Mathematics | The Knowledge Dynasty


The only way to prove you aren’t a robot is to solve this chess puzzle

Check it out, Mate!

Image: James Tagg

Check out the chess board above: looks wrong, right?

If you’ve ever played chess, you know something’s amiss, here. For one thing, someone chose to exchange a pawn for another bishop instead of a queen. For another, virtually all the action’s moved to the left side of a board.

It’s hard to imagine how the game got here: it’s even harder to imagine what happens next, let alone a scenario in which four white pawns and a white king could play to a draw, or even win this game.

The Penrose Chess Puzzle: Can you find the solution that results in either a white win or a game draw?

Image: James tagg

Yet: scientists at the newly-formed Penrose Institute say it’s not only possible, but that human players see the solution almost instantly, while chess computers consistently fail to find the right move.

“We plugged it into Fritz, the standard practice computer for chess players, which did three-quarters of a billion calculations, 20 moves ahead,” explained James Tagg Co-Founder and Director of the Penrose Institute, which was founded this week to understand human consciousness through physics.

“It says that one-side or the other wins. But,” Tagg continued, “the answer that it gives is wrong.”

Tagg and his co-founder, Mathematical Physicist and professor Sir Roger Penrose who successfully proved that black holes have a singularity in them cooked up the puzzle to prove a point: Human brains think differently.

(Those who figure out the puzzle can send their answers to Penrose to be entered to win the professor’s latest book.)

“Humans can look at a problem like this strange chess board configuration,” said Tagg, “and understand it. What a computer does is brute force calculation, which is different.” This is set up, rather exquisitely, to show the difference, he added.

They forced the computer out of its comfort zone by, at least in part, making an unusual choice: the third bishop.

“All those bishops can move in lots of different ways, so you get computation explosion. To calculate it out would suck up more computing power than is available on earth,” claimed Tagg.

Tagg told us that there is, in fact, a natural way to get to this board configuration.

We’re trying to figure it out here, but lacked an extra black bishop. So we tagged one to keep track.

Image: lance ulanoff/mashable

Sir Richard Penrose’s brother is, according to Tagg, a very strong chess player. He assures me that it’s a position you can get to, but I have not played it through. Question is, is there a rational game that gets you there?

In fact, those who can figure out that second puzzle and get the answer to Penrose, could also receive a free copy of Professor Penrose’s book.

Chess computers fail at Penrose’s chess puzzle because they have a database of end-games to choose from. This board is not, Tagg and Penrose believe, in the computer’s playbook. “We’re forcing the chess machine to actually think about the position, as opposed to cheat and just regurgitate a pre-programmed answer, which computers are perfect at,” said Tagg.

So far, Tagg and the Penrose Institute haven’t heard from any artificial intelligence experts refuting their claims. “I’m quite surprised,” said Tagg.

Mashable has contacted several AI experts for comment and will update this post with their response.

Aside from the fun of solving this puzzle (Tagg said hundreds already have and claim they have done so in seconds), it poses a deeper question: Are we executing some fiendishly clever algorithm in our brain, that cuts through the chaff? It is just a higher level of computation, one that computers can still aspire to or something unique to brain-matter-based thought?

Tagg said Penrose Institute falls into the latter camp.

Penrose and Tagg don’t think you can simply call a brain a machine. It sits in skull, made of gray matter and we don’t understand how it works. “Simply calling it a clever computer, this sort of puzzle shows that it clearly is not,” he said.

You can send your Chess Puzzle solution to the Penrose Institute here:

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The Weather Channel takes on stereotypes in STEM

Female meteorologists have a simple request: Stop labeling us educated, highly trained scientists as “weather girls.” And maybe stop commenting on our dresses and lipstick and start asking about our forecasts.

“Women on television can also be scientists. It’s that simple,” said Ginger Zee, a meteorologist on Good Morning America and ABC World News Tonight.

On Sunday, The Weather Channel (TWC) debuted a two-part series exploring the unique challenges that women face in science, technology, engineering and mathematics (STEM).

In the clip shown here, Zee and two other leading meteorologists Janice Huff of WNBC-New York and Jen Carfagno join Marshall Shepard, host of TWC’s Weather Geek program, to discuss ways to encourage young women to embrace science from an early age.

Left to right: Marshall Shepard, Jen Carfagno, Janice Huff and Ginger Zee on the set of “Women in Science Wx Geeks.”

Image: the weather channel

“More than ever, there are opportunities for girls and women who are interested in science to … find extra help and support along the way,” Carfagno, who hosts TWC’s AMHQ program, told Mashable.

She noted that stereotypes about women in science extend far beyond the 1950s-notion of the well-heeled weather girl. Think about the recurring movie plot, in which a frizzy, bespectacled geek suddenly gets hair gel, contact lenses and stops droning on about all that “dull” science stuff.

“Society still hangs on to that, and it’s a sort of underlying theme I feel that’s hard to shake,” Carfagno said.

Given those lingering cultural perceptions, and the fact that more men work in meteorology than women, “We decided we really need to do a show about women in science,” she said.

Part one of TWC‘s “Wx Geeks” special on women in science airs Jan. 29 at 12 p.m. ET. Part two airs Feb. 5 at the same time.

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Stephen Hawking Fast Facts

(CNN)Here’s a look at the life of the world renowned theoretical physicist, cosmologist, astronomer and mathematician, Stephen Hawking.

