Alex’s Adventures in Numberland (44 page)

Demaine is tall and skinny, with a fluffy beard and fuzzy dark-blond ponytail. On to a big screen behind him he projected an image of the Haberdasher’s Puzzle. He said that he had recently decided to attack the problem with his PhD students. ‘I didn’t believe it was true,’ he said. Contrary to his expectation, however, he and his students found that you
can
transform any polygon to any other polygon of equal area through a Haberdasher’s Puzzle-style hinged dissection. The hall started clapping – a rare occurrence in the upper reaches of computational geometry. But in puzzle land this was about as exciting a breakthrough as you can get – the solution to an iconic problem by one of the cleverest minds of his generation.

The Atlanta conference, called the Gathering for Gardner, was the most appreciative audience possible for Demaine’s talk. The Gathering is the world’s premier jamboree for mathematicians, magicians and puzzle people. It is a biannual homage to the man who revolutionized recreational mathematics in the second half of the last century. Martin Gardner, now 93 years old, wrote a monthly maths column in
Scientific American
between 1957 and 1981. This was a period of great scientific advances – space travel, information technology and genetics – yet it was Gardner’s lively and lucid prose that really caught readers’ imaginations. His column covered subjects from board games to magic tricks, from numerology to early computer games, and often ventured into tangential areas such as linguistics and design. ‘I thought [Gardner] had a playful respect for mathematics that is often lost in mathematical circles,’ Demaine told me when I spoke to him after his talk. ‘People tend to be too serious. My aim is to make everything I do fun.’

As a boy, Demaine was introduced to Gardner’s columns through his father, a glass-blower and sculptor. The Demaines, who often publish mathematical papers together, embody Gardner’s interdisciplinary spirit. Erik is a pioneer of computational origami, a field both mathematical and artistic, and some of the Demaines’ origami models have even been exhibited in New York’s Museum of Modern Art. Demaine considers maths and art parallel activities, which share an ‘aesthetic about simplicity and beauty’.

In Atlanta Demaine didn’t explain the details of his proof of the universality of Haberdasher’s Pzzle-style dissections to the audience, but he did say that dissecting one polygon so it can be rearranged and hinged to form another polygon isn’t always pretty – and will often be completely impracticable. Demaine is now applying his theoretical work on hinged dissections to make robots that can transform from one shape into another through folding – just like the heroes of the comic book and movie franchise
Transformers
, where robots morph into different types of machine.

The conference was the eighth Gathering for Gardner, or G4G, and its logo, designed by Scott Kim, is known as an inversion, or ambigram.

If you turn it upside-down, it reads exactly the same. Kim, a computer scientist turned puzzle-designer, invented this style of symmetrical calligraphy in the 1970s. Ambigrams do not have to be the same when rotated 180 degrees – any symmetry, or concealed writing, will do.

The writer Isaac Asimov called Kim ‘the Escher of the alphabet’, comparing him to the Dutch artist who played with perspective and symmetries to create self-contradictory images, most famously steps that appear to rise and rise until they reach where they began. Another similarity between Escher and Kim is that their work first reached a mass audience thanks to Martin Gardner.

Ambigrams were independently, and contemporaneously, conceived by the typographer and artist John Langdon. Mathematicians especially love this type of lettering since it is a witty take on their own search for patterns and symmetry. The author Dan Brown was introduced to ambigrams through his father Richard Brown, a maths teacher. Dan Brown commissioned Langdon to design the phrase Angels & Demons as an ambigram for his bestselling book of the same title, and named the lead character Robert Langdon in his honour. Langdon reappeared as the hero of
The Da Vinci Code
and
The Lost Symbol
. Ambigrams have also found a new niche – as body art. The quasi-gothic flourishes, often added to aid symmetry, together with the mystic energy of reading a name backwards and forwards, or upside-down and the right way up, coincides perfectly with the aesthetics of tattoos.

At the G4G it was impossible not to think that maths wards off the onset of dementia. Many of the guests were over 70 – some were in their eighties and even nineties. For more than half a century Gardner corresponded with thousands of readers, many of them famous mathematicians, and some became close friends. Raymond Smullyan, aged 88, is the world’s foremost expert on logical paradoxes. He began his talk: ‘Before I begin speaking, there is something I want to say.’ Willowy and charmingly scruffy, with flowing white hair and a feathery beard, Smullyan was frequently entertaining guests on the hotel piano. He also performed magic tricks on unsuspecting passers-by, and over dinner one evening brought the house down with a stand-up comedy routine.

In this tattoo designed by Mark Palmer, Angel becomes Devil when upside-down.

Aged 76, Solomon Golomb was less physically energetic than Smullyan but able to converse without talking in paradoxes. A soft-spoken grandfatherly figure, Golomb has made important discoveries in space communications, mathematics and electrical engineering. With the helping hand of Martin Gardner, he has also contributed to global pop culture. Early in his academic career Golomb came up with the idea of polyominoes, which are dominoes made out of more than two squares. A triomino is made of three, a tetromino out of four, and so on. An early Gardner column on how they fit together caused such international interest that Golomb’s book,
Polyominoes
, was translated into Russian, where it became a bestseller. One fan made a game that involved falling tetrominoes. That game, Tetris, became one of the world’s most enduring and best-loved computer games. Golomb, of course, has played Tetris for no longer than half an hour.

