Tag: Mathematics

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
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.

NO SPOILERS PLEASE

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.

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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.

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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
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
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
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.

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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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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.

undefined

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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
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
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
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.

undefined
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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The hidden history of Nasas black female scientists

The diversity of Nasas workforce in 1940s Virginia is uncovered in a new book by Margot Lee Shetterly. She recalls how a visit to her home town led to a revelation

Mrs Land worked as a computer out at Langley, my father said, taking a right turn out of the parking lot of the First Baptist church in Hampton, Virginia. My husband and I visited my parents just after Christmas in 2010, enjoying a few days away from our full-time life and work in Mexico.

They squired us around town in their 20-year-old green minivan, my father driving, my mother in the front passenger seat, Aran and I buckled in behind like siblings. My father, gregarious as always, offered a stream of commentary that shifted fluidly from updates on the friends and neighbours wed bumped into around town to the weather forecast to elaborate discourses on the physics underlying his latest research as a 66-year-old doctoral student at Hampton University.

He enjoyed touring my Maine-born-and-raised husband through our neck of the woods and refreshing my connection with local life and history in the process.

As a callow 18-year-old leaving for college, Id seen my home town as a mere launching pad for a life in worldlier locales, a place to be from rather than a place to be. But years and miles away from home could never attenuate the citys hold on my identity and the more I explored places and people far from Hampton, the more my status as one of its daughters came to mean to me. That day after church, we spent a long while catching up with the formidable Mrs Land, who had been one of my favourite Sunday school teachers. Kathaleen Land, a retired Nasa mathematician, still lived on her own well into her 90s and never missed a Sunday at church.

Read more: https://www.theguardian.com/books/2017/feb/05/hidden-figures-black-female-scientists-african-americans-margot-lee-shetterly-space-race

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Mathematician Eugenia Cheng: Yes, I am an anarchist!

Eugenia Cheng combines home baking with higher-dimensional category theory. She talks about pudding, infinity, and why geeks are the new alphas

Eugenia Cheng is a British mathematician who is senior lecturer at the School of the Art Institute of Chicago. Her main interest is higher-dimensional category theory but she has also written a book about the maths of baking entitled How To Bake Pi. Her latest book is Beyond Infinity: An Expedition into the Outer Limits of Mathematics.

What is higher-dimensional category theory? Can you describe it in a sentence?
It is the mathematics of mathematics. It does for mathematics the same thing that mathematics does for the world it makes connections between things and it highlights patterns between things, so that we can be more efficient about how we use our brain power.

Youve declared your vision is to rid the world of mathematics phobia. How do you erode it once it has taken hold?
Unfortunately, the kind of maths we teach in school is often not in any way useful for most peoples lives people say When am I ever going to need to solve a quadratic equation in my life? The kind of maths I teach is about logical thinking, thinking your way through situations, understanding what is causing something to happen and working out how things fit together.

Youve said mathematicians are a bunch of rebels are you an anarchist?
Yes, definitely! Mathematicians really like making up their own rules that make sense for particular situations, and we hate having rules imposed on us.

How to Bake Pi was quite a hit. Why did you use cooking to explore maths?
It started because I always tell an anecdote when I am teaching, because I want everyone to be able to relate it to something in normal life. I realised that whenever an anecdote involved food, my students perked up. One day, one of my students called out Explain some maths using Oreo cookies, and I realised they represented something we were going to do in the lecture that day. It was this thing called conjugation, where you multiply A by B and A inverse you sandwich B between two As, one of which is the other way around. The cookie demonstrated that perfectly, because you have the cream filling between two cookies, but one of them is the other way around from the other. Suddenly they all got it, and I realised I could explain anything using a food analogy.

Whats your favourite maths-based recipe?
The one about millefeuille, because it was the one I did with [US talk show host] Stephen Colbert, and we had a rolling-pin fight. Puff pastry is one of those thingswhich is notoriously difficult to make. It also demonstrates the principle of exponentials.

Are you worried that by turning to cookery, you send the message that maths can only be exciting through analogies?
Mathematics is actually all analogies. What I am trying to do is provide the ideas and the way into something. Unfortunately, a lot of people derive their feeling of self-worth from the fact that they can understand things other people cant. I dont believe in that.

Presumably, your favourite idiom is the proof of the pudding is in the eating?
One of my students at the University of Chicago brought some pudding a bit like Angel Delight and we ate the chocolate pudding and at the bottom was a mathematical proof. It was hilarious!

Your latest book, Beyond Infinity, tackles one of the most mind-boggling concepts in maths. What is the weirdest thing about infinity?
It is one of those things, like optical illusions, where I enjoy not being comfortable with it: you can sort of swim in the weirdness of it. I dont like understanding it too much because then the illusion goes away. There is the thing about some infinities being bigger than others, but one of my favourite things is that one plus infinity is different from infinity plus one. It is like that Shakespeare thing of forever and a day that for ever and a day is longer than for ever.