Birth date:
January 8, 1942
Birthplace: Oxford, England (grew up in and around London)
Birth name: Stephen William Hawking
Father: Frank Hawking, a doctor and research biologist
Mother: E. Isobel Hawking
Marriages: Elaine Mason (1995-2006, divorce); Jane Wilde (1965-1991, divorce)
Children: with Jane Wilde: Timothy, Lucy and Robert
Education: Oxford University, B.A., 1962; Cambridge University, Ph.D., 1966
Other Facts:
Stephen Hawking’s birthday (January 8, 1942) is the 300th anniversary of the death of astronomer and physicist Galileo Galilei.
He is the 17th Lucasian Professor of Mathematics, an academic chair at Cambridge University. From 1669 to 1702, the position was held by Sir Isaac Newton.
Has guest-starred, as himself, on “The Big Bang Theory,” “Star Trek: The Next Generation” and “The Simpsons.”
Lou Gehrig’s Disease (Amyotrophic Lateral Sclerosis or ALS), is usually fatal after three years. Hawking has survived it for 50 years.
The disease has left him paralyzed and completely dependent on others and/or technology for everything: bathing, dressing, eating, mobility and speech. He’s able to move only a few fingers on one hand.
His speech synthesizer has an American accent.
1963 –
Is diagnosed with the motor neuron disease, amyotrophic lateral sclerosis (ALS).
1966 – Completes doctoral work in theoretical physics, submitting a thesis on black holes.
1970 – Combines the theory of relativity with quantum theory and finds that black holes emit radiation.
1979 – Becomes the 17th Lucasian professor of mathematics at Cambridge University.
1982 – Awarded CBE – Commander of the Order of the British Empire.
1985 – Hospitalized with pneumonia Hawking requires an emergency tracheotomy, causing permanent damage to his larynx and vocal cords. A keyboard operated electronic speech synthesizer is refined and adapted to his wheelchair by David Mason, an engineer married to Elaine Mason, one of Hawking’s nurses (and future wife).
1988 – His book, “A Brief History of Time: From the Big Bang to Black Holes,” is published.
2004 – Reverses the 1966 theory that black holes swallow everything in their path forever and declares that black holes will never support space travel to other universes.
April 26, 2007 – Becomes the first quadriplegic to experience zero gravity, aboard a Zero Gravity Corporation flight.
October 2007 – “George’s Secret Key to the Universe,” the first in a series of children’s books to help explain the universe, written with daughter Lucy is published.
November 30, 2008 – Is appointed by the Perimeter Institute for Theoretical Physics in Waterloo, Ontario to be its first Distinguished Research Chair.
May 19, 2009 – “George’s Cosmic Treasure Hunt,” the second in the series of children’s books written with daughter Lucy, is published.
July 30, 2009 – Is awarded the 2009 Presidential Medal of Freedom by President Barack Obama.
September 30, 2009 – Steps down as Lucasian Professor of Mathematics at Cambridge University after 30 years. Hawking will continue to work at the university.
2009-present – Director of Research at the Institute for Theoretical Cosmology at the Department of Applied Mathematics and Theoretical Physics at Cambridge University.
September 7, 2010 – “The Grand Design,” written with Leonard Mlodinow, is published.
August 28, 2012 – “George and the Big Bang,” the third installment in a series of children’s books written with daughter Lucy, is published.
December 2012 – Wins the Fundamental Physics Prize and is awarded $3 million for his theory on black holes emitting energy.
September 10, 2013 – Hawking publishes “My Brief History,” a biography that looks at his life and the development of his intellect.
June 5, 2014 – “George and the Unbreakable Code” the fourth installment in a series of children’s books written with daughter Lucy, is published.

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First edition of Isaac Newton’s Principia set to fetch $1m at auction

Rare European copy of key mathematics text is going under hammer at Christies in New York with record guide price.

A first edition of Sir Isaac Newton’s Principia Mathematica could become the most expensive print sold of the revolutionary text when it goes under the hammer with a guide price of at least $1m (790,000) this month.

The extremely rare continental copy being sold by auction house Christies in New York is one of a handful of texts thought to have been destined for Europe and has minor differences from those distributed in England by Newton and the book’s editor, Edmond Halley.

The list price of between $1m and $1.5m is thought to be a record for the book. An English version also bound in red morocco leather, which was said to have been presented to King James II, sold for more than $2.5m in 2013. Its list price was $600,000.

About 400 copies of Principia’s first edition were printed, of which the continental versions accounted for about 20%. Halley, the astronomer best known for the comet named after him, encouraged Newton to organise his theories into a text and paid for the printing because the Royal Society of which he and Newton were members had run out of funds.

The society retains two copies of the book, including the original manuscript on which the first print run in 1687 was based, which is described as its greatest treasure.

Written in Latin, the books full title is Philosophi Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). It laid out Newton’s groundbreaking theories in areas such as gravity and the forces of motion, and introduced a more rigorous mathematical method to physical science.

Keith Moore, the head of the Royal Society library, described it as a benchmark in human thought.

“It’s not just the history and development of science; it’s one of the greatest books ever published,” he said. “It was hugely influential in terms of applying mathematics to basic physical problems.”

Moore said the large sum set to be attracted by the book could be in part due to the growing influence of science within culture, as well as the huge earnings of some technology entrepreneurs.

