exciting game for me is the space game, the search of possible space shapes, that is to say the logical and concrete building of various layouts.

Of course you can’t give the exact time of an idea’s birth, it seems impossible to me, to me particularly, for who time, from this point of view, has very little interest. It can have been in spring, 1974 that the idea came to me, as a noteworthy possibility. I have a nature attached to experiences, so, since the beginning, I studied variations of a 2×2x2 cube. I was immediately struck by the wealth that could be sensed only from this start. The final technical solution, which is the simplest form 3×3x3, the most easily workable in models, after a few tries, came to me toward the end of autumn, 1974. Several models ready to work were made for me and my friends, it has been exciting to play with them for the first time. We were all surprised to discover gradually that we had made something original, new.
The question of the patent of the invention was immediately raised, so I began the necessary process the 30 January, 1975. Almost at the same, sensing something of the importance of the invented game, of its possibilities and of its real value, I began searching for a partner for the manufacturing and by an extraordinary chance, I indeed found one. The following is rather simple : after its launch on the market (1977) the game became, rapidly and as if by magic, very popular in Hungary, then, from 1980 in the whole world. I feel like the story only begins, and that we can’t predict the end, as well as nobody, I think, could have guessed its future.

London, 31 January, 1981,
Ernö RUBIK

Translated from Hungarian to French by Paul G. Bozsoky and from French to English by me.
Source : "Réussir le Rubik’s Cube"(Succeed in solving the Rubik’s Cube) by André Warusfel
Editions Denoël, 1981

Rubik’s Cube History

May 26th, 2005 by rubikscubesolverssociety

Every
invention has an official birth date. For the Cube this date is 1974 when the
first working prototype came into being and a patent application was initially
drafted. The place was
Budapest, the capital of Hungary. The inventor’s name is now a
household word, Rubik’s Cube.

Although 1974 marks the
inauguration of the Cube, the processes that led to the invention began a few
years earlier. At the time, Erno Rubik was a lecturer at the Department of
Interior Design at the
Academy of Applied Arts and Crafts in Budapest.

He had a
passionate interest in geometry, in the study of 3D forms, in construction and
in exploring the hidden possibilities of combination of forms and materials,
not just in theory, but also in practice.

In the course of his
teaching, Erno Rubik preferred to communicate his ideas by the use of actual
models, made from paper, cardboard, wood or plastic, challenging his students
to experiment by manipulating clearly constructed and easily interpreted forms.
It was the realisation that even the simplest elements, cleverly duplicated and
manipulated, yield an abundance of multiple forms that was the first step on
the long road that led finally to the Cube.

When the Cube was complete,
Erno Rubik demonstrated it to his students and let some of his friends play
with it. The effect was instantaneous. Once somebody laid his hands on the Cube
it was difficult to get it back! The compulsive interest of friends and
students in the Cube caught its creator completely by surprise and it was
months before any thought was given to the possibility of producing it on an
industrial scale.

During 1978, without any
promotion or publicity, the Cube began very slowly to make its way through the
hands of fascinated youths into homes, playgrounds and schools. The word of
mouth spread and by the beginning of 1979 there were enthusiastic circles of
Cube devotees in various parts of
Hungary.

Undeterred by the universal
rejection, spurred on by his firm belief in the exceptional quality of the toy,
Tom Kremer, now armed with a convincing marketing plan, continued his search
for a viable partner. After many disappointments, he succeeded in persuading
Stewart Sims, Vice President of Marketing of the Ideal Toy Corporation, to come
to
Hungary, to see with his own eyes the Cube
in play. It was now September 1979, by which time the Cube has gained a
sufficient degree of popularity to be seen occasionally in the street, on
trams, in the cafes, each time in the hand of someone turning and twisting and
completely absorbed. After five days of convoluted negotiations between a
sceptical American capitalist and an obstinate communist organization largely
ignorant of the operation of a free market, with Laczi and Kremer holding
desperately the two sides together, an order for one million cubes was signed
amidst much handshaking and great relief all round.

The challenge of trying to
master the Cube, to be able to restore all of its six sides to the original
colours seemed to have a mesmeric effect on an amazing variety of individuals
right across age, occupation, wealth and social standing. Grandmothers, bank
managers, baseball players, pilots, librarians, park attendants could be seen
working away at their Cubes at any hour of the day. In restaurants the Cube
would feature on tables side by side with salt and pepper pots, handled with
greater frequency than either. But it was the young, schoolboys and students,
who were in the vanguard of what was fast becoming a massive movement that
swept through the world. They were the ones who proved most adept at solving
the puzzle, they were the ones to form special cubists clubs, to organise
competitions, to suffer from Rubik’s wrist playing continuously for hours and
days with an object that simply could not put down.

But now, in its second
incarnation, the Cube is part of a family of puzzles and games which bear the
stamp of the genius who created the greatest three dimensional puzzle the world
has ever known.

Erno Rubik has not changed
much over the years. Working closely with Seven Towns, he is still deeply
engaged in creating new games and puzzles, and remains one of the principal
beneficiaries of what proved to be a spectacularly successful invention.