Alex Bellos, Author at ��è��Ƶ

Test yourself against these superpuzzles from Japan

Alex Bellos — Tue, 19 Dec 2017 18:00:00 +0000

How much brain power have you got left after floating on a sea of indulgence this holiday? Find out with some ninja puzzles from Alex Bellos

Mathematics has a bad hair day

Alex Bellos — Mon, 10 Jul 2017 08:00:35 +0000

Seaweed? Worms? Bad hair day? The animation below is a new way to look at the Collatz conjecture, one of the most high-profile open problems in mathematics.

The conjecture is easy to explain, given that all you need to know is how to add 1, how to divide by 2, and how to multiply by 3. Such simplicity, however, stands in striking contrast to the difficulty of proving the conjecture itself. “Mathematics is not yet ready for such problems,” the late Paul Erdös said, and the animation gives an insight into why it is such a tough nut to crack.

To begin: take a number, any number, and apply this rule:

If the number is even, halve it;
If the number is odd, triple it, add one, and halve it;
Repeat.

Let’s get going with 13:

13 is odd, so we triple it to get 39, add one to get 40, and halve it to get 20;
Now we repeat with 20, which is even, so we halve it to get 10;
10 is even, so we halve it again to get 5;
5 is odd, so we triple it and add one, halve the result, and get 8.

The full sequence from 13 looks like this:

13 → 20 → 10 → 5 → 8 → 4 → 2 → 1

The Collatz conjecture is the statement, first made by the German mathematician Lothar Collatz in 1937, that whatever number you start with, by applying these rules you will always, eventually, end up at 1.

Most mathematicians believe he was correct. Computers have checked it out up to pretty high numbers and found no counterexamples – but no one has a clue how to prove it.

The left-hand image below shows the sort of tree diagram you get in books or papers that discuss the conjecture. For each number in the tree – look at the 13, for example – the numbers along the branch leading downwards are those in its Collatz sequence. Since all numbers investigated so far reach 2 (and then 1), all the numbers are branches in the same tree.

To illustrate why the problem is so complicated, mathematical artist Edmund Harriss at the University of Arkansas in Fayetteville took the standard diagram, but added a rule for the direction of the branch (illustrated above, centre), to reflect which half of the algorithm was operating to generate the number.

This wrinkle reveals how unpredictable the path can be from a given number to 1. Whichever number you are at, if the one above it in the tree is double that value, the direction of the branch turns at a fixed clockwise angle – and if not, it turns at a fixed anticlockwise angle. In the right-hand image, the numbers have been taken out and the lines thickened.

The animation shows how the tree grows when it includes every number under 10,000. It displays brilliantly how a simple, orderly rule can create havoc: the structure looks organic and wild. Harriss’s image makes us understand instantly why solving the Collatz conjecture is so hard.

The image of the Collatz conjecture comes from Visions of Numberland: A colouring journey through the mysteries of maths, by Alex Bellos and Edmund Harriss (published in the US as Visions of the Universe)

Mathematics ‘Nobel’ rewards boundary-busting work

Alex Bellos — Thu, 19 Aug 2010 15:37:00 +0000

The Fields medal

Elon Lindenstrauss was awarded the 2010 Fields medal for his results on measure rigidity in ergodic theory, and their applications to number theory

Cédric Villani was awarded the medal for his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation

Stanislav Smirnov was awarded the medal for the proof of conformal invariance of percolation and the planar Ising model in statistical physics

Ngô Bảo Châu was awarded the medal for for his proof of the fundamental lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods

Four mathematicians this morning received the most prestigious award in their subject, the Fields medal, for work that helps unify the traditionally disparate fields of pure and applied mathematics.

The Fields medal is widely considered the mathematics equivalent of the Nobel prize and is awarded by the International Mathematical Union on the opening day of the , which takes place every four years.

The last time the event was held, in 2006, it was somewhat overshadowed by winner Grigori Perelman’s refusal to accept his prize for solving the Poincaré conjecture.

ICM 2010 kicked off today in Hyderabad, India, with the Indian president Prathibha Patil awarding the prizes. This time, all four medallists accepted, and remarked that their work helps demonstrate how the traditional distinction between pure and applied maths no longer holds.

“The most interesting problems are on both sides of the division,” said one medallist, , of the Hebrew University, Jerusalem.

From ergodic theory to soil

Aside from Lindenstrauss, this year’s winners were of the University of Paris-South, France, of the University of Geneva, Switzerland, and of the Henri Poincaré Institute, Paris, France.