“People who have big books these days maybe are the kinds of people who have made their money on the internet or the web … If you have a few million quid to spend, why wouldnt you buy a copy of Principia Mathematica?

“If you’ve made your money from a really cool algorithm, you will probably appreciate Newtonian physics.”

Despite its wide-ranging impact, and the books use as a foundational physics text being unsurpassed until Einstein’s general theory of relativity, Principia did not make a list last year of the top 20 most important academic books of all time. The list was topped by Charles Darwins On the Origin of Species.

But because it was published almost two centuries earlier, first editions of Principia are rarer and likely to continue selling for far larger amounts. One of the highest prices paid for a first edition of Darwins book laying out the theory of evolution was 103,000 in 2009, and subsequent sales have been lower.

While the prices differ, the impact of the two texts was comparable, Moore said. What Newton does in the 1680’s is revolutionise the physical sciences. The fundamental laws of physics.

Darwin’s great work published in 1859 revolutionised the biological sciences in the same way. They are similar books in the impact they had.

  • The picture caption on this article was amended on 5 December 2016 to clarify that the copy of Principia Mathematica up for sale is not the one held by Cambridge University.

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Kazuo Ishiguro: ‘Were coming close to the point where we can create people who are superior to others’

Social changes unleashed by new technologies could undermine core human values unless we engage with science, warns author.

Imagine a two-tiered society with elite citizens, genetically engineered to be smarter, healthier and to live longer, and an underclass of biologically run-of-the-mill humans. It sounds like the plot of a dystopian novel, but the world could be sleepwalking towards this scenario, according to one of Britain’s most celebrated writers.

Kazuo Ishiguro argues that the social changes unleashed by gene editing technologies, such as Crispr, could undermine core human values.

“We’re going into a territory where a lot of the ways in which we have organised our societies will suddenly look a bit redundant,” he said. “In liberal democracies, we have this idea that human beings are basically equal in some very fundamental way. We’re coming close to the point where we can, objectively in some sense, create people who are superior to others.”

Ishiguro spoke to the Guardian ahead of the opening of a new permanent mathematics gallery at the Science Museum in London, which features a machine to predict coastal storm surges built by his oceanographer father, Shizuo Ishiguro.

The author hopes that the 5 million exhibition, and others like it, will encourage people to engage with the process of science and its future trajectory, rather than simply tuning in for the headline results of research and only then worrying about the implications.

Despite the atom bomb and things like this, were still in the habit of compartmentalising scientific endeavour, he said. Its important that we, as a society, get much more interested in science and maths, that we dont silo it off in our minds … until theres some breakthrough product that turns up.

Ishiguro cites three areas – gene editing, robotics and Artificial Intelligence – that he believes could transform the way we live and interact with each other over the next 30 years.


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We are on the brink of all kinds of discoveries that will completely alter the way we run our lives, said the author, whose 2005 book, Never Let Me Go, imagines a dark future in which human clones are raised to be organ donors.

The gene editing tool, Crispr, allows scientists to cut, paste and delete single letters of the genome with unprecedented precision, meaning aberrant genes can be overwritten with working copies, and, potentially, functional genes replaced with enhanced versions. Chinese scientists are already trialling the technology in patients to treat lung cancer.

“When you get to the point where you can say that person is actually intellectually or physically superior to another person because you have removed certain possibilities for that person getting ill or because they’re enhanced in other ways, that has enormous implications for very basic values that we have,” said Ishiguro.

He also has concerns that in AI and robotics the bulk of intellectual capital lies with the Silicon Valley masters of the universe rather than universities or government-funded labs.

“There are some very powerful and rich people who want to do enormous research in this area,” he said. “Some of them might want to come out with things that are very beneficial, but it’s probably outside of regulation and so, yes, I think society as a whole needs to be more engaged.”

Ishiguros father, an oceanographer originally based in Nagasaki, moved with the family to Guildford, Surrey, to work at the National Institute of Oceanography in 1957, when Ishiguro was five.

Dr Shizuo Ishiguro with his electronic analogue machine, which converted meteorological and ocean data into electrical signals on a series of wire meshes. This allowed the height of storm surges, and where and when they would make coastal impact, to be predicted. Photograph: Image courtesy of NOC Archive.

“Despite the two countries having been at war just a decade previously, the family were made welcome,” he said. “The British people of that era had a very sophisticated sense of the international community because they had come through the war,” he said. “They knew the difference between serious things and less serious things, [having] lived through a period when they thought they were going to be under Nazi occupation.”

Ishiguro contrasts this with the anti-immigrant rhetoric that dominated the Brexit and US election campaigns.

“We have become much more multi-cultural and much more cosmopolitan in many ways, but the attitude to, say, the refugee crisis, I think is quite different to what I remember from the Britain I grew up in,” he said.

His father’s electronic analogue machine, which is the size of a large wardrobe, converted meteorological and ocean data (wind speed, tidal motion, water depth and so on) into electrical signals on a series of wire meshes. This allowed the height of storm surges, and where and when they would make coastal impact, to be predicted. The system was originally developed to help Japanese fishing fleets, but modified to be applied to the North Sea, where floods following a storm surge in 1953 led to more than 2,000 deaths.

Ultimately, the machine was superseded by digital computers, but the scientist continued to perfect his creation in the family garden shed in Guildford until his death in 2007; a fact Ishiguro describes as entirely unsurprising.

