Month: July 2017

Can you solve it? Are you smarter than a forester?

A puzzle about planting trees

Hello guzzlers,

Your mission today is to design an arrangement of trees on a desert island, like the one below.

An aerial view of five trees on an island.

When there is a single tree, no matter where you stand on the island you will always be able to see exactly one tree.

An island with a single tree. From each of the two black dots you can see a single tree.

With two trees, however, there are some places where you can see two trees, and there are some places where you can see only a single tree, since the other one is blocked from view.

Read more:

Technorati Tags: ,

The world has lost a great artist in mathematician Maryam Mirzakhani

She was the only woman to have won the Fields medal, maths equivalent of the Nobel prize

The mathematician Maryam Mirzakhani died two weeks ago. Shewas 40. I had never heard of her before reading about her death in the papers. Its a piercingly sad story: Iranian-born, and latterly a professor at Stanford University, Mirzakhani was the only woman to have won the Fields medal, the equivalent for a mathematician of the Nobel prize, and is survived, in newspaper-speak, by a husband and a daughter.

I always find the locution survived by too cruel to bear. So final the rupture, no room for error: shes gone, theyre left. And, in this case, how young the mother and the wife.

It is a sad story for other reasons, too, not least the intensity of Mirzakhanis expression in the photograph most of the papers used. There is a beauty that can onlybe described as that of the minds migration to the face, the transfiguring beauty of exceptional intelligence. So its a double loss: thepremature loss of a person and the premature loss of her genius.

I remember there being an unspoken qualitative distinction atschool between those who were good at maths and science the priests of numbers and symbols and the more poetical of us, whose medium, as Wordsworth had it, was the language of men talking to men. The assumption, at least on the part of us Wordsworthians, was that creativity was all on our side. I have since come to think the word creative has much to answer for. Among the freedoms it sometimes gave us was the freedom from structure, knowledge and the obligation to convince.

Mirzakhani, it is said, considered being a writer before turning to mathematics. It is unlikely she believed shed made a choice in favour of an inferior, or less artistic, discipline. And she expressed her immersion in mathematics in language every writer will recognise like being lost in a jungle and trying to use all the knowledge you can gather to come up with some new tricks, and with luck you might find a way out.

The luck, of course, is no such thing. Its the mystery Keats called negative capability, the trust that the work will do itself if only we dareto plunge without irritability orinsistence into the dark, not sure we will find a way out at all. The bestwriting happens in this way, unintended, unknowing, grateful and surprised. Such abnegation of will is what we mean by creativity. So the mathematician and the artist are companioned in the same dark, and do obeisance to the same gods. The pity of Mirzakhanis death will be felt by poets as well as mathematicians.

Read more:

Technorati Tags: , , , , ,

Why theres no such thing as a gifted child

Even Einstein was unexceptional in his youth. Now a new book questions our fixation with IQ and says adults can help almost any child become gifted

When Maryam Mirzakhani died at the tragically early age of 40 this month, the news stories talked of her as a genius. The only woman to win the Fields Medal the mathematical equivalent of a Nobel prize and a Stanford professor since the age of 31, this Iranian-born academic had been on a roll since she started winning gold medals at maths Olympiads in her teens.

It would be easy to assume that someone as special as Mirzakhani must have been one of those gifted children who excel from babyhood. The ones reading Harry Potter at five or admitted to Mensa not much later. The child that takes maths GCSE while still in single figures, or a rarity such as Ruth Lawrence, who was admitted to Oxford while her contemporaries were still in primary school.

But look closer and a different story emerges. Mirzakhani was born in Tehran, one of three siblings in a middle-class family whose father was an engineer. The only part of her childhood that was out of the ordinary was the Iran-Iraq war, which made life hard for the family in her early years. Thankfully it ended around the time she went to secondary school.

Mirzakhani, did go to a highly selective girls school but maths wasnt her interest reading was. She loved novels and would read anything she could lay her hands on; together with her best friend she would prowl the book stores on the way home from school for works to buy and consume.

