Rubik’s Cube

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.

Number

The more I pushed Pica for facts and figures, the more reluctant he was to provide them. I became exasperated. It was unclear if underlying his responses was French intransigence, academic pedantry or simply a general contrariness. I stopped my line of questioning and we moved on to other subjects. It was only when, a few hours later, we talked about what it was like to come home after so long in the middle of nowhere that he opened up. ‘When I come back from Amazonia I lose sense of time and sense of number, and perhaps sense of space,’ he said. He forgets appointments. He is disoriented by simple directions. ‘I have extreme difficulty adjusting to Paris again, with its angles and straight lines.’ Pica’s inability to give me quantitative data was part of his culture shock. He had spent so long with people who can barely count that he had lost the ability to describe the world in terms of numbers.

**

It is Pica’s belief that understanding quantities approximately in terms of estimating ratios is a universal human intuition. In fact, humans who do not have numbers—like Indians and young children—have no alternative but to see the world in this way. By contrast, understanding quantities in terms of exact numbers is not a universal intuition; it is a product of culture. The precedence of approximations and ratios over exact numbers, Pica suggests, is due to the fact that ratios are much more important for survival in the wild than the ability to count. Faced with a group of spear-wielding adversaries, we needed to know instantly whether there were more of them than us. When we saw two trees we needed to know instantly which had more fruit hanging from it. In neither case was it necessary to enumerate every enemy or every fruit individually. The crucial thing was to be able to make quick estimates of the relevant amounts and compare them, in other words to make approximations and judge their ratios.

**

There are tribes whose only number words are ‘one’, ‘two’ and ‘many’. The Munduruku, who go all the way up to five, are a relatively sophisticated bunch.

Numbers are so prevalent in our lives that it is hard to imagine how people survive without them. Yet while Pierre Pica stayed with the Munduruku he easily slipped into a numberless existence. He slept in a hammock. He went hunting and ate tapir, armadillo and wild boar. He told the time from the position of the sun. If it rained, he stayed in; if it was sunny, he went out. There was never any need to count.

**

In 1992, Karen Wynn, at the University of Arizona, sat a five-month-old baby in front of a small stage. An adult placed a Mickey Mouse doll on the stage and then put up a screen to hide it. The adult then placed a second Mickey Mouse doll behind the screen, and the screen was then pulled away to reveal two dolls. Wynn then repeated the experiment, this time with the screen pulling away to reveal a wrong number of dolls: just one doll or three of them. When there were one or three dolls, the baby stared at the stage for longer than when the answer was two, indicating that the infant was surprised when the arithmetic was wrong. Babies understood, argued Wynn, that one doll plus one doll equals two dolls.

 The Mickey experiment was later performed with the Sesame Street puppets Elmo and Ernie. Elmo was placed on the stage. The screen came down. Then another Elmo was placed behind the screen. The screen was taken away. Sometimes two Elmos were revealed, sometimes an Elmo and an Ernie together and sometimes only one Elmo or only one Ernie. The babies stared for longer when just one puppet was revealed, rather than when two of the wrong puppets were revealed. In other words, the arithmetical impossibility of 1 + 1 = 1 was much more disturbing than the metamorphosis of Elmos into Ernies. Babies’ knowledge of mathematical laws seems much more deeply rooted than their knowledge of physical ones.

The Swiss psychologist Jean Piaget (1896–1980) argued that babies build up an understanding of numbers slowly, through experience, so there was no point in teaching arithmetic to children younger than six or seven. This influenced generations of educators, who often preferred to let primary-age pupils play around with blocks in lessons rather than introduce them to formal mathematics. Now Piaget’s views are considered outdated. Pupils come face to face with Arabic numerals and sums as soon as they get to school.

Dot experiments are also the cornerstone of research into adult numerical cognition. A classic experiment is to show a person dots on a screen and ask how many dots he or she sees. When there are one, two or three dots, the response comes almost instantly. When there are four dots, the response is significantly slower, and with five slower still.