Theres been a lot of discussion about the best way to teach maths, with the east Asian approach taking off in the UK. What did the west get wrong?
There is that stereotype that east Asian people are really good at maths, and because I am Chinese by origin I get this a lot. It is a bit frustrating and a bit racistthanks for removing all my agency in the things I have done in my life! But I now teach arts students, and many of them are from China and Korea, and many of them say I was put off maths because of the Asian system.

Youve said that people often tell you that you dont look like a mathematician. Are you optimistic that societal stereotypes will fade?
The stereotype is based on some reality, but I think the reality is an accident, and it is self-perpetuating. You dont have to reject looking nice in photos just because you are intelligent, and it is not a proof of intelligence if you reject wearing nice clothes and looking nice in photos. It does frustrate me when the depictions of intelligent people, especially mathematically intelligent people, in things like films are all socially weird white guys. Also, in a very pedantic way it doesnt make sense. I am a mathematician, so I look like one I am me. It is like saying That is not very feminine, but everything I do is feminine because I am female.

Theres only been one female winner of the Fields medal since it was first awarded in 1936 Maryam Mirzakhani. Does maths suffer from an old boys club mentality?
I am happy to say I have not experienced that. On the other hand, maths cares about solving big problems and proving big theorems rather than making a theory that connects things together. There is a great female mathematician, Emmy Noether, who is very neglected. She suffered for many reasons: she was Jewish in Germany in the 1930s, and she couldnt get a position because she was female, but she just carried on anyway. One of her great theorems brought together maths and physics [but] it didnt solve a particular problem, it wasnt relativity. I think it is going to be a very long time before anyone gets a Fields medal for category theory, because it brings things together rather than solving a particular problem. It is not an old boys club, it is more of an old theorems club.

Is the era of big data, coding, and the inherent reliance on numbers changing the reputation of maths? Will it ever be considered as glamorous and powerful as, say, genetics?
I have said for a while that the day of the geek is coming. I dont like the word geek particularly, because I dont think I am one. But I like thinking about the fact that when we were cave people, the important thing was to be able to defend ourselves from woolly mammoths. So we evolved to think that was a thing we needed to be attracted to; and I like to think that now we depend on computers all the time, the most important thing is to be able to fix your computer or code and therefore that is the new beating off a woolly mammoth.

You are an accomplished concert pianist does that come back to a love of maths, or is it a very different discipline?
Its partly that it is quite abstract, playing the piano. Singing is very visceral and, because you are using words, very direct. There is also so much structure in pianos and piano music. It is a mental shortcut, so you can produce more things using less brain power. It is also about a balance. Music balances out the sheer mathematical thinking that I do because it is abstract. The things it is expressing and exploring are emotions. Mathematics is doing the opposite.

Does logic have its limits?
Definitely, but the limits move. I have this image that logic is a sphere at the centre of our thoughts, and all the time we understand more mathematics we are putting more things into that central part, and it is growing. For me, the most beautiful part is the boundary between what we understand logically and what we dont. The more we understand, the more of that boundary we have, because the surface of the sphere grows. So as we go, we get more access to beauty.

Beyond Infinity by Eugenia Cheng is published by Profile (12.99). To order a copy for 11.04 go to bookshop.theguardian.com or call 0330 333 6846. Free UK p&p over 10, online orders only. Phone orders min p&p of 1.99

Read more: https://www.theguardian.com/science/2017/feb/26/eugenia-cheng-interview-observer-nicola-davis

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Nominations For The “Science Oscars” Have Opened With More Than $20m Up For Grabs

Its an argument youve probably heard many times before. Why do actors get huge salaries and glitzy award shows complete with all their envelope dramas when the real heroes like scientists go unnoticed behind the scenes?

Well, thats where the Breakthrough Prize comes in. Founded by Facebooks Mark Zuckerberg and physicist-turned-venture capitalist Yuri Milner in 2012, this annual competition seeks to award scientists with multi-million dollar prizes for groundbreaking research and discoveries. And the nominations for this years awards have just opened.

Nominations can be submitted online from now until May 31, with anyone allowed to nominate anyone else for one of the prizes (note, you cannot nominate yourself). The awards, which last year saw celebrities like will.i.am. and Vin Diesel attend, will take place at a televised ceremony in Silicon Valley in December 2017.

There will be seven prizes with an award of $3 million each. Five of these are in Life Sciences, awarded to people who have made transformative advances in understanding living systems and extending human life, according to a statement.

content-1488453929-breakthrough.jpgThe other two will be awarded in Fundamental Physics and Mathematics. The former recognizes one or more individuals who have made profound contributions to human knowledge, and is open to all physicists. The latter is awarded to someone who has made outstanding contributions to the field of mathematics.