Despite having taken a different career path, Ishiguro inherited an obsessive attitude towards work from his father, he said, recalling him mulling over equations every evening while watching American thrillers on television.

“Looking back now I can see that the whole approach to his work is quite like my approach to my work as a writer,” he said. “He didn’t think of it as a job at all. It was something he obsessively thought about the whole time. That was my model.”

Mathematics: The Winton Gallery, designed by Zaha Hadid Architects, opens on 8 December. It spans 400 years of mathematics, focusing on ideas and objects that have influenced everyday lives.

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Use your wits! Can you solve these fiendishly difficult puzzles?

Isn’t it time we learned to reconnect with the idea of problem-solving for fun? After all, argues Alex Bellos, the best puzzles are pieces of poetry. In an exclusive extract from his new book, he selects some of his favourites.

For those of us whose school memories include struggling with long division and quadratic equations, it may be surprising to learn that there was a time when almost all mathematics was recreational.

In the medieval world, for example, the role of maths was mostly as an intellectual diversion. (Apart from its use for technical, real-life tasks like measuring land area or calculating tax.) When, in 799, the British scholar Alcuin of York sent a letter containing 50 or so maths problems to Charlemagne, he did so not to infuriate the king but to amuse him.

In ancient times, too, many arithmetical problems were designed to entertain. Indeed, for the Greeks, a principle motivation for studying geometry was intellectual stimulation and discovery, with no concern for practical applications.

Maths got serious during the scientific revolution. Blame Isaac Newton. The discovery of his laws of motion and gravitation and the infinitesimal calculus, which provided the tools for dealing with these new ideas created new, difficult jobs for mathematicians to do, like finding the equations that describe as much of the world as possible.

The consequences are still felt today. Maths has mostly lost its reputation as a vehicle for playfulness and diversion. (When a piece of mathematics becomes successful, as in the case of sudoku, people make the excuse that this isnt maths, as if it stops being so merely by being popular.) And this is a shame, because we are depriving ourselves of a lot of fun.

Maths and logic puzzles are pleasurable and life-affirming because they force us to use our wits. A good puzzle is also never too hard to solve, thus presenting us with an achievable goal, supremely gratifying when it comes. In a puzzle we are proving to ourselves that we can do something we doubted we could. Deductive reasoning in simple logical steps is comforting, especially when real life is so illogical.

Beginning with Alcuin, Britain has produced many illustrious puzzlesmiths. Alcuin is notable as the first writer to introduce humour into mathematical problems, and his teasers have stood the test of time. Few adults can be unaware of his most famous riddle, about a traveller trying to cross a river with a wolf, a goat and some cabbages, which was once even featured in an episode of The Simpsons. I like the other types of puzzle he introduced too, such as questions about strange family relationships.

In the 19th century Lewis Carroll popularised the fun to be had with logic in Alices Adventures in Wonderland and Through the Looking-Glass. But Britains greatest ever puzzle inventor was Henry Ernest Dudeney, whose puzzles appeared for decades in British newspapers and magazines. In April 1930, the month Dudeney died, one of his most influential puzzles appeared in Strand Magazine the literary journal that published Conan Doyles first Sherlock Holmes stories. Smith, Jones and Robinson inspired a whole new genre of inferential logic puzzles in which the solver must deduce conclusions from a list of whimsical statements. These puzzles invite you to become a detective. On first reading it looks like there is far too little information to find the answer. But slowly you will piece together the clues.

Even when maths got serious in the late 17th century, however, it did not totally lose its element of play. As well as finding out the equations for commonplace phenomena like the vibration of string, mathematicians also used Newtons new tools to investigate unusual and bizarre curves. In fact, many great discoveries have come from mathematicians trying to understand simple puzzles and games. Most obviously, perhaps, the field of probability and therefore all of statistics was the result of Blaise Pascal and Pierre de Fermats correspondence about gambling games.

Not only have great mathematicians used simple brainteasers to inspire new research, but they have also devised many of their own puzzles. Edouard Lucas, who in the 19th century made important advances in our understanding of prime numbers, wrote a book of puzzles for the general reader and also invented a few classic problems.

Despite the current poor image of maths, the ubiquity of sudoku attests that maths puzzles do remain a popular pastime for many. I discovered quite how much a good puzzle can go viral when I posted a puzzle from a Singapore maths exam on my Guardian puzzle blog: the post was the ninth most read story across the Guardian online for last year, and the post with the solution was the sixth. The success of that puzzle inspired me to delve into puzzle history and rediscover many puzzles from the last thousand years. One of my favourites is only a few decades old, from the Japanese puzzle maven Nob Yoshigahara.

The best puzzles, like this one, are pieces of poetry. With elegance and brevity they pique our interest, kindle our competitive spirit, and in some cases reveal universal truths. Puzzles appeal to our impulse to make sense of the world, but most importantly they indulge our intellectual playfulness. Yet no matter how frivolous or contrived they are, the strategies we use to solve them expand our armoury for tackling other challenges in life.

Ever since the birth of mathematics, we have enjoyed posing and answering puzzles. And long may we continue to do so.

Puzzle 1: qwerty

The 10 letter keys on the top line of a typewriter are
Can you find a 10-letter word that uses only these keys?

Puzzle 2: business card

Jasper Jason works for local radio. This is his business card. Can you spot the pattern?