So what! you might say. Well, this probably explains why in several cultures the numerals for 1, 2 and 3 have been one, two and three lines, while the number for 4 is not four lines. When there are three lines or fewer we can tell the number of lines straight away, but when there are four of them our brain has to work too hard and a different symbol

Indian-Arabic Numerals

Owing to its ease of use, the Indian method spread to the Middle East, where it was embraced by the Islamic world, which accounts for why the numerals have come to be known, erroneously, as Arabic. From there they were brought to Europe by an enterprising Italian, Leonardo Fibonacci, his last name meaning ‘son of Bonacci’. Fibonacci was first exposed to the Indian numerals while growing up in what is now the Algerian city of Béjaïa, where his father was a Pisan customs official. Realizing that they were much better than Roman ones, Fibonacci wrote a book about the decimal place-value system called the Liber Abaci, published in 1202. It opens with the happy news:

The nine Indian figures are:

9     8     7     6     5     4     3     2     1

With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written, as will be demonstrated.

More than any other book, the Liber Abaci introduced the Indian system to the West. In it Fibonacci demonstrated ways to do arithmetic that were quicker, easier and more elegant than the methods the Europeans had been using. Long multiplication and long division might seem dreary to us now, but at the beginning of the thirteenth century they were the latest technological novelty.

Not everyone, however, was convinced to switch immediately. Professional abacus operators felt threatened by the easier counting method, for one thing. (They would have been the first to realize that the decimal system was essentially the abacus with written symbols.) On top of that, Fibonacci’s book appeared during the period of the Crusades against Islam, and the clergy was suspicious of anything with Arab connotations. Some, in fact, considered the new arithmetic the Devil’s work precisely because it was so ingenious. A fear of Arabic numerals is revealed through the etymology of some modern words. From zephyr came ‘zero’ but also the Portuguese word chifre, which means ‘[Devil] horns’, and the English word cipher, meaning ‘code’. It has been argued that this was because using numbers with a zephyr, or zero, was done in hiding, against the wishes of the Church.

In 1299 Florence banned Arabic numerals because, it was said, the slinky symbols were easier to falsify than solid Roman Vs and Is. A 0 could easily become a 6 or 9, and a 1 morph seamlessly into a 7. As a consequence, it was only around the end of the fifteenth century that Roman numerals were finally superseded, though negative numbers took much longer to catch on in Europe, gaining acceptance only in the seventeenth century, because they were said to be used in calculations of illegal money-lending, or usury, which was associated with blasphemy. In places where no calculation is needed, however, such as legal documents, chapters in books and dates at the end of BBC programmes, Roman numerals still live on.

With the adoption of Arabic numbers, arithmetic joined geometry to become part of mathematics in earnest, having previously been more of a tool used by shopkeepers, and the new system helped open the door to the scientific revolution.

Non-Euclidean Geometry

Gauss’s final contribution to research on the fifth postulate came shortly before he died, when, already seriously ill, he set the title for the probationary lecture of one of his brightest students, 27-year-old Bernhard Riemann: ‘On the hypotheses that lie at the foundations of geometry’. The cripplingly shy son of a Lutheran pastor, Riemann at first had some kind of breakdown struggling with what he would say, yet his solution to the problem would revolutionize maths. It would later revolutionize physics too, since his innovations were required by Einstein to formulate his general theory of relativity.

Riemann’s lecture, given in 1854, consolidated the paradigm shift in our understanding of geometry resulting from the fall of the parallel postulate by establishing an all-embracing theory that included the Euclidean and non-Euclidean within it. The key concept behind Riemann’s theory was the curvature of space. When a surface has zero curvature, it is flat, or Euclidean, and the results of The Elements all hold. When a surface has positive or negative curvature, it is curved, or non-Euclidean, and the results of The Elements do not hold.

The simplest way to understand curvature, continued Riemann, is by considering the behaviour of triangles. On a surface with zero curvature, the angles of a triangle add up to 180 degrees. On a surface with positive curvature, the angles of a triangle add up to more than 180 degrees. On a surface with negative curvature, the angles of a triangle add up to less than 180 degrees.