Another attendee, Ivan Moscovich, is the spitting image of an elderly Vincent Price. Impeccably dressed in a sharp dark suit, he had sparkling eyes, a pencil moustache and full head of brushed-back grey hair. For Moscovich, the attraction of puzzles is the creative thinking they require. He was born in what is now Serbia, and during the Second World War was interned at both Auschwitz and Bergen-Belsen. He believes he survived because of an innate creativity – he was continuously creating situations that ended up saving him. After the war, he turned into a workaholic puzzle-inventor. He likes to think constantly outside the box, to sidestep the inevitable. The motivation to continually come up with new ideas, he said, was an after-effect of the trauma of his own lucky escape.

Moscovich has had about 150 puzzles licensed and produced over the last half century, and compiled a book of puzzles that has been hailed as the greatest collection since the era of Loyd and Dudeney. Now 82, he clutched his latest creation: a sliding block puzzle called You and Einstein. The idea of the game is to slide blocks around a square grid to create a picture of Einstein. Moscovich’s clever twist is that each block has a slanted mirror that reflects the box to its side, meaning that what you think is the block is actually the reflection of something else. Moscovich told me he was excited that You and Einstein could be a global success.

The dream of Moscovich, like everyone in his industry, is, of course, to discover a new puzzle craze. There have been only four international puzzle crazes with a mathematical slant: the tangram, the Fifteen puzzle, the Rubik’s Cube and Sudoku. So far, the Cube has been the most lucrative. More than 300 million have been sold since Ernö Rubik came up with the idea in 1974. Apart from its commercial success, the gaudily coloured cube is a popular-culture evergreen. It is the nonpareil of puzzledom and, unsurprisingly, its presence was felt at the 2008 G4G. A talk on the Rubik’s Cube in four dimensions drew huge rounds of applause.

The original Rubik’s Cube is a 3 × 3 × 3 array made up of 26 smaller cubes, or cubies. Each horizontal and vertical ‘slice’ can be rotated independently. Once the pattern of the cubies is jumbled, the aim of the puzzle is to twist the slices so that each side of the cube has cubies of just one colour. There are six colours, one for each side. Moscovich told me Ernö Rubik was doubly brilliant. Not only was the idea of the cube a stroke of genius, but the way he made the blocks fit together was an outstandingly clever piece of engineering. When you dismantle a Rubik’s Cube there is no separate mechanical device holding it all together – each cubie contains a piece of a central, interlocking sphere.

As an object, the cube itself is sexy. It is a Platonic solid, a shape that has had iconic, mystical status since at least the ancient Greeks. The brand name was also a dream: catchy, with delicious assonance and consonance. The Rubik’s Cube had an Eastern exoticism too, not from Asia this time but from Cold War Eastern Europe. It sounded a lot like Sputnik, the original showpiece of Soviet space technology.

Another ingredient in its success was the fact that while solving the cube was not easy, the challenge did not put people off. Graham Parker, a builder from Hampshire, kept at it for 26 years until he achieved his dream. ‘I have missed important events to stay in and solve it and I would lay awake at night thinking about it,’ he said, after an estimated 27,400 hours of cube time. ‘When I clicked that last bit into place and each face was a solid colour I wept. I cannot tell you what a relief it was.’ Those who solved it over a more manageable period invariably wanted to solve it again, but quicker. Reducing one’s Rubik’s record became a competitive sport.

Speedcubing has only really taken off, however, since around 2000. One of the reasons is thanks to a sport even more quirky than the timed solving of mechanical puzzles. Speedstacking is the practice of stacking plastic cups in set patterns as fast as you can. It is both mesmerizing and awesome – the top stackers move so fast it is as if they are painting the air with plastic. The sport was invented in California in the 1980s as a way of improving children’s hand-eye coordination and general fitness. It is claimed that 20,000 schools worldwide now include it in their physical education curriculum. Speedstacking uses specialized mats that have a touch sensor linked to a stopwatch. The mats provided the speedcubing community for the first time with a standardized method to measure the time it takes to solve the cube, and are now used in all competitions.

Every week or so, somewhere around the world now hosts an official speedcubing tournament. To make sure that the starting position is sufficiently difficult in these competitions, the regulations stipulate that cubes must be scrambled by a random sequence of moves generated by a computer program. The current record of 7.08 seconds was set in 2008 by Erik Akkersdijk, a 19-year-old Dutch student. Akkersdijk also holds the record for the 2 × 2 × 2 cube (0.96secs), the 4 × 4 × 4 cube (40.05secs) and the 5 × 5 × 5 cube (1min 16.21 secs). He can also solve the Rubik’s Cube with his feet – his time of 51.36secs is fourth-best in the world. However, Akkersdijk really must improve his performance at solving the cube one-handed (33rd in the world) and blindfolded (43rd). The rules for blindfolded solving are as follows: the timer starts when the cube is shown to the competitor. He must then study it, and put on a blindfold. When he thinks it is solved he tells the judge to stop the stopwatch. The current record of 48.05secs was set by Ville Seppänen of Finland in 2008. Other speedcubing disciplines include solving the Rubik’s Cube on a rollercoaster, under water, with chopsticks, while idling on a unicycle, and during freefall.

The most mathematically interesting cube-solving category is how to solve it in the fewest moves possible. The contestant is given an officially scrambled cube and has 60 minutes to study the position before describing the shortest solving sequence he can come up with. In 2009 Jimmy Coll of Belgium claimed the world record: 22 moves. Yet this was just how many moves a very smart human needed to solve a jumbled-up cube after 60 minutes of thinking about it. Might he have been able to find a solution from the same configuration in a smaller number of moves if he had had 60 hours? The question that has most intrigued mathematicians about the Rubik’s Cube is this: what is the smallest number of moves,
n
, such that every configuration can be solved in
n
moves or less? As a mark of reverence,
n
in this instance is nicknamed ‘God’s number’v hei>