Tag: Mathematics

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: https://www.theguardian.com/science/2017/jul/31/can-you-solve-it-are-you-smarter-than-a-forester

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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: https://www.theguardian.com/science/2017/jul/29/maryam-mirzakhani-great-artist-mathematician-fields-medal-howard-jacobson

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10 Truth Bombs to Drop at your Next Dinner Party

1. There are more trees on Earth
than stars in the Milky Way Galaxy


Photograph by Layne Lawson

In a September 2015 paper published in the scientific journal Nature entitled “Mapping tree density at a global scale†an estimate for the number of trees on Earth was approximately 3.04 trillion.
Meanwhile, according to NASA the generally accepted answer for number of stars in the Milky way is between 100 and 400 billion. [source]

2. Speaking of stars, guess how many miles (or km)
the nearest star (after our Sun) to Earth is?


Photograph by ESO/Y. Beletsky

Like objects in your side-view mirror, stars in the night sky seem a lot closer than they are. Alpha Centauri is the closest ‘star system’ to us at an approximate distance of 4.37 light-years which works out to roughly 25 trillion miles or 40 trillion km away. 😳

3. Alaska is simultaneously the most northern,
the most western, and the most eastern state in the US


Wait, what? Look on a map and it’s easy to see that Alaska is the United States’ most northern and western state. But eastern? That’s because the Aleutian Islands are part of Alaska and stretch beyond the 180° line of longitude (which is measured from Greenwich) thus placing some of the islands technically in the Eastern hemisphere, since the dividing line for the eastern/western hemisphere is at 180° (source)

4. ‘Oxymoron’ is an oxymoron


The term was first recorded as latinized Greek oxymÅrum and is derived from the Greek where ‘oxys’ means “sharp, keen, pointed” and ‘moros’ means “dull, stupid, foolish”. Oxymoron is also an autological word, which means it expresses a property that is also possesses (e.g. the word “noun” is a noun, “English” is English, “pentasyllabic” has five syllables, and “word” is a word) [source]

5. If you start counting at one and spell out the numbers
as you go, you won’t use the letter “A” until you reach 1,000


6. Oxford University is (way) Older than the Aztec Empire


Photograph by Chensiyuan

Older, like it’s not even close. As the oldest university in the English-speaking world, Oxford is a unique and historic institution. While there is no clear date of foundation, teaching existed at Oxford in some form in 1096 and developed rapidly from 1167, when Henry II banned English students from attending the University of Paris. [source]
Conversely, the Aztec Empire, or the Triple Alliance, began as an alliance of three Nahua “altepetl” city-states: Mexico-Tenochtitlan, Texcoco, and Tlacopan. These three city-states ruled the area in and around the Valley of Mexico from 1428 until they were defeated by the combined forces of the Spanish conquistadores and their native allies under Hernán Cortés in 1521. [source]

7. The official animal of Scotland is… the Unicorn


Royal Coat of Arms of the Kingdom of Scotland used from the 12th century to 1603


According to The Scotsman: in Celtic mythology, the Unicorn of Scotland symbolized innocence and purity, healing powers, joy and even life itself, and was also seen as a symbol of masculinity and power. It has been a Scottish heraldic symbol since the 12th century and today, the Royal Coat of Arms of the United Kingdom of Great Britain and Northern Ireland still has the English lion on the left and the Scottish unicorn on the right. [source]

8. There was a third Apple co-founder, Ronald Wayne.
He sold his 10% stake for $800 in 1976.
Today it would be worth roughly $75.5 billion


Ronald Wayne worked with Steve Jobs at Atari before he, Jobs, and Wozniak founded Apple Computer on April 1, 1976. Serving as the venture’s “adult supervision”, Wayne drew the first Apple logo, wrote the three men’s original partnership agreement, and wrote the Apple I manual.
Wayne received a 10% stake in Apple. Less than two weeks later, on April 12, 1976 he relinquished his equity for US$800. Legally, all members of a partnership are personally responsible for any debts incurred by any partner; unlike Jobs and Wozniak, then 21 and 25, Wayne had personal assets that potential creditors could seize. The failure of a slot machine company, which he had started five years earlier also contributed to his decision to exit the partnership.
Later in 1976, venture capitalist Arthur Rock and Mike Markkula helped develop an Apple business plan and converted the partnership to a corporation. A year after leaving Apple, Wayne received $1,500 for his agreement to forfeit any claims against the new company. [source]

