Beijing (CNN)Imagine taking a five-hour exam just to get a job interview.
Beijing (CNN)Imagine taking a five-hour exam just to get a job interview.
Newcombs problem has split the world of philosophy into two opposing camps. Two philosophers explain – then take the test yourself
Two boxes or not two boxes? That is the question.
For almost half a century Newcombs problem has been one of the most contentious conundrums in philosophy, with ramifications in economics, politics and computer science.
Vast amounts have been written about it, yet thinkers cannot agree on the right answer. Rather uncharacteristically for the gentle and cerebral world of philosophy, here is a debate in which each side is extremely confident they are correct and that the other side is wrong.
On which side do you sit? Read the problem and submit your decision below. In order to help you make up your mind, Ive enlisted two philosophers with opposing views to persuade you as to the most sensible course of action.
Two closed boxes, A and B, are on a table in front of you. A contains 1,000. B contains either nothing or 1 million. You dont know which. You have two options:
You keep the contents of the box/boxes you take, and your aim is to get the most money.
But heres the thing. The test was set by a Super-Intelligent Being, who has already made a prediction about what you will do. If Her prediction was that you would take both boxes, She left B empty. If Her prediction was that you would take B only, She put a 1 million cheque in it.
Before making your decision, you do your due diligence, and discover that the Super-Intelligent Being has never made a bad prediction. She predicted Leicester would win the Premier League, the victories of Brexit and Trump, and that Ed Balls would be eliminated yesterday from Strictly Come Dancing. She has correctly predicted things you and others have done, including in situations just like this one, never once getting it wrong. Its a remarkable track-record. So, what do you choose? Both boxes or just box B?
Why you should take B only.
By Dr Arif Ahmed, Reader in Philosophy at Cambridge university and a Fellow of Gonville and Caius College. His most recent book is Evidence, Decision and Causality.
You know that the Super-Intelligent Being is always right. So whatever you do, She will have predicted it! If you take box B only, She will have predicted that and you will get 1 million. If you take both boxes for any reason for instance, because of what David is about to say She will have predicted that and you will get only 1,000. Clearly, you should take box B only.
Why you should take both boxes.
The Super-Intelligent Being has already made Her prediction before you make your decision. So, she has either put 1 million in Box B or not. What do you have to lose by choosing both boxes? You cannot influence a decision made in the past by a decision made in the present! Think of it this way. Suppose box B has transparent glass on the far side the side I cant see. And suppose I have a friend who can see through that glass side, and knows whether or not the 1 million is there. If my friend were allowed to communicate with me, what would his advice be? Surely it would be to take both boxes. If the 1 million is there, it is not going to disappear in a puff of smoke by my decision to take both boxes. Taking both boxes will always enrich me by an extra 1,000 in comparison to taking B only.
Now its your turn. What will you do?
Ill be back later with the results of the decisions you made, and a further discussion of this problem.
Meanwhile, I encourage fierce debate a boxing match? – below the line about the pros and cons of each of the choices. Explain why you chose what you did.
Margaret Hamilton, a computer scientist who was pivotal in humans first landing on the Moon, was awarded the Presidential Medal of Freedom this week, honoring her pioneering work.
In the ceremony led by President Barack Obama on Tuesday, Hamilton was one of 21 recipients of the award, the highest civilian award in the United States. Another female computer scientist, Grace Hopper, was also posthumously given an award.
Everyone on this stage has touched me in a powerful personal way, Obama said at the ceremony, reported the New York Times. These are folks who have helped make me who I am and think about my presidency.
Hamilton began working with NASA in the 1960s as part of the Massachusetts Institute of Technology (MIT), which she had joined as a computer scientist after completing her degree in mathematics from the University of Michigan.
MIT was given a contract by NASA in 1961 to develop the guidance and navigation system for the Apollo spacecraft that would go to the Moon, and Hamilton was put in charge of the Software Engineering Division. She worked long hours, aware of how pivotal her code was to the success of the mission.
I was always imagining headlines in the newspapers, and they would point back to how it happened, and it would point back to me, she told Wired.
Hamilton pictured in the Apollo Command Module. NASA
This work would prove crucial during the descent of Apollo 11 to the lunar surface. Just minutes before the landing, the software overrode a command, causing some confusion. But the resulting 1202 alarm from the software let everyone know it was simply shedding less important tasks to give more focus to the engine. The landing was thus able to continue without aborting.
(In fact, the story behind this is incredibly interesting, as it was thanks to a young computer engineer called Jack Garman that the decision to ignore the alarm and proceed with the landing was made. He passed away in September this year.)