Puzzle 3: the double date

If two men each take the others mother in marriage, what would be the relationship between their sons?

Puzzle 4: from Le Havre to New York

The following story about the 18th-century French mathematician Edouard Lucas is absolument authentique, according to a 1915 French maths text book. It took place many years ago, the author writes, at a scientific conference. Several well-known mathematicians were milling around after lunch. Lucas piped up and challenged them to the puzzle below. A few replied with the wrong answer. Most were silent. No one got it right.

Every day at noon in Le Havre an ocean liner sails to New York, and (simultaneously) in New York an ocean liner sails to Le Havre. The crossing takes seven days and seven nights in either direction. How many ocean liners will an ocean liner leaving Le Havre today pass at sea by the time it arrives in New York?

Puzzle 5: Smith, Jones and Robinson

Smith, Jones and Robinson are the driver, fireman and guard on a train, but not necessarily in that order. The train carries three passengers, coincidentally with the same surnames, but identified with a Mr: Mr Jones, Mr Smith and Mr Robinson.

Mr Robinson lives in Leeds.

The guard lives halfway between Leeds and Sheffield.

Mr Joness salary is 1,000 2s. 1d. per annum.

Smith can beat the fireman at billiards.

The guards nearest neighbour (one of the passengers) earns exactly three times as much as the guard.

The guards namesake lives in Sheffield.

What is the name of the engine driver?

(I have kept the original phrasing of the puzzle, which uses the old British currency. The importance of the value 1,000 2s. 1d., or one thousand pounds, two shillings and one pence, is that you cannot divide it by three to produce an exact amount.)


Puzzle 6: free the coin

Coins and matchsticks are historically the two most popular puzzle props. This puzzle contains both a coin and a matchstick. Consider it the recreational mathematics equivalent of a rare duet between two famous old crooners.

Two upturned glasses are positioned as above. A match rests between them and a coin is trapped in the left glass. Can you remove the coin without letting the match fall?

Puzzle 7: eight neat sheets

Eight sheets of identically sized paper are placed on a table. Their edges form the following pattern, with only one sheet, marked 1, completely visible:

eightCan you number the other sheets from top to bottom where 2 is the second layer, 3 is the third layer, and so on?

Puzzle 8: the number tree

The Japanese puzzle inventor Nob Yoshigahara considered this puzzle his masterpiece.

The numbers below are arranged according to a certain rule. Once youve worked out the rule, fill in the missing number. The number seven in the final circle is not a typographical error.

QUESTIONPlease, no spoilers in the comments since that spoils the fun for others. See the solutions here.

Can You Solve My Problems? A Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles by Alex Bellos is published by Guardian Faber at 14.99. To order a copy for 10.99 go to, or call 0330 333 6846.

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Did you solve it? Where’s the best place to tie your shoe in an airport?

The solution to today’s travel teaser.

Earlier today I asked you the following questions:

You are in an airport, and you are walking from security to your gate. The distance includes stretches of travelator, or moving walkway, and stretches of floor. You have a constant walking speed, u, and the travelators also have constant speed, v. So, when you are on the travelator you are traveling at u + v.

1. You need to stop to tie your shoe. If tying your shoe always takes the same time, will you get to the gate quicker if you tie your shoe on the travelator, off the travelator or does it make no difference?

2. You have enough energy to walk at faster speed, w, for a fixed time. In order to get to the gate quicker, do you increase your speed when you are on the travelator, off the travelator or does it make no difference?

The answers:

  1. Tie your shoelace on the travelator.
  2. Speed up off the travelator.

The explanations (simple):

Lets simplify the details and suppose that there is a single stretch of floor followed by a single stretch of travelator. There’s a lovely intuitive explanation of why it is better to tie your laces on the travelator.

Imagine two people, say, Angela and Beatrice, start walking from security at the same time. Angela stops to tie her shoe immediately before the travelator, but Beatrice ties her shoe as soon as she steps on the travelator. By the time Angela has finished tying her shoe and steps on the travelator, Beatrice will now be way ahead of her and Angela will never make up the distance.

There is a similar intuitive explanation of why it is better to fast-walk before you get to the travelator. Imagine again that Angela and Beatrice start together. Say, to keep it simple, that Angela fast-walks straight away, which puts her at a distance D in front of Beatrice. When Angela steps on the travelator she is D ahead, but while she is on the travelator, and Beatrice is not, Angela increases her distance from Beatrice to more than D. When Beatrice reaches the travelator, she can start fast-walking, but so long as both her and Angela remain on the travelator she will never make up more than D.

The history

The shoelace puzzle was first posed by Terence Toe, I mean Terence Tao. (Couldnt resist, sorry). He is one of the most celebrated mathematicians alive. He featured this problem on his blog in 2008, two years after winning the Fields Medal.

How well did you do?

About 15,000 of you submitted answers. The majority – just – got the first one correct. For the second question, however, the correct answer was the one chosen by the fewest people. Interesting.

Q1: where do you tie your shoe?

on the travelator: 47.3%

off the travelator: 11.4%

doesnt matter: 41.3%

Q2: where do you speed up?

on the travelator: 28.7%

off the travelator: 17.2%

doesnt matter: 54.1%

The explanations (trickier)

Here is a more thorough explanation of the answers using some algebra. Again lets assume there is a single stretch of floor and a single stretch of travelator. Lets start by defining the terms.