As for maths, she did rather poorly at it for the first couple of years in her middle school, but became interested when her elder brother told her about what hed learned. He shared a famous maths problem from a magazine that fascinated her and she was hooked. The rest is mathematical history.

Is her background unusual? Apparently not. Most Nobel laureates were unexceptional in childhood. Einstein was slow to talk and was dubbed the dopey one by the family maid. He failed the general part of the entry test to Zurich Polytechnic though they let him in because of high physics and maths scores. He struggled at work initially, failing to get academic post and being passed over for promotion at the Swiss Patent Office because he wasnt good enough at machine technology. But he kept plugging away and eventually rewrote the laws of Newtonian mechanics with his theory of relativity.

Lewis Terman, a pioneering American educational psychologist, set up a study in 1921 following 1,470 Californians, who excelled in the newly available IQ tests, throughout their lives. None ended up as the great thinkers of their age that Terman expected they would. But he did miss two future Nobel prize winners Luis Alvarez and William Shockley, both physicists whom he dismissed from the study as their test scores were not high enough.

There is a canon of research on high performance, built over the last century, that suggests it goes way beyond tested intelligence. On top of that, research is clear that brains are malleable, new neural pathways can be forged, and IQ isnt fixed. Just because you can read Harry Potter at five doesnt mean you will still be ahead of your contemporaries in your teens.

According to my colleague, Prof Deborah Eyre, with whom Ive collaborated on the book Great Minds and How to Grow Them, the latest neuroscience and psychological research suggests most people, unless they are cognitively impaired, can reach standards of performance associated in school with the gifted and talented. However, they must be taught the right attitudes and approaches to their learning and develop the attributes of high performers curiosity, persistence and hard work, for example an approach Eyre calls high performance learning. Critically, they need the right support in developing those approaches at home as well as at school.

Maryam Mirzakhani won the Fields Medal, the mathematical equivalent of the Nobel prize, but showed little maths ability to begin with. Photograph: Clay Mathematics Institute

So, is there even such a thing as a gifted child? It is a highly contested area. Prof Anders Ericsson, an eminent education psychologist at Florida State University, is the co-author of Peak: Secrets from the New Science of Expertise. After research going back to 1980 into diverse achievements, from music to memory to sport, he doesnt think unique and innate talents are at the heart of performance. Deliberate practice, that stretches you every step of the way, and around 10,000 hours of it, is what produces the expert. Its not a magic number the highest performers move on to doing a whole lot more, of course, and, like Mirzakhani, often find their own unique perspective along the way.

Ericssons memory research is particularly interesting because random students, trained in memory techniques for the study, went on to outperform others thought to have innately superior memories those you might call gifted.

He got into the idea of researching the effects of deliberate practice because of an incident at school, in which he was beaten at chess by someone who used to lose to him. His opponent had clearly practised.

But it is perhaps the work of Benjamin Bloom, another distinguished American educationist working in the 1980s, that gives the most pause for thought and underscores the idea that family is intrinsically important to the concept of high performance.

Blooms team looked at a group of extraordinarily high achieving people in disciplines as varied as ballet, swimming, piano, tennis, maths, sculpture and neurology, and interviewed not only the individuals but their parents, too.

He found a pattern of parents encouraging and supporting their children, in particular in areas they enjoyed themselves. Blooms outstanding adults had worked very hard and consistently at something they had become hooked on young, and their parents all emerged as having strong work ethics themselves.

While the jury is out on giftedness being innate and other factors potentially making the difference, what is certain is that the behaviours associated with high levels of performance are replicable and most can be taught even traits such as curiosity.

Eyre says we know how high performers learn. From that she has developed a high performing learning approach that brings together in one package what she calls the advanced cognitive characteristics, and the values, attitudes and attributes of high performance. She is working on the package with a group of pioneer schools, both in Britain and abroad.

But the system needs to be adopted by families, too, to ensure widespread success across classes and cultures. Research in Britain shows the difference parents make if they take part in simple activities pre-school in the home, supporting reading for example. That support shows through years later in better A-level results, according to the Effective Pre-School, Primary and Secondary study, conducted over 15 years by a team from Oxford and London universities.