Also awarded the Medal of Freedom was Grace Hopper, a computer scientist who helped make coding languages more practical, along with creating the first compiler for code. She was known as Amazing Grace and the first lady of software, and remained at the forefront of programming from the 1940s through the 1980s. Hopper passed away in 1992, but was given the award posthumously.
Both Hamilton and Hopper were wholly deserving of their awards, and its great to see what might be regarded as unsung heroes given some deserved attention. Hamiltons software would continue to be used through the later Apollo missions, and was adapted for NASAs first space station (Skylab) and the Space Shuttle.
A bake off for the brain
To celebrate my birthday this week I thought Id serve up three cake-based puzzles. Ent-icing! No soggy cerebellums please. Ready. Steady. THINK
1.You have a square cake, and four friends. How do you divide the cake into five slices of equal size? Each slice must be slice-like, meaning that the knife cuts vertically through the cake and the tip of each slice is at the centre of the cake. You have no ruler or tape measure, but you can use the horizontal grid here.
Yes, Pi Day is a day we put aside our cake-loving differences and take a few moments to show some appreciation to those flaky, buttery, delicious bundles of heaven.
Whether you’re looking for a recipe for a sweet dessert pie or a savory one (think quiche), people are flocking to all-things-DIY haven Pinterest to find new ones to enjoy.
The top trending Pinterest pies around the world are sure to make your Pi Day much more delicious. Check them out below, and head to Pinterest to see all the most popular pies on one glorious board.
Most Pinned: Maple Bacon Pie
Germany: Strawberry Pie
Brazil: Tortinha de Morango no Copinho
South Korea: Tomato Pie
Japan: Rose Pie (an apple pie made to look like roses)
Mexico: Lime Pie
Australia: No-Bake Vegan French Silk Pie
Spain: Apple Pie Chimichangas
UK: Vegetarian Shepherd’s Pie
U.S.: Boston Cream Pie
The solution to todays topical teaser
Earlier today I set you the following puzzle:
Illustrated below is a quarter-circle, containing two semicircles of smaller circles. Prove that the red segment has the same area as the blue.
Children in Swindon are being failed by schools “at every level”, according to education inspectors.
In a letter to Swindon Borough Council, head teachers and local MPs, Ofsted’s Bradley Simmons said the town’s schools were a “cause for serious concern”.
He said immediate action was needed and urged “all involved” to unite so pupils could get the “education they deserve”.
The council said the criticism was misplaced, and work was continuing to improve standards where necessary.
The letter was published following an Ofsted inspection of the borough council’s arrangements for supporting school improvement.
In it Mr Simmons said he had raised concerns with the borough council “on at least three separate occasions”.
He said Swindon was in the “bottom 10 local authorities nationally” in phonics in 2016, while the town’s seven-year-olds were the “joint lowest performers in reading in the South West”.
At Key Stage 2, he said only 44% of 11-year-olds reached the “new expected standard in reading, writing and mathematics”, while at GCSE level only 17.3% of pupils – compared to 22.8% nationally – achieved the English Baccalaureate.
The Ofsted letter leaves no doubt as to the scale of concern about schooling in this large industrial town in the south-west of England.
Only a handful of such warning letters addressed to every organisation involved in an area’s education have ever been issued.
Mr Simmons exhorts academy bosses, head teachers, the local education authority and the Regional Schools Commissioner, which oversees academies, to join with local politicians and governors to make improvements.
This range of addresses reflects the complex modern education landscape.
Of the 15 secondary level schools in the area with GCSE results, only one is a traditional local authority-maintained school.
Twelve are academies, the privately run but state-funded schools ministers see as the engine of school improvement in England. The other two are special schools.
If the education for the area’s pupils, said to be failing at every level, is to be turned around, a concerted effort from a large range of “key players” will be needed.
“Pupils in Swindon are being failed at every level. Primary school performance which had previously shown a positive trend of improvement in Swindon is now a concern,” Mr Simmons said.
“Recent inspections of five secondary schools in the town also indicate a trend of decline, with only one of these schools being rated good.
“Of the others, one went from good to ‘requires improvement’, one failed to improve from requires improvement and two went from requires improvement to inadequate.”
In an open letter response, the borough council said it felt the data released had been “used selectively”.
It said that claims it had failed its pupils were “overly harsh and indeed unfair”.
Allison Standley is among those who have reacted on social media. She wrote: “Another blow for our very hard working teachers, how much more can they take? Parents, you have a huge responsibility too you know.”
Jacob Samuel Allinson agreed, saying he wished “more parents worked as hard at educating their children as teachers do”.