In its explanation of its decision, the International Mathematical Union praised Lindenstrauss for his work on ergodic theory, which concerns dynamical systems that change over time and which originated as a branch of applied mathematics. His results have yielded important results in the completely different field of number theory, a branch of pure mathematics.

Ngô won for his proof of the fundamental lemma, a conjecture posed by of the Institute of Advanced Study in Princeton, New Jersey. The conjecture is part of the “Langlands programme”, a collection of conjectures that aim to unify certain elements of algebra with number theory. US magazine Time included Ngô’s proof in its “.

Smirnov’s area is mathematical physics. His award was given for highly theoretical work that touches on very practical problems, such as how water flows through soil.

Villani’s work is also related to physics, in particular the mathematical interpretation of the concept of entropy. He has applied this to solve long-standing problems, such as how fast the motions of gas particles converge to equilibrium.

Youth and age

The Fields medal is only awarded to mathematicians under 40, but that does not mean mathematicians only do their best work when young, said Smirnov. “I look forward to proving more theorems. I hope the weight of this prize doesn’t slow me down.”

When asked why mathematics has been so successful at finding applications in the real world, Villani – sporting a burgundy silk cravat and a palm-sized spider brooch – said: “It is a very pleasant mystery. Let’s continue to enjoy it and explore.”

Also presented at the ICM today were three other awards, including the first ever Chern medal, for outstanding lifetime achievement in mathematics, which went to Louis Nirenberg, aged 85, of the Courant Institute of Mathematical Sciences in New York.

Meanwhile, the Nevanlinna prize, for contributions to information sciences, also awarded only to under-40s, was given to of Yale University. The Gauss prize for applied maths was won by Yves Meyer, aged 71, of the École Normale Supérieure de Cachan, France

Each Field’s winner receives a cash prize of CAN$15,000, which is a much smaller sum than the US$1 million or so given to Nobel laureates, or even the amount received by winners of the Chern medal, who get US$250,000 for themselves and another US$250,000 to donate to organisations that support mathematics.

Both the Nevanlinna and the Gauss prizes are worth €10,000 to their winners.

Magic numbers: A meeting of mathemagical tricksters

Alex Bellos — Mon, 24 May 2010 12:44:00 +0000

Getting together for some mathematical fun Gallery: Mathemagical visions: a Gathering for Gardner album Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?” The event was the earlier this year, a convention held every two years in Atlanta, Georgia, uniting mathematicians, magicians and puzzle enthusiasts. The audience was silent as they pondered the question. “The first thing you think is ‘What has Tuesday got to do with it?'” said Foshee, deadpan. “Well, it has everything to do with it.” And then he stepped down from the stage. The gathering is the world’s premier celebration of recreational mathematics. Foshee’s “boy born on a Tuesday” problem is a gem of the genre: easy to state, understandable to the layperson, yet with a completely counter-intuitive answer that can leave you with a smile on your face for days. If you have two children, and one is a boy, then the probability of having two boys is significantly different if you supply the extra information that the boy was born on a Tuesday. Don’t believe me? We’ll get to the answer later. As a melting pot of outside-the-box abstract thinking, this gathering is one of a kind. Attendees were invited to make the world’s first out of balloons, shown how to solve the Rubik’s cube while blindfolded and given tips on how to place a lemon under a handkerchief without anyone knowing. The 300 guests included magicians, origamists, artists, maze designers, puzzle writers, toy inventors and cognitive psychologists, as well as some of the world’s most gifted mathematicians.

Origami letters

Erik Demaine, for example, is a former winner of the MacArthur fellowship, aka the “genius award”, and the youngest person in recent years to be made a professor at the Massachusetts Institute of Technology. Now aged 29, he presented to the gathering some typefaces that he had invented as a result of his academic work in computational geometry. The is a font in which each letter can be made from the same linked chain of 128 identical isosceles triangles. (A hinged dissection is a technique in which a large shape is divided up into smaller shapes, linked together with “hinges”, and then refolded into another large shape.) The letters looked distinctive, if a little clunky. Demaine reckoned the typeface had another more serious problem: “It’s way too easy to read!” So he revealed based on his work in origami: the “origami maze font”, for which each letter of the alphabet is the origami fold-pattern that, once folded, would make that letter protrude from the page. When he started to write in this font, and the screen at the front of the hall filled with an impenetrable grid of red and blue lines – red lines are the “mountain” folds and the blue ones the “valley” folds – the audience clapped and guffawed. Recreational mathematics may be the “mathematics of fun” but it often inspires serious science. Hinged dissections were invented by the British puzzle creator a century ago. Though Demaine became interested in them and in origami for purely playful purposes, they have resulted in some of his most important academic research. Hinged dissections form the basis of his designs for reconfigurable robots, in which blocks hinged together in a chain can be made to fold into any three-dimensional shape. The pop-up lettering technique behind his origami font could, he says, be used with paper that can pucker up to form tactile shapes: to make, for example, a map you can read in the dark. “There is no way to predict from the recreational side to the product side,” Demaine says.