In addition to these, there will be six New Horizons prizes, each awarding $100,000 to promising early-career researchers in Fundamental Physics and Mathematics.

Last year, winners included the team behind the LIGO experiment that discovered gravitational waves and three scientists who advanced quantum field theory (Joseph Polchinski, Andrew Strominger, and Cumrun Vafa), while Jean Bourgain scooped the Mathematics prize for work in number theory and other areas.

So, if you know someone who might be particularly deserving of one of the awards, why not go and nominate them. Hopefully La La Land wont accidentally scoopone of these prizes too.

Image in text: The Breakthrough Prize trophy, courtesy of Breakthrough Prize

Read more: http://www.iflscience.com/editors-blog/nominations-for-the-science-oscars-have-opened/

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Study reveals why so many met a sticky end in Boston’s Great Molasses Flood

In 1919, a tank holding 2.3m gallons of molasses burst, causing tragedy. Scientists now understand why the syrup tsunami was so deadly

It may sound like the fantastical plot of a childrens story but Bostons Great Molasses Flood was one of the most destructive and sombre events in the citys history.

On 15 January 1919, a muffled roar heard by residents was the only indication that an industrial-sized tank of syrup had burst open, unleashing a tsunami of sugary liquid through the North End district near the citys docks.

As the 15-foot (5-metre) wave swept through at around 35mph (56km/h), buildings were wrecked, wagons toppled, 21 people were left dead and about 150 were injured.

Now scientists have revisited the incident, providing new insights into why the physical properties of molasses proved so deadly.

Presenting the findings last weekend at the American Association for the Advancement of Science annual meeting in Boston, they said a key factor was that the viscosity of molasses increases dramatically as it cools.

This meant that the roughly 2.3m US gallons of molasses (8.7m litres) became more difficult to escape from as the evening drew in.

Speaking at the conference, Nicole Sharp, an aerospace engineer and author of the blog Fuck Yeah Fluid Dynamics said: The sun started going down and the rescue workers were still struggling to get to people and rescue them. At the same time the molasses is getting harder and harder to move through, its getting harder and harder for people who are in the wreckage to keep their heads clear so they can keep breathing.

As the lake of syrup slowly dispersed, victims were left like gnats in amber, awaiting their cold, grisly death. One man, trapped in the rubble of a collapsed fire station, succumbed when he simply became too tired to sweep the molasses away from his face one last time.

Its horrible in that the more tired they get its getting colder and literally more difficult for them to move the molasses, said Sharp.

Leading up to the disaster, there had been a cold snap in Boston and temperatures were as low as -16C (3F). The steel tank in the harbour, which had been built half as thick as model specifications, had already been showing signs of strain.

Two days before the disaster the tank was about 70% full, when a fresh shipment of warm molasses arrived from the Caribbean and the tank was filled to the top.

One of the things people described would happen whenever they had a new molasses shipment was that the tank would rumble and groan, said Sharp. People described being unnerved by the noises the tank would make after it got filled.

Ominously, the tank had also been leaking, which the company responded to by painting the tank brown.

There were a lot of bad signs in this, said Sharp.

Sharp, and a team of scientists at Harvard University, performed experiments in a large refrigerator to model how corn syrup (standing in for molasses) behaves as temperature varies, confirming contemporary accounts of the disaster.

Historical estimates said that the initial wave would have moved at 56km/h [35mph], said Sharp. When we take models … and then we put in the parameters for molasses, we get numbers that are on a par with that. Horses werent able to run away from it. Horses and people and everything were all caught up in it.

The giant molasses wave follows the physical laws of a phenomenon known as a gravity current, in which a dense fluid expands mostly horizontally into a less dense fluid. Its what lava flows are, its what avalanches are, its that awful draught that comes underneath your door in the wintertime, said Sharp.

The team used a geophysical model, developed by Professor Herbert Huppert of the University of Cambridge, whose work focuses on gravity currents in processes such as lava flows and shifting Antarctic ice sheets.

The model suggests that the molasses incident would have followed three main stages.

The current first goes through a so-called slumping regime, said Huppert, outlining how the molasses would have lurched out of the tank in a giant looming mass.

Then theres a regime where inertia plays a major role, he said. In this stage, the volume of fluid released is the most important factor determining how rapidly the front of the wave sweeps forward.

Then the viscous regime generally follows, he concluded. This is what dictates how slowly the fluid spreads out and explains the grim consequences of the Boston disaster.

It made a difference in how difficult it would be to rescue people and how difficult it would be to survive until you were rescued, said Sharp.

Read more: https://www.theguardian.com/science/2017/feb/25/study-reveals-why-so-many-met-a-sticky-end-in-bostons-great-molasses-flood

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