But Jessy Webster, a politics graduate, tweeted the criticism was “long overdue” as “Swindon schools were notoriously bad” when she was in Wiltshire.
Some parents and teachers have contacted the BBC to share their views.
Christel Stevens said she and her husband were teachers in other boroughs and had just transferred their son out of a primary school in Swindon to one elsewhere in Wiltshire after a “poor experience”.
She added: “He is thriving now despite reception year being wasted.”
Parent Sarah-Kate Tonkin disagreed with the findings, saying: “If you want to talk academic achievement [measurable] one of my older children has been in NATIONAL finals for Maths and STEM [Science, Technology, Engineering and Maths] this year.
“That is *not* a sign of failing pupils as far as I can see.”
But parent James Garfield described his child’s primary school as an “utter disgrace”.
He said: “When challenged as to why the school did not set homework I was informed that the teacher in question did not have homework when he was at school and he did all right – hardly the point.
“I believe the reason the schools in north Swindon in particular are so bad is that the head teachers are prepared to play social experiments with our children such as the no homework policy.”
Key Stage 2 results, published in September 2016, suggest Swindon had one of the lowest levels of attainment, but not the worst.
The average, based on the new assessment criteria, was for 53% of children to meet the required standard in all of reading, writing and mathematics by the time they finished year six. In Swindon it was 44%. The same was true of Liverpool and West Sussex.
Luton and Dorset did worse with 43%, while Bedford achieved 42% and Peterborough 39%.
However, a look at the test results suggests Swindon was only slightly out of step with the national average when it came to individual subjects.
The England average for reading was 66% meeting the expected standard. In Swindon it was 65%.
Grammar, spelling and punctuation saw 72% of pupils in England meet the expected standard. Swindon was exactly the same.
Mathematics saw 69% of children in Swindon meet the expected standard, compared with an England average of 70%.
The problem as far as Key Stage 2 goes is that not enough children did well enough across the board.
20,368 Academy pupils
10,858 Primary school pupils
1,064 Secondary school pupils
Swindon Borough Council said, overall, phonics test results were lower than it would have liked at the end of year one, but “children have caught up and are above the national average” by the end of year two.
It said 11-year-olds were meeting the national average for reading, mathematics and grammar, but admitted writing results were “lower this year”.
And despite it sharing Ofsted’s “concerns about secondary education”, the council said “GCSE results have continued to improve”.
“By making his views so public in this way, Mr Simmons must have recognised the demotivating impact they would have on teachers,” the authority said.
“In fact it has really angered and annoyed many head teachers of good and outstanding schools who are doing an excellent job.”
North Swindon Tory MP Justin Tomlinson said it was “very disappointing news”.
“I will do all I can support both Swindon Borough Council and the schools highlighted to deliver immediate improvements,” he said.
Happy Pi Day, where we celebrate the worlds most famous number. The exact value of =3.14159 has fascinated people since ancient times, and mathematicians have computed trillions of digits. But why do we care? Would it actually matter if somebody got the 11,137,423,895,285th digit wrong?
Probably not. The world would keep on turning (with a circumference of 2r). What matters about isn’t so much the actual value as the idea, and the fact that seems to crop up in lots of unexpected places.
Lets start with the expected places. If a circle has radius r, then the circumference is 2r. So if a circle has radius of one foot, and you walk around the circle in one-foot steps, then it will take you 2 = 6.28319 steps to go all the way around. Six steps isnt nearly enough, and after seven you will have overshot. And since the value of is irrational, no multiple of the circumference will be an even number of steps. No matter how many times you take a one-foot step, you’ll never come back exactly to your starting point.
Calculating the area of a circle with wedges. Jim.belk
From the circumference of a circle we get the area. Cut a pizza into an even number of slices, alternately colored yellow and blue. Lay all the blue slices pointing up, and all the yellow slices pointing down. Since each color accounts for half the circumference of the circle, the result is approximately a strip of height r and width r, or area r2. The more slices we have, the better the approximation is, so the exact area must be exactly r2.
Pi in other places
You don’t just get in circular motion. You get in any oscillation. When a mass bobs on a spring, or a pendulum swings back and forth, the position behaves just like one coordinate of a particle going around a circle.
Simple harmonic motion is another view of circular motion.
If your maximum displacement is one meter and your maximum speed is one meter/second, its just like going around a circle of radius one meter at one meter/second, and your period of oscillation will be exactly 2 seconds.
Pi also crops up in probability. The function f(x)=e-x, where e=2.71828 is Eulers number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of .