9. With just 70 people, there is a 99.9% chance
that two people share the same birthday


23 people is all it takes for there to be a 50/50 chance that two of the people share a birthday. The ‘birthday paradox‘ provides a valuable lesson in probability and reveals our tendency to think linearly instead of exponentially.
You can find a thorough mathematical explanation of the birthday paradox here, but at it’s core, we tend to think of our birthday compared to the 22 other people so there are 22 chances. But when all 23 birthdays are compared against each other, it makes for much more than 22 comparisons.
So the first person has 22 comparisons to make, but the second person was already compared to the first person, so there are only 21 comparisons to make. The third person then has 20 comparisons, the fourth person has 19 and so on. If you add up all possible comparisons (22 + 21 + 20 + 19 + … +1) the sum is 253 comparisons, or combinations. Check out the table below to see how the probability increases as the number of people do. [source]

The following table shows the probability for some other values of n (this table ignores the existence of leap years, as described above, as well as assuming that each birthday is equally likely)

10. There’s enough water in Lake Superior
to cover North and South America in a foot of water


Photograph by Lorie Shaull

To talk of Lake Superior is to talk in superlatives. Its 3 quadrillion gallons are enough to cover both North and South America under a foot of water; it holds 10% of the world’s surface fresh water supply; at 31,700 square miles (82,100 sq km) it’s roughly the size of Maine.
If all 7 billion people on Earth drank a gallon of water per day it would collectively take us 1,174 years to drain it. [source]

Read more: http://twistedsifter.com/2017/07/10-truth-bombs-to-drop-at-your-next-dinner-party/

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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: https://www.theguardian.com/science/2017/jul/17/can-you-solve-it-are-you-smarter-than-an-architect

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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: https://www.theguardian.com/us-news/2017/jul/15/maryam-mirzakhani-mathematician-dies-40

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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: http://twistedsifter.com/2017/07/profs-use-this-puzzle-to-teach-lesson-about-problem-solving/

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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: https://www.theguardian.com/science/2017/jul/03/can-you-solve-it-are-you-smarter-than-a-cat

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Can you solve it? Pythagoras’s best puzzles

Three teasers from the vaults

Hi guzzlers,

The most famous theorem in maths is named after the Greek thinker Pythagoras. So is the most famous recreational mathematics publication in the Netherlands.

Pythagoras Magazine was founded in 1961, and to celebrate its half century it recently published a selection of its best brainteasers in English. Ive selected three of them here, in increasing order of difficulty.

1) Dollar bills. In a bag are 26 bills. If you take out 20 bills from the bag at random, you have at least one 1-dollar bill, two 2-dollar bills, and five 5-dollar bills. How much money was in the bag?

2) Yin and Yang. The Yin-Yang symbol is based on the figure below, bordered by three semi-circles. How can you divide this shape into two identical shapes?

Big yin

3) Huge pie. A huge pie is divided among 100 guests. The first guest gets 1% of the pie. The second guest gets 2% of the remaining part. The third guest gets 3% of the rest, etc. The last guest gets 100% of the last part. Who gets the biggest piece?

Ill be back later today with the solutions.


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.

Thanks to the editors of Pythagoras Magazine for todays puzzles. You can check out more of them in Half a Century of Pythagoras Magazine.


Football School, which I which I co-wrote with Ben Lyttleton, is a book for 7 to 13-year olds children that uses football to explain subjects like English, maths, physics, geography, philosophy and zoology. You (by which I mean any 7-13-year-olds you may know) can check out the Football School YouTube channel, in which Ben and I answer all questions about football and life. Submit your questions and subscribe!

Read more: https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles

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Can you solve it? The incredible sponge puzzle

This brainteaser will wring out your brain

Hi guzzlers.

For todays puzzle, let me introduce you to the Menger sponge, a fascinating object first described by the Austrian mathematician Karl Menger in 1926. Well get to the problem as soon as I explain what the object is.

The Menger sponge is a cube with smaller cubes extracted from it, and is constructed as follows: Step A: Take a cube. Step B: Divide it into 27 smaller subcubes, so it looks just like a Rubiks cube.



Step C: Remove the middle subcube in each side as well as the subcube at the centre of the cube, so if you looked through any hole you would see right through it. Step D: Repeat steps A to C for each of the remaining subcubes, that is, imagine that each subcube is made from 27 even smaller cubes and remove the middle one in each side and the central one.

We could carry on repeating steps A to C ad infinitum, on smaller and smaller subcubes, but here lets do it just once more:

Menger sponge. Illustration: Edmund Harriss/Visions of Numberland

Menger sponges are so loved within the maths community that building origami models of them out of business cards is a thing.