T = time it takes to tie laces.

u:human speed

v: travelator speed

Df: Floor distance

Dtr: travelator distance

The relevant equation:

Speed = distance/time, and so, time = distance/speed

Question 1: Total time = time on floor + time on travelator + time tying laces = (distance on floor/speed on floor) + (distance on travelator/speed on travelator) + T = Df/u + Dtr/(u + v) + T

If laces tied on floor, the total time: Df/u + T + Dtr/(u + v)

If laces tied on the travelator, the total time: (Df/u) + T + (Dtr Tv)/(u + v)

Since Dtr > Dtr Tv, the total time if laces tied on floor is greater, so you get there quicker if you tie laces on the travelator.

Question 2: Now let T = time spent at higher speed w.

Total time if speeding on floor: T + (Df Tw)/u + Dtr/(u + v)

Total time is speeding on travelator: (Df/u) + T + (Dtr T(w+ v))/(u + v)

Since w > u then w/u > (w + v)/(u + v) and (I promise) this shows that total time is less when you walk fast on the floor.

Because of the time and space constraints of this blog I havent been able to explain each step in detail. Im sure some commentators will be able to help out below the line if you are still confused. Thanks all!

I post a puzzle here on a Monday every two weeks. If you are reading this on the Guardian app, click where it says Follow Alex Bellos and youll get an alert when I publish the next puzzle. If you want to propose a puzzle for this column, please email me Id love to hear it.

Im the author of three popular maths books including Alexs Adventures in Numberland and the maths colouring book Snowflake Seashell Star. You can check me out on Twitter, Facebook, Google+, my personal website or my Guardian maths blog.


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Genius by numbers: why Hollywood maths movies don’t add up

From A Beautiful Mind to The Theory of Everything and The Man Who Knew Infinity, Hollywood loves a mathematician. So why cant it get beyond the fevered prodigy scribbling equations on windows?

In the Tina Fey sitcom Unbreakable Kimmy Schmidt, wealthy Manhattanite Jacqueline Vorhees wails to her assistant that she cant afford to get divorced. Even though shed get $1m for every year of her marriage.

I spend 100 grand a month. Ill be broke in 10 years, she wails. No, thats wrong, counters Kimmy (Ellie Kemper), who scribbles some sums with a marker on Mrs Vorheess window. So $100,000 times 12 months. Thats $1.2m a year. Divide that into $12m, and yes, youd be broke in 10 years. But if you invest some of it, assuming a 7% rate of return, using the compound interest formula, your money would almost double.

Kimmy turns round triumphantly: Mrs Voorhees, I mathed, and you can get divorced! Mrs Vorhees eyes Kimmy narrowly. Those are not, she complains, erasable markers. What she doesnt mention is that math isnt a verb. Not yet.

The scene is, among other things, Feys satire of the Hollywood cliche of genius squiggling on glass. In A Beautiful Mind (2001), for instance, Russell Crowe, playing troubled maths star John Forbes Nash Jr, writes formulae on his dorm window. This scene is echoed in The Social Network (2010), where Andrew Garfield sets out the equations for Facebooks business model on a Harvard window while Jesse Eisenbergs Mark Zuckerberg looks on. In the opening scene of Good Will Hunting (1997), janitor prodigy Matt Damon writes equations on a bathroom mirror.

Why do so many Hollywood maths whizzes forego paper? Stanford mathematician Keith Devlin explains. Depicting a mathematician scribbling formulas on a sheet of paper might be more accurate, but it certainly doesnt convey the image of a person passionately involved in mathematics, as does seeing someone write those formulas in steam on a mirror or in wax on a window, nor is it as cinematographically dramatic.

Good point. When we watch A Beautiful Mind and look through the window at our Russ, Hollywoods most built mathematician (counterexamples on postcards, please show your workings), we pass beyond incomprehensible equations and convince ourselves were seeing Genius at Work. Even if, as some critics have complained uncharitably, Russs pi glyphs, greater-than and less-than symbols and such dont make sense.

But theres another way maths movies can confound the Boredom Equation, namely by leaving a black hole where the maths should be. The Man Who Knew Infinity, the new film starring Dev Patel and Jeremy Irons about the great Indian mathematician Srinivasa Ramanujan, is intriguing in this respect. Although we see Ramanujan doing maths, mostly the film is interested in other things how he falls in love with his wife, the pain of separation when he travels from Madras to study at Cambridge, the racism he suffers in England and, most stirringly, the narrative arc from lowly clerk to globally recognised mathematician.

Hollywoods most built mathematician Russell Crowe in A Beautiful Mind. Photograph: Universal Studios

That said, the film has its charming moments. When Hardy visits Ramanujan in a nursing home, he complains about the boring number of the cab that brought him there. Ramanujan begs to differ: 1,729 is the smallest that is expressible as the sum of two cubes in two different ways. Today 1,729 is known as the Hardy-Ramanujan number. How does that work, you may be wondering? Like this: 1729 = 13 + 123 = 93 + 103.

Ramanujans mentor GH Hardy (Irons) is an atheist and rationalist, exasperated that this Indian prodigy cannot produce proofs for his work and, worse, is doubtful that proofs can explain the inexplicable. You wanted to know how I get my ideas, says Ramanujan. God speaks to me. But while the film may sketch two different mathematical philosophies, we leave the cinema with a warm glow that comes from anything but hard thinking.