Games master

The four-day Gathering for Gardner, or G4G, owes its name to the journalist Martin Gardner, who died as this article was going to press, aged 95. Between 1957 and 1981 Gardner wrote the monthly “Mathematical games” column in Scientific American, which inspired a cult following. A decade after he put down his pen, Atlanta businessman and puzzle collector Tom Rodgers decided to pay tribute to Gardner by organising a gathering in his honour. This year the gathering mostly took place in the windowless conference room of an Atlanta hotel, apart from one afternoon when everyone relocated to Rodgers’s residence on the outskirts of Atlanta. It’s a modern house surrounded by a Japanese garden, containing geometrical sculptures and a pathway based on the . Here the guests are free to indulge in all manner of mathematical recreations: University of Arkansas mathematician led a group in building a Hilbert-space-filling curve out of 320 kilograms of steel; performance artist Caspar Schwabe constructed a “kinetic bamboo hyperboloid”; and composer demonstrated how to make polyhedra out of balloons.

“As a melting pot of outside-the-box abstract thinking, this gathering is one of a kind”

Of the three main professions represented – mathematicians, magicians and “puzzle people” – the magicians were the easiest to spot: they tended to wear black jackets. was a prime example, and he also wore a dandyish black shirt. He was displaying a copy of a puzzle he had invented called Jigazo, currently the top-selling puzzle in Japan. Setteducati is a performing magician who curates the magic side of the G4G. Magic was chosen as a theme because Gardner himself was an amateur magician and his first foray into mathematics writing in the 1950s was a book on the mathematics of magic. The two disciplines are happy bedfellows: both exhibit a wonder at the structure of the world, and seek playful ways to subvert it.

Mathemagics

The G4G facilitates the free exchange of ideas between the magical and mathematical communities. The Jigazo – which will be released in Europe and the US later this year – was a product of this traffic. At a previous G4G, Setteducati met the computer graphics pioneer , and the puzzle is a collaboration between them. The Jigazo is a jigsaw puzzle that can be customised to display any picture you want. It contains 300 identically shaped pieces that are all a different shade of blue, and all have a unique symbol on the back. If you take a portrait of yourself and email the image to the Jigazo website, you will receive an email back with a map of the arrangement of the symbols such that when the pieces are assembled in this way the jigsaw shows your portrait. “What I love about this is the magic behind it,” says Setteducati. “With 300 pieces you can make any face in the world.” In this context, “magic” means a good deal of fiendishly clever number-crunching. Setteducati gave me his business card, which is written in backwards-script and incorporates a mirror to read it the right way round. Judged on the originality of its guests’ business cards, the G4G is a pretty special event: several were origami models or unusual shapes. The card of Pablos Holman, futurist and security expert, was completely black, his details only visible in UV light. Another charming aspect of the gathering is the age range of those present. Many of Gardner’s contemporaries still come, including 93-year-old mathematician Richard Guy and Solomon Golomb, aged 88, whose work on – dominoes with more than two squares – gained prominence thanks to a Gardner column in the late 1950s. (And, many years later, inspired the computer game Tetris, in which falling tetrominoes must be stacked together.) At the opposite end of the age spectrum was , aged 12, a mathematics blogger from San Jose, California.

Reptilian designs

In 1964, Golomb coined the term “reptile” – short for repetitive tiling – to mean any tile that can reproduce a larger copy of itself. An example of this is the L-shaped tile. Carolyn Yackel, who is best known for her book , demonstrated that the number of L-tiles you need to make into a bigger L is always a square: 4, 9, 16 and so on. And she showed a table made of ceramic reptiles that she made with her father. Tiling is a major theme in recreational mathematics – possibly because as well as being conceptually rich it is also aesthetically gratifying. For example, computer scientist Craig Kaplan from the University of Waterloo in Ontario, Canada, spoke of different ways to tiles that morph from one shape to another as they cross the floor. The evolution of tiles, he said, suggested “an inexhaustible source of inspiration for art”. Mathematicians are especially fascinated by sets of tiles that can only produce a pattern that never repeats. Such sets are called “aperiodic” and were discovered in the 1960s and later popularised by Roger Penrose in the 1970s. Aperiodicity is interesting because it is so counter-intuitive, and because if patterns never repeat it means that every tile in an arrangement somehow influences the position of every other tile. Until now, at least two different shapes of tiles were needed, such as Penrose’s famous “kite” and “dart” tiles. Which is why a new design known as the “einstein” tile was such a significant mathematical breakthrough. It is the first aperiodic tile that works on its own, with no need for any other shape. Designed by Joshua Socolar, a physicist from Duke University in Durham, North Carolina, and Joan Taylor from Burnie, Tasmania, Australia, it looks like . Identical einsteins can only fit together without any gaps or overlaps in a non-periodic way.