Eye-opening spin-off research, which looked in detail at 24 of the 3,000 individuals being studied who were succeeding against the odds, found something remarkable about what was going in at home. Half were on free school meals because of poverty, more than half were living with a single parent, and four in five were living in deprived areas.

The interviews uncovered strong evidence of an adult or adults in the childs life who valued and supported education, either in the immediate or extended family or in the childs wider community. Children talked about the need to work hard at school and to listen in class and keep trying. They referenced key adults who had encouraged those attitudes.

Einstein, the epitome of a genius, clearly had curiosity, character and determination. He struggled against rejection in early life but was undeterred. Did he think he was a genius or even gifted? No. He once wrote: Its not that Im so smart, its just that I stay with problems longer. Most people say that it is the intellect which makes a great scientist. They are wrong: it is character.

And what about Mirzakhani? Her published quotations show someone who was curious and excited by what she did and resilient. One comment sums it up. Of course, the most rewarding part is the Aha moment, the excitement of discovery and enjoyment of understanding something new the feeling of being on top of a hill and having a clear view. But most of the time, doing mathematics for me is like being on a long hike with no trail and no end in sight.

The trail took her to the heights of original research into mathematics in a cruelly short life. That sounds like unassailable character. Perhaps that was her gift.

Great Minds and How to Grow Them, by Wendy Berliner & Deborah Eyre. You can order it from the Guardian Bookshop for 14.99, with free P&P. Telephone orders (0330 333 6846) attract minimum p&p of 1.99.

  • Wendy Berliner will be taking part in an online live chat next Thursday, discussing the idea that genius is made rather than born. Details to follow on the Teacher Network soon

Read more:

Technorati Tags: , , , , ,

Afghan girls finish in 99th place in global robotics competition

The Afghan girls who were originally denied entry into the United States for a robotics competition finished in 99th place.

The six teenage girls, who were denied U.S. visas twice, outshined 67 other countries including America in the FIRST Global Challenge on Tuesday in Washington, D.C.

President Donald Trump granted the girls, between ages 14 and 16, and their chaperone access to the United States on July 13 after their visa rejections sparked an international backlash.


It is unclear why their visas were denied, but the State Department previously responded by saying all applications are adjudicated on a case-by-case basis in accordance with U.S. law.

The first annual robotics competition, which hopes to inspire the youth to pursue careers in science, technology, engineering and mathematics (STEM), drew 163 teams from around the globe. Poland won the competition, while the United States ranked 155th.

Fox News Jennifer Griffin contributed to this report.

Read more:

Technorati Tags: , , ,

Can you solve it? Are you smarter than an architect?

A puzzle that tests 3D thinking

Hi guzzlers,

Todays puzzle was sent in by a reader who remembers it from his days as an architecture student.

Draw a 3-dimensional picture of a shape that goes through each of these holes, exactly touching all sides as it passes through.

A triangle with sides 1 unit. A square with sides 1 unit. A circle with diameter 1 unit.

Architects will surely find the answer obvious. The heads of the rest of us will look rather like the house in the picture above, since it requires you to visualise an object in three dimensions, which is a challenge if your brain isnt trained to do it.

If you want to email me your answer, or post it on Twitter with the hashtag #MondayPuzzle, Ill send the author of my favourite image a copy of my puzzle book Can You Solve My Problems?

Ill be back at 5pm UK time with the solution.


I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. Im always on the look-out for great puzzles. If you would like to suggest one, email me.

My puzzle book Can You Solve My Problems? is just out in paperback.

Read more:

Technorati Tags: ,

Maryam Mirzakhani, first woman to win mathematics’ Fields medal, dies at 40

Stanford professor, who was awarded the prestigious prize in 2014, had suffered breast cancer

Maryam Mirzakhani, a Stanford University professor who was the first and only woman to win the prestigious Fields medal in mathematics, has died. She was 40.