If you want to learn some more about Ramanujans contribution to mathematics, rent High School Musical. Freeze-frame it at the moment brainy Gabriella Montez challenges her teacher. On the board are two of the equations of the inverse of the constant pi (1/) that Ramanujan offered in his first paper published in England. Shouldnt the second equation read 16 over pi? asks Gabriella. Of course it should.

Cinema often struggles with dramatising difficult ideas, particularly if they are abstract. One way of overcoming that problem is by metaphorical explanation. For instance, in Nicholas Roegs Insignificance (1985), a Marilyn Monroe-like character demonstrates relativity using toy trains and flashing lights. In The Theory of Everything, Jane Hawking uses a pea and a potato to explain the difference between quantum theory and general relativity, while her husbands friends explain Hawking Radiation with beers and crisps.

Movie explanations of difficult stuff, though, may obscure rather than enlighten. Whats more, some directors know this and have fun pointing out the shortcomings of their medium and those of their audiences. In Adam McKays The Big Short (2016), for instance, Margot Robbie sits in a tub sipping champagne and describing how sub-prime loans work. Her explanation is doubtless coherent, but when Im looking at a beautiful woman in a bubble bath, Im not thinking about credit default swaps. So sue me. Later in the film, chef Anthony Bourdain chops fish in his kitchen while describing how collateralised debt obligations work. Finally, Selena Gomez plays roulette to illustrate the idea of gambling on other peoples gambles.

Each scene serves as a parody of explanation. They are part of a film that mocks you, you poor jerk, and your intellectual aspirations. You are never ever going to understand how difficult stuff works from watching movies, however much youd like to.

Sometimes, though, cinema can give a real insight into the intellectual process. In Agora (2009), Rachel Weisz as ancient philosopher Hypatia does an experiment on a ship to test relative motion. If, she hypothesises, you drop a heavy sack from the mast while the ship is moving forward, it will fall on the deck several feet behind the mast. The sack is dropped and falls much closer to the mast than she predicted. Hypatia claps her hands in delight. But you were wrong! says the ships captain. Yes, but it is definitive proof! The sack behaves as if the boat were stationary.

What does that mean?

I dont know. But the same principle could be applied to the Earth. It could be moving around the sun without us realising.

Hypatia, that is to say, infers a revolutionary heliocentric cosmology from her falsified hypothesis. The film thus generously gives us what we are effectively denied in Good Will Hunting or A Beautiful Mind the inside track on how someone clever is thinking about a problem. Whats more, its an antidote to Hollywoods vision of genius. It suggests that getting stuff wrong is at least as important in the story of human intellectual progress as being right all the time.

Maths is often reduced to nothing more than a MacGuffin. In Rushmore (1998), for instance, Max Fischer (Jason Schwartzman) is reading the newspaper while his teacher tells his class that on the blackboard is the hardest geometry equation in the world. What credits would anyone solving it get, asks one student. Well, considering Ive never seen anyone get it right, including my mentor Dr Leaky at MIT, I guess if anyone here can solve that problem, Id see to it that none of you ever have to open another math book again for the rest of your lives.

Thus tempted, Fischer folds his paper and goes to the blackboard, and squiggles his solution while nonchalantly sipping espresso. The film at this point has nothing to declare but Fischers genius. Do we really believe Jason Schwartzmann can compute the area of an ellipse? Sure. Whatever.

Genius squiggling is there once again merely to help Hollywood tell the sentimental story it never tires of: namely the story of someone usually borderline demented and by definition insufficiently recognised sticking it to the establishment.

Genius squiggling Rushmore

None of this should suggest we cant learn maths from movies. In Tina Feys Mean Girls (2004), for example, Lindsay Lohan plays a finalist in the Illinois high school mathletes state championship. Will her North Shore High team stick it to those prep school toffs opposite? Heres the first question: Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers? Got it yet? 14 and 5. In the end, Lohans team become the new state champs because she wins the sudden death tie-break. What does the scene prove? That those of you who thought Lindsay Lohan cant do maths should really have a word with yourselves.

Perhaps the most resonant maths scene in Hollywood cinema, though, comes in a very old comedy. In the Abbott and Costello movie In the Navy (1941), Lou is a ships cook. Hes baked 28 doughnuts, which he reckons is just enough to give 13 to each of his seven officers. But seven goes into 28 four times, objects Lous straight man. Not so, says Lou, who goes on to prove it on the blackboard in a masterclass of cheating and illusion. The scene demonstrates a general truth, namely that when Hollywood does maths, it doesnt necessarily add up.

The Man Who Knew Infinity is released on 8 April.

Read more:

GCSE Mathematics Compared to GCE – A Betrayal of Standards

In the U.K prior to 1986 there were two separate exams for 16 year-olds, running parallel to each other. These were the C.S.E.(Certificate of secondary education) and the G.C.E.(General Certificate of Education).
The G.C.E. was as an exam-only based qualification that would allow students to carry on with secondary education to do A-level courses. The C.S.E. on the other hand was a slightly inferior qualification. It had a course work element to it and the exams were less rigorous. That is the questions were seen as easier and there was less emphasis on recalling facts. However, the two qualifications did have some overlap to some degree. The highest result at C.S.E( a grade 1) equated roughly to the third highest at G.C.E(grade C).