How did get into it?! The two-dimensional function f(x)f(y) stays the same if you rotate the coordinate axes. Round things relate to circles, and circles involve .
Another place we see is in the calendar. A normal 365-day year is just over 10,000,000 seconds. Does that have something to do with the Earth going around the sun in a nearly circular orbit? Actually, no. Its just coincidence, thanks to our arbitrarily dividing each day into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.
Whats not coincidence is how the length of the day varies with the seasons. If you plot the hours of daylight as a function of the date, starting at next weeks equinox, you get the same sine curve that describes the position of a pendulum or one coordinate of circular motion.
Advanced appearances of
More examples of come up in calculus, especially in infinite series like
1 – (13) + (15) – (17) + (19) + = /4
12 + (12)2 + (13)2 + (14)2 + (15)2 + = 2/6
(The first comes from the Taylor series of the arctangent of 1, and the second from the Fourier series of a sawtooth function.)
Also from calculus comes Eulers mysterious equation
ei + 1 = 0
relating the five most important numbers in mathematics: 0, 1, i, , and e, where i is the (imaginary!) square root of -1.
At first this looks like nonsense. How can you possibly take a number like e to an imaginary power?! Stay with me. The rate of change of the exponential function f(x)=ex is equal to the value of the function itself. To the left of the figure, where the function is small, its barely changing. To the right, where the function is big, its changing rapidly. Likewise, the rate of change of any function of the form f(x)=eax is proportional to eax.
We can then define f(x)= eix to be a complex function whose rate of change is i times the function itself, and whose value at 0 is 1. This turns out to be a combination of the trigonometric functions that describe circular motion, namely cos(x) + i sin(x). Since going a distance takes you halfway around the unit circle, cos()=-1 and sin()=0, so ei=-1.
Finally, some people prefer to work with =2=6.28 instead of . Since going a distance 2 takes you all the way around the circle, they would write that ei = +1. If you find that confusing, take a few months to think about it. Then you can celebrate June 28 by baking two pies.
Its often said that if you want to understand the universe, you need to think in equations. While mathematics can easily set off a high school flashback, its often only the raw power ofnumbers thathave the ability to express the inexpressible complexity and beauty of our universe.
So, surely with that in mind, BBC Earth asked a group of mathematicians and physicists what their favourite equation was. Their picks included: The Dirac equation, Riemann’s formula, Pi (how it relates to the circumference of a circle), and Einstein’s field equation.
This shortlist was then put up for the public to choose their favourite in an online poll. After nearly 60,000 votes, the winners were announced back in January, but as it’s Pi Day we thought we’d take another look.
The most popular equation, with nearly 20,000 votes, was the Dirac Equation. The equation is loved both for its elegance and as a symbol of 20th century physics. The equation was proposed in 1928 by the British physicist Paul Dirac, in an attempt to weld together ideas of Einsteins relativity with quantum mechanics. In essence, Dirac managed to explain how electrons behave when they travel close to the speed of light. This work went on to explain and predict the existence of antimatter; the idea that every particle has a mirror-image antiparticle. This, of course, turned out to be right on the money.
Coming in at number two was Eulers Identity. The equation was first described in the 1748 book, Introduction to Analysis of the Infinite by Leonhard Euler. This cult favorite of equations has cropped up in The Simpsons on more than one occasion, and was also Richard Feynman’s personal favourite, describing it as “the most remarkable formula in mathematics.”
It combines five of maths most fundamental constants: 0, 1, e, i, and . On top of that is has three mathematical operations: addition, multiplication, and exponentiation. It also has a lot of applications outside the ivory tower, from communication, meteorology, medicine, navigation, energy, robotics, manufacturing, and finance. These characteristics have given the equation a reputation of being the key to something an infinitely connected universe and something deeply transcendental.
The third most popular was Pi (). Its beauty stems from the idea it is simultaneously simple yet profoundly deep. Most simply put, the irrational number Pi explains the relationship between a circles diameter and its circumference. That number is 3.141… and so on, to infinity, in a non predictable pattern.
Here are the rest of the results, going from most popular to least: 4) Einstein’s field equation5) Riemann’s formula 6) the wave equation 7) Euler-Lagrange equation 8) Bayes’s theorem 9) a “simple” arithmetic progression 10) Hamilton’s quaternion formula 11) the logistic map and 12) the Yang-Baxter equation.
Make sure you check out BBC Earth for more information behind these little mathematical windows to the universe.
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.
The 10 letter keys on the top line of a typewriter are
Q W E R T Y U I O P
Can you find a 10-letter word that uses only these keys?
Jasper Jason works for local radio. This is his business card. Can you spot the pattern?