“Everyone finds probability questions mind-bending, even the world’s best mathematicians”

Perhaps the mathematician whose career most embodies the spirit of recreational mathematics is G4G stalwart John Horton Conway of Princeton University in New Jersey. “I absolutely love this. It’s just so interesting being here with such different people.” Yet Conway’s presence in Atlanta was not merely physical. In 1970 Gardner wrote one of his most widely read columns, on a Conway invention called the . Conway’s “Life” is an example of what is known as a cellular automaton, a grid in which the cells can be either alive (white) or dead (black) and where four simple rules determine what happens to the cells in successive generations. The “game” is to choose a pattern of live cells and see how they evolve. As well as providing something fun for early computer programmers, the wonder of seeing how complexity emerges from such simple beginnings captured many people’s imaginations in the early 1970s.

Life and Stephen Wolfram

The Game of Life seemed to be everywhere at the G4G. California-based mathematician Tom Rokicki showed some new patterns of mind-boggling ingenuity, including one which produced the image of the numeral 3, then the numeral 1, then 4 and continued writing out the digits of pi. William Gosper, who once held the record for discovering pi to the greatest number of decimal places and is considered one of the first computer hackers, was showing new Life forms on his laptop: “The ingenuity is staggering,” he says. Life appeared again in a talk by Stephen Wolfram, the mathematician and software entrepreneur behind the Wolfram Alpha computational engine and the blockbuster software Mathematica. He believes that cellular automata can recreate the universe and is trying to find the ones that do. Wolfram attended the G4G with his wife and two of his four children. Which brings us back to the opening question: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”

Trouble with boys

The first thing to remember about probability questions is that everyone finds them mind-bending, even mathematicians. The next step is to try to answer a similar but simpler question so that we can isolate what the question is really asking. So, consider this preliminary question: “I have two children. One of them is a boy. What is the probability I have two boys?” This is a much easier question, though a controversial one as I later discovered. After the gathering ended, Foshee’s Tuesday boy problem became a hotly discussed topic on blogs around the world. The main bone of contention was how to properly interpret the question. The way Foshee meant it is, of all the families with one boy and exactly one other child, what proportion of those families have two boys? To answer the question you need to first look at all the equally likely combinations of two children it is possible to have: BG, GB, BB or GG. The question states that one child is a boy. So we can eliminate the GG, leaving us with just three options: BG, GB and BB. One out of these three scenarios is BB, so the probability of the two boys is 1/3.

“Judged on the originality of attendees’ business cards, this is a pretty special event”

Now we can repeat this technique for the original question. Let’s list the equally likely possibilities of children, together with the days of the week they are born in. Let’s call a boy born on a Tuesday a BTu. Our possible situations are:

When the first child is a BTu and the second is a girl born on any day of the week: there are seven different possibilities.
When the first child is a girl born on any day of the week and the second is a BTu: again, there are seven different possibilities.
When the first child is a BTu and the second is a boy born on any day of the week: again there are seven different possibilities.
Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a BTu – and this is where it gets interesting. There are seven different possibilities here too, but one of them – when both boys are born on a Tuesday – has already been counted when we considered the first to be a BTu and the second on any day of the week. So, since we are counting equally likely possibilities, we can only find an extra six possibilities here.

Summing up the totals, there are 7 + 7 + 7 + 6 = 27 different equally likely combinations of children with specified gender and birth day, and 13 of these combinations are two boys. So the answer is 13/27, which is very different from 1/3. It seems remarkable that the probability of having two boys changes from 1/3 to 13/27 when the birth day of one boy is stated – yet it does, and it’s quite a generous difference at that. In fact, if you repeat the question but specify a trait rarer than 1/7 (the chance of being born on a Tuesday), the closer the probability will approach 1/2. Which is surprising, weird… and, to recreational mathematicians at least, delightfully entertaining. Gallery:Mathemagical visions: a Gathering for Gardner album

Alex Bellos, Author at ��è����Ƶ