The G.C.E pass mark and grades were decided upon using a statistical method. All the marks converted to percentages from were written along the x-axis of a graph, with the frequency of each mark plotted above. The result was what is called a Normal Distribution curve. This is a symmetrical curve that rises from zero, reaches a maximum and descends to zero again. In a perfect curve, the maximum corresponds to the average mark for the test.

To cut through a lengthy statistical explanation, the pass mark was decided upon from the shape of the curve and the relative numbers of students above and below this mark. In this way, assuming intelligence is a constant, approximately the same percentage of pupils would pass each year. As a result of the huge numbers of students involved, the exam was deemed to be fair and did seem to maintain standards.

G.C.S.E. on the other hand depends on actual marks to divide its grade boundaries. Unlike G.C.E, there is no pass mark in G.C.S.E. Everyone passes. So if the exam gets easier more students get higher grades; and this has been the state of affairs since the beginning of G.C.S.E. Year on year the numbers of A and A-star grades in mathematics has steadily increased.

The beginning of the debate into the lowering of standards in schools centred around the inclusion of ‘course work’, some of it as high as 30% of the total marks. The marking of this course work was open to abuse by teachers, as was the creation of it by students in the first place. Parents eager to improve the grades of their offspring were also drawn into the game. It was only in 2007 that the U.K. government decided that course work in G.C.S.E. mathematics should be abandoned (eleven years after its inception).

I am a retired maths teacher and feel well placed to make a measured call on the issue of standards. I have taught all three exams (C.S.E., G.C.S.E. & G.C.E.) in my time, over 25 years. Recently two G.C.E. maths papers (1957, 1968) came into my possession. Objectively I made the comparison with this year’s G.C.S.E. paper (2009). Unsurprisingly G.C.S.E. did not fair well. This result is more than just disappointing. It is a sad indictment of our education system. I shall be making a detailed subjective study of how these papers compare and will publish my findings in due course.

As if to confirm my lack of confidence in this qualification, I read that in the coming school year many
public(private) schools in the U.K. are starting their 15 year olds(year 10’s) on A-level mathematics courses, and completely side stepping G.C.S.E.

GCSE Mathematics – Hints and Tips to Help You Succeed

If you do nothing else, read this document carefully. Especially important is advice given at the bottom of the page on the exam itself.

Be prepared!

Have pens and sharpened pencils at the ready. Don’t use pens that leak or ones with broad tips. By the same measure, don’t use untried new pens. They may let you down. Use a pen that you know is reliable.

Write clearly!

You might be the next Einstein or Dirac but unless you write clearly we will never know! Anything the examiner cannot read he/she cannot mark. If you are a poor writer, please take the time to write neatly.

Read the question !

This is not wasted time. Read the whole question once to get a rough idea what it is about. Then read it again from the beginning – slowly. It is a good idea to tick off each line, including information in diagrams. Do not start until you are sure you thoroughly understand what is asked.

Be Accurate!

Again, read the question! If it asks for 2 decimal places or 3 significant figures, then that is what it wants. Before going on to the next question, check you answer with the original question.

Simultaneous Equations

The answers to these types of exam questions tend to be nice rounded numbers. If they have a decimal, it usually ends in .5, but these are rare. Answers with a line of decimals are probably wrong, so check for errors.

Inverse % questions

These are ‘backward looking’ problems. With current values given in the question, you have to calculate some original value before the decrease/increase occurred. Again, the answers tend to be rounded numbers. If you get a train of decimals, check back in your working.


Simply check that your answer is between 1 and 0; and of course, there are no negative values.

The Mean

Common sense here! Check that the number is not silly – too high/too low, and that it is between the highest and lowest values, and not outside them.


Don’t round up calculator display numbers unnecessarily. If you do this, and the calculation has a number of stages, you final answer will be incorrect.

Quadratic Equations

A question asking for significant figures or decimal places indicates you should use the quadratic formula.


Look at your answer. You should never get a length longer than the hypotenuse.


Common sense here. Look out for silly answers. Again, lengths should not be longer than the hypotenuse.

Use of Bold Print

This means that the question is not straight forward. Bold print is a cue to read the question again carefully, so you understand exactly what is asked.


Tracing paper is useful for rotating shapes. Remember if you have forgotten something, don’t worry. Put your hand up and wait. The invigilator will come and supply you with the item.

Your answers

-write small
-show all working
-use words to describe your calculations/equation changes
-where space is at a premium, draw a vertical line and work either side of it, rather than sprawl work across a page

The Exam

Do not work late the night before, but do get up early to go through your final revision notes.

Revise right up to the point when you are told to come into the exam room, but please do not bring any revision materials into the exam room itself. Leave them in your bag outside.

Go through the paper quickly reading or scanning it to get an idea of what is asked.

Work your way through the paper. If you find any question particularly formidable, do not carry on with it. Come back to it later. You will lose valuable time being bogged down with one question. Get some easy marks in the bag first.

When you have finished do not sit arms folded looking up at the ceiling! You will not have scored 100%! There are some marks still to be had. Spend every last minute going through the paper carefully looking for errors. Trust me. There will be some!

There only remains for me to say good luck. But exams have little to do with luck. If you have done the work and revised thoroughly, you will undoubtedly do well. The fact that you are reading this shows your intent. So don’t look at exams as impossible hurdles to jump. Look at them as opportunities for you to shine and show everyone just what you can do!

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