Mirzakhani, who had breast cancer, died on Saturday, the university said. It did not indicate where she died.

In 2014, Mirzakhani was one of four winners of the Fields medal, which is presented every four years and is considered the mathematics equivalent of the Nobel prize. She was named for her work on complex geometry and dynamic systems.

Mirzakhani specialized in theoretical mathematics that read like a foreign language by those outside of mathematics: moduli spaces, Teichmller theory, hyperbolic geometry, Ergodic theory and symplectic geometry, the Stanford press announcement said.

Mastering these approaches allowed Mirzakhani to pursue her fascination for describing the geometric and dynamic complexities of curved surfaces spheres, doughnut shapes and even amoebas in as great detail as possible.

Her work had implications in fields ranging from cryptography to the theoretical physics of how the universe came to exist, the university said.

Mirzakhani was born in Tehran and studied there and at Harvard. She joined Stanford as a mathematics professor in 2008. Irans president, Hassan Rouhani, issued a statement praising Mirzakhani.

The grievous passing of Maryam Mirzakhani, the eminent Iranian and world-renowned mathematician, is very much heart-rending, Rouhani said in a message that was reported by the Tehran Times.

Irans foreign minister, Mohammad Javad Zarif, said her death pained all Iranians, the newspaper reported.

The news of young Iranian genius and math professor Maryam Mirzakhanis passing has brought a deep pang of sorrow to me and all Iranians who are proud of their eminent and distinguished scientists, Zarif posted in Farsi on his Instagram account.

I do offer my heartfelt condolences upon the passing of this lady scientist to all Iranians worldwide, her grieving family and the scientific community.

Mirzakhani originally dreamed of becoming a writer but then shifted to mathematics. When she was working, she would doodle on sheets of paper and scribble formulas on the edges of her drawings, leading her daughter to describe the work as painting, the Stanford statement said.

Mirzakhani once described her work as like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.

Stanford president Marc Tessier-Lavigne said Mirzakhani was a brilliant theorist who made enduring contributions and inspired thousands of women to pursue math and science.

Mirzakhani is survived by her husband, Jan Vondrk, and daughter, Anahita.

Read more:

Technorati Tags: , , , , , , , , , ,

Teachers raise concerns over Sats marking – BBC News

Image caption Primary head teacher Cathryn Throup highlighted the guidance given to markers

Primary school teachers in England have taken to social media to vent their anger about what they claim are inconsistencies in the marking of this year’s national curriculum test (Sats).

Using the hashtag #SATsshambles on Twitter, teachers have listed a range issues and are calling on all schools to go through their pupils’ marked papers to check for errors.

The Department for Education said results of the tests were “robust and accurate” but head teachers could apply for a review of contested marks.

According to the teachers tweeting, 10- and 11-year-olds were asked to put punctuation in a pre-written sentence and – even though they got the right answer – did not get a mark because their commas were not curved the right way or their semi-colon was too large or not in precisely the right place.

They also complained about marking guidance which they claimed only markers, not teachers, had access to.

Primary head teacher Cathryn Throup tweeted some of the issued guidance which gave details of the “origin, height, depth and orientation” of semi-colons – or where pupils’ should write their answers.

In particular it said:

  • Neither element of the semi-colon should start higher than the letter ‘l’
  • The dot of the semi-colon must not be lower than the bottom of the letter ‘w’ in the word ‘tomorrow’
  • The orientation of the comma element of the semi-colon must be inclined to the left or straight down. It cannot incline to the right.

Primary teacher Liz Hindley, who tweets as @Leaping_liz, put up pictures of four answers all featuring the semi-colon in the correct place, but two were given a mark and two were not.

“The lack of consistency is so frustrating,” she said.

Other teachers raised similar issues, such as pupils’ answers straying outside of the box.

Brian Walton, head teacher of Brookside Academy in Somerset, told the BBC that schools had not been told that markers would mark pupils down for misshapen semi-colons and answers straying outside of a box.