A surface with negative curvature is called hyperbolic. So, the surface of a Pringle is hyperbolic. The Pringle, however, is only an hors d’oeuvre in understanding hyperbolic geometry since it has an edge. Show a mathematician an edge and he or she will want to go over it.

Vedic Mathematics

In India in those days [shortly after Independence] there was a strong feeling that we needed to get back [from the British] what we lost by hook or by crook. It was mostly in terms of artefacts, stuff that the British might have taken away. Because we lost such a lot, I thought we should have the equivalent back of what we lost.

‘Vedic Mathematics is a misguided attempt to claim arithmetic back for India.’

Randomness and IPod

The human brain finds it incredibly difficult, if not impossible, to fake randomness. And when we are presented with randomness, we often interpret it as non-random. For example, the shuffle feature on an iPod plays songs in a random order. But when Apple launched the feature, customers complained that it favoured certain bands because often tracks from the same band were played one after another. The listeners were guilty of the gambler’s fallacy. If the iPod shuffle were truly random, then each new song choice is independent of the previous choice. As the coin-flipping experiment shows, counterintuitively long streaks are the norm. If songs are chosen randomly, it is very possible, if not entirely likely, that there will be clusters of songs by the same artist. Apple CEO Steve Jobs was totally serious when he said, in response to the outcry: ‘We’re making [the shuffle] less random to make it feel more random.’

Abacus

Until the availability of cheap calculators in the 1980s abacuses were commonly seen on shop counters from Moscow to Tokyo. In fact, during the transition between the manual and electronic eras, a product combining both calculator and abacus was sold in Japan. Addition is usually faster on the abacus since you get your answer as soon as you input the numbers. With multiplication the electronic calculator gives you a slight speed advantage.

“Abacus use has dropped in Japan since the 1970s when, at its peak, 3.2 million pupils a year sat the national soroban proficiency exam. Yet the abacus still remains a defining aspect of growing up, a mainstream extra-curricular activity like swimming, violin or judo. Abacus training, in fact, is run like a martial art. Levels of ability are measured in dans, and there is a competitive structure of local, provincial and national competitions. One Sunday I went to see a regional event. Almost 300 children, aged between 5 and 12, sat at desks in a conference hall with an array of special soroban accessories, like sleek abacus bags. An announcer stood at the front of the hall and dictated, with the intonation of an impatient muezzin, numbers to be added, subtracted or multiplied. It was a knock-out competition that lasted several hours. A chorus of military brass-band music was pumped through the sound system when the trophies—each with a winged figure holding an abacus aloft—were presented to the victors.

At Miyamoto’s school he introduced me to one of his best pupils. Naoki Furuyama, aged 19, is a former national soroban champion. He was dressed casually, with a light checked shirt over a black T-shirt, and seemed a relaxed and well-adjusted teenager—certainly not the cliché of a socially awkward übergeek. Furuyama can multiply two six-digit numbers together in about four seconds, which is about as long as it takes to say the problem. I asked him what the point was of being able to calculate so fast, since there is no need for such skills in daily life. He replied that it helped his powers of concentration and self-discipline. Miyamoto was standing with us, and he interrupted. What was the point of running 26 miles, he asked me? There was never any need to run 26 miles, but people did it as a way of pushing human performance to the limit. Likewise, he added, there was a nobility in training one’s arithmetical brain as far as one could.

Some parents send their children to abacus club because it is a way to improve school maths results. But that does not completely explain the abacus’s popularity. Other after-school clubs provide more targeted maths tuition—Kumon, for example, a method of ploughing through worksheets that started in Osaka in the early 1950s, is now followed by more than four million children around the world. Abacus club is fun. I saw that in the faces of the pupils at Miyamoto’s school. They clearly enjoyed their dexterity at flicking the beads with speed and precision. The Japanese heritage of the soroban generates national pride. Yet the real joy of the abacus, I thought, is more primal: it has been used for thousands of years and, in some cases, is still the fastest way to do sums.

What is Math?

“Maths is not sums, calculations and formulae. It is pulling things apart to understand how things work.” 

Colin Wright, Professional juggler