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A Brilliant Young Mind


Maths at the Movies – A Brilliant Young Mind
~A socially awkward teenage math prodigy finds new confidence and new friendships when he lands a spot on the British squad at the International Mathematics Olympiad~

Paradox – an absurd truth


By definition, a Paradox is an absurd statement which may prove to be true. The amazingly impossible designs of the artist Escher are excellent examples and his work is shown in the sculpture above. Mathematics is one large system of logic – a kind of universal language that transcends individual civilizations and specific languages. As such, certain paradoxes have arisen that have troubled mathematicians from ancient times to the present. Some are false paradoxes in that they do not present actual contradictions, but are merely slick logic tricks. Others have shaken the very foundations of Mathematics – requiring brilliant, creative mathematical thinking to resolve. Others remain unresolved to this day. Check some of these out in the Math Forum.


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Trick Math Question Upsets Beethoven


Frustrated maths fans were recently seething at a puzzling question. The trick question about Beethoven’s 9th Symphony was trending on Twitter after an alarmed person posted it.

“An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th symphony,” the question goes. “How long would it take for 60 players to play the symphony?”

The incensed tweeter wrote: “That’s not how it works. That’s not how any of this works.” Read more here.

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John Sims knows the Art of Math


“I am inspired by Pythagoras, who saw maths sitting at the centre of art, life and nature.”

This picture of John Sims was taken at a recent exhibition where he displayed 13 math/art quilts, nine dresses based on the number Pi, a blues composition based on Pi and many other mathartifacts. In his own words, “I grew up in Detroit, Michigan, and became interested in maths through a high-school science-fair project on Pythagorean triples. It was in graduate school that I started to connect maths and art. I taught a calculus course where I allowed the students to make a ‘cheat sheet’ of notes and formulae to take into the exam. One was visually stimulating, so I bought it. Later, I met mathematician John Horton Conway and sculptor Brent Collins who got me excited about visual maths and art. Soon after, I went to Ringling College of Art and Design in Sarasota, Florida, to develop a maths curriculum for art students.

I admire the work of the sixteenth-century painter Albrecht Dürer, particularly his use of magic squares [number grids in which every row, every column and the diagonals sum to the same constant]. I like the way that M. C. Escher was able to draw on the tradition of Islamic geometric art in a representational context, and I like his lithograph of an impossible waterfall inspired by the work of British mathematician Roger Penrose. In the conceptual realm, I like the surrealist artist Marcel Duchamp for his subversive audacity. However, my greatest influence is the unfolding system of structures, patterns and cycles of nature itself.

It is art that embraces the spirit, language and process of mathematics. Both maths and art are concerned with truth, but they differ in their ways of searching for it. Maths uses analysis and proof; art uses the senses and emotions. But maths can harness the spirit of creativity and art can be analytical. Together they form a great alliance for understanding the world around us.” From an interview in Nature.

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The God Number Solves Rubik’s Cube


The world has waited with bated breath for 30 years, and now finally a group of academics, engineers, and math geeks has discovered the number that explains life, the universe, and everything. That number is 20, and it’s the maximum number of moves it takes to solve a Rubik’s Cube.

Known as God’s Number, the magic number required about 35 CPU-years and a good deal of man-hours to solve. Why? Because there’s 43,252,003,274,489,856,000 possible positions of the cube, and the computer algorithm that finally cracked God’s Algorithm had to solve them all. (The terms “God’s Number/Algorithm are derived from the fact that if God was solving a Cube, he/she/it would do it in the most efficient way possible. The Creator did not endorse this study, and could not be reached for comment.) Read more here.

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You need to build a pen for your dog…


‘Sadie’ is such a lovely dog – playful, cuddly and, well, just cute! But, your parents are going out to buy 24 meters of fencing to make a pen for dear Sadie. What will you do while they are out shopping? Why, play with Sadie of course? No, no, no – you have to put your math skills to use and work out the maximum size that Sadie’s pen can be made using the 24 meters of fencing. But, being the helpful parents they are, they have suggested that you use an Excel spreadsheet to find out the maximum area using different lengths and widths for the pen. A snapshot of the spreadsheet is set up for you and one attached for you to complete. The solution will be posted in the ‘Post Support’ shortly, or you can comment back to this post. Good luck!

Area of Dog Pen-1cs7z3a

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Math Jokes – you either love them or h8 them!


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An ancient tablet that demonstrates undoubted genius


For nearly 100 years, the mysterious tablet above (no, it’s not an iPad) has been referred to as Plimpton 322. It was first discovered in Iraq in the early 1900s by Edgar Banks, the American archaeologist on which the character Indiana Jones is thought to have been largely based.

Now researchers from the University of New South Wales are calling it one of the oldest and possibly most accurate trigonometric tables of the ancient world.

Findings published in the journal Historia Mathematica, the official journal for the International Commission on the History of Math, reveal how researchers dated the ancient clay tablet and came to conclusions about its use.

The tablet is arranged in a series of 15 rows intersected by four columns. According to the UNSW researchers the tablet uses a base number