“The markers had guidance that none of the teachers, none of the schools knew about, so a lot of this guidance about the size and the shape and the orientation and how we form letters – we didn’t know that,” he told the BBC’s Victoria Derbyshire programme.

“Remember, they [pupils] are putting written semi-colons in text type with no gaps between the writing at the same time – we’re really getting pernickety when we’re getting to that level.”

Mr Walton said he did know the scale of the problem, but had already had 50 or 60 heads in his area contact him with concerns.

Writer and poet Michael Rosen tweeted: “The punctuation police demand that the mark has to be drawn correctly and at the right angle.”

‘Robust and accurate’

In a statement, Pearson, the company which administered the Sats, said: “Marking quality is extremely important and is something we monitor continuously.

“In the unusual circumstance that there is an error, there is a review process in place which ensures a fair and transparent system and enables Pearson to correct any discrepancies and ensure pupils receive a fair mark.”

A spokesman for the DfE said there were “a number of measures in place to ensure that schools’ Key Stage 2 writing teacher assessment judgements are robust and accurate”.

“The Standards and Testing Agency takes any issues with the accuracy of schools’ teacher assessment judgments very seriously.

“Any concerns about particular schools should be reported to the STA so that they may be properly investigated.”

But teacher unions criticised the marking for being inconsistent, saying pupils were being marked down on a technicality when it was clear they knew the correct answer to the question.

Russell Hobby, general secretary of school leaders’ union NAHT, said: “We now operate within a testing culture which appears focused on catching young children out rather than recording their achievements.

“Such a culture will swiftly erode the confidence of parents and teachers that the system is operating in the best interests of pupils.

“The stakes are so high that we seem unable to apply reasonable common sense.”

Image caption This pupil received no marks, despite appearing to answer the question correctly

Kevin Courtney, general secretary of the National Union of Teachers, said: “”We already know that moderation is inconsistent and open to gaming.

“Now, teachers are finding out that marking is unreliable too.

“The system does not deserve anyone’s trust, and it should not be the basis on which schools are held to account.”

Last week, official figures showed two-fifths (39%) of primary school pupils in England had failed to meet the expected standard in reading, writing and mathematics.

However, this summer’s results were an improvement on the success rate last year (53%), which was the first year of new, more rigorous tests.

Read more:

Technorati Tags: , , ,

New Form Of Carbon Discovered That Is Harder Than Diamond But Flexible As Rubber

Scientists have found a way to make carbon both very hard and very stretchy by heating it under high pressure. This compressed glassy carbon, developed by researchers in China and the US, is also lightweight and could potentially be made in very large quantities. This means it might be a good fit for several sorts of applications, from bulletproof vests to new kinds of electronic devices.

Carbon is a special element because of the way its atoms can form different types of bonds with each other and so form different structures. For example, carbon atoms joined entirely by sp bonds produce diamond, and those joined entirely by sp bonds produce graphite, which can also be separated into single layers of atoms known as graphene. Another form of carbon, known as glassy carbon, is also made from sp and has properties of both graphite and ceramics.

But the new compressed glassy carbon has a mix of sp and sp bonds, which is what gives it its unusual properties. To make atomic bonds you need some additional energy. When the researchers squeezed several sheets of graphene together at high temperatures, they found certain carbon atoms were exactly in the right position to form sp bonds between the layers.

Bond, sp bond.Timothy Strobel

The researchers made the compressed glassy carbon using a relatively simple method that could be reproduced on a large scale easily and cheaply. In simple terms, they used a sort of machine press that applies high-pressure loads to the carbon. But this must have involved several tricks to control the pressure and temperature in exactly the right way. This would have been a time-consuming process but should still be achievable for other people replicate the results.

New surprises

Carbon materials are continually surprising us and the emphasis of research has been to find or cook things in between its natural forms of diamond and graphite. This new form is the latest of what seem like limitless ways you can bond carbon atoms, following on from the discovery of graphene, cylindrical carbon nanotubes and spherical buckminsterfullerene molecules.