Menger sponge made as part of Matt Parker and Laura Taalmans MegaMenger project. Photograph: MegaMenger

There are lots* of reasons why Menger sponges are cool and one of them is illustrated by todays puzzle.

How to slice a cube in two.

On the left here is how you slice a cube in half such that the cross section is a hexagon.

When you slice a Menger sponge in two like this, what does the hexagonal slice look like?

This question is probably the most difficult one I have ever set in this column, as it requires phenomenal levels of spatial intuition. But I urge you to give it a go, even if just a basic sketch. Send me some images, or post them to me on social media. You may draw something along the right lines…

Please forgive me, though, for posing this toughie. The answer is jaw-droppingly amazing. In fact, I was told about the Menger slice by a respected geometer who told me it gave him probably his biggest wow moment in maths. Come back at 5pm BST and see for yourself.

NO SPOILERS PLEASE! Please talk about Karl Menger and origami instead.

Photograph: Bloomsbury

Both the Menger sponge and the Menger slice are included in my latest book, Visions of Numberland: A Colouring Journey Through the Mysteries of Maths. The book is a gallery of the most spectacular images that Edmund Harriss, my co-author, and I could find in maths. You can colour them in, or just contemplate them in black and white.

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.

* Here are a couple. 1) Each time you follow the iteration described in steps A to C you decrease the volume of the sponge, but increase its surface area. After an infinite number of iterations, you will have removed an infinite number of cubes. The sponge will then have zero volume and infinite surface area. 2) After an infinite number of iterations, the object is a fractal, that is, it contains parts that are identical to the whole thing.

Read more: https://www.theguardian.com/science/2017/apr/10/can-you-solve-it-the-incredible-sponge-puzzle

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‘Granny style’ is best way to take a basketball free throw, study shows

Mathematical analysis reveals that for players with good control, using an unorthodox underarm technique gives better odds of scoring

It might invite ridicule, but it gets results. A scientific analysis has concluded that using a granny style underarm technique is the optimal way to take a free throw in basketball.

Adopting the unorthodox strategy could result in marginal gains for professional players, the research suggests. And, as sporting doctrine goes, marginal gains can lead to remarkable results.

Madhusudhan Venkadesan, who led the work at Yale University, said: Our mathematical analysis shows that if the thrower is capable of controlling the release angle and speed well, the underarm throw is slightly better for a basketball free throw.

However, it remains to be seen whether science will prove more persuasive than professional advocates of the underarm style.

The retired NBA player Rick Barry, a pioneer of the underarm free throw, was one of the most effective shooters of all time and when he retired in 1980 his 90% free throw record ranked first in NBA history. But he struggled to convince his teammates due to the inescapable fact that shooting underarm makes you look like a sissy, Barry said.

Venkadesan acknowledges that it is a difficult case to make.

One suspects there are social and cultural reasons you dont see that practised too often, he said. So what if some call it the granny throw? What matters is that the ball goes through the hoop! Rick Barrys record does support the underarm throw.

The study, published in the journal Royal Society Open Science, considered the chances of the ball being on target, depending on the style, speed and accuracy of a throw.

It found that if the player is capable of controlling the release angle and speed well, the underarm throw has slightly better odds of going in. But for amateurs who have only crude control, the release of the ball overarm is safer, sparing casual players the dilemma of choosing style or results.

An important factor in comparing the two strategies was how the ball approaches its target. When the ball approaches the net from directly above, as in a typical underarm throw, the cross-section of the target is large from the balls vantage point. This is good, as it means that if a throw is close to being exactly on target it has a very high chance of going in.

However, in trying to achieve this straight down entry, the amateur risks lobbing the ball extremely high due to their mediocre control. In this scenario, a small error in the timing of the release can cause the ball to grossly overshoot or undershoot the hoop.

So the overarm shot, where the ball sees a smaller cross-section of the hoop, but is less likely to go wildly off course, is a more conservative strategy.

This competition between the entry angle and speed underlies both the speed-accuracy trade-off and the relative accuracy of one style versus another, said Venkadesan.

For the professional player, the analysis predicts, this trade-off is finely balanced and probably within the margins of error of the model, which did not consider the backboard.

Barry, no doubt, would view the findings as confirmation of what he has argued all along. From the physics standpoint, its a much better way to shoot, he told the author Malcolm Gladwell in a recent interview. You have a little bit more margin for error than when you shoot overhand.

Read more: https://www.theguardian.com/science/2017/apr/26/granny-style-is-best-way-to-take-a-basketball-free-throw-study-shows

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