A material like this that is strong, hard, lightweight and flexible will be in high demand and could be used for all sorts of applications. For example, military uses could involve shields for jets and helicopters. In electronics, lightweight, cheaply manufactured materials with similar properties to silicon that could also have new abilities could provide a way to overcome the limitations of existing microchips.

The ConversationThe dream is to find a carbon material that could replace silicon altogether. What is needed is something that allows electrons to move through it quickly and whose electrons can easily be placed into an excited state to represent the on and off functions of a transistor. The researchers behind glassy carbon havent studied these properties in the new material so we dont yet know how suitable it might be. But it might not be that long until another of carbon is found. So far, decades of hunting hasnt turned up what we need, but maybe we just have to look deep down to find it.

Elton Santos, Research Fellow, School of Mathematics and Physics, Queen’s University Belfast

This article was originally published on The Conversation. Read the original article.

Read more:

Technorati Tags: , , ,

In Math, Profs Use This Puzzle To Teach a Valuable Lesson About Problem Solving

Graphic by Krauss

The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry.
It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it. [source]

Graphic by Trekky0623

The key to the puzzle is the fact that neither of the 13×5 “triangles” is truly a triangle, because what appears to be the hypotenuse is bent. In other words, the “hypotenuse” does not maintain a consistent slope, even though it may appear that way to the human eye. [source]

Graphic by Krauss

According to Martin Gardner, this particular puzzle was invented by a New York City amateur magician, Paul Curry, in 1953. However, the principle of a dissection paradox has been known since the start of the 16th century. [source]

Graphic by Krauss

Read more:

Technorati Tags: , , , ,

Can you solve it? Are you smarter than a cat?

Feline clever? This moggy mystery will mess with your mind

Hi guzzlers,

Todays puzzle requires you to demonstrate superior intelligence to a contrary cat.

A straight corridor has 7 doors along one side. Behind one of the doors sits a cat. Your mission is to find the cat by opening the correct door. Each day you can open only one door. If the cat is there, you win. You are officially smarter than a cat. If the cat is not there, the door closes, and you must wait until the next day before you can open a door again.

If the cat was always to sit behind the same door, you would be able to find it in at most seven days, by opening each door in turn. But this mischievous moggy is restless. Every night it moves one door either to the left or to the right.

How many days do you now need to make sure you can catch the cat?

A cat sits behind one of these doors. Whats the best strategy to find it?

(First some clarifications. The 7 doors are in a line, so if the cat is behind the first or the last door, it has only one option for where it can move during the night. Otherwise, each night it decides randomly whether to move to the left or to the right.)

I purr with delight at this puzzle. At first it appears almost impossible that you will be able to get your hands on the furtive feline. But if you begin by trying the puzzle with a smaller number of doors, you will hopefully be able to work out the correct strategy.

Ill get you started. If there are only THREE doors, then it is possible to catch the cat in two days:

  • Day 1: open the middle door.
  • Day 2: open the middle door.

This strategy guarantees you will get the cat, since if it is not behind the middle door on Day 1, then it must be behind either of the end doors. And if it is behind either of the end doors on Day 1, then in both cases it will move to behind the middle door on Day 2. Caught!

If there are FOUR doors, it is possible to catch the cat in four days. But now its up to you to work out how.

The cat puzzle originally appeared in the New York Times now defunct Numberplay column as The Princess Problem, where a prince was knocking on doors and a flighty princess moving from room to room. This version has become a staple problem for maths teachers in Singapore. Toh Pee Choon, of Singapores National Institute of Education, told me that the princess context had great effect in stirring up interests in young girls.

I rephrased the puzzle with a cat to make it non gender specific, and also because people on the internet like looking at pictures of cats.


Ill be back at 5pm with the solution.

UPDATE: Read the solution here.

I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. Im always on the look-out for great puzzles. If you would like to suggest one, email me.


My puzzle book Can You Solve My Problems is out in paperback this week. You can get it from the Guardian bookstore or other online retailers.

Thanks to Charlie Gilderdale from maths resource project NRICH for first alerting me to this puzzle.

Read more:

Technorati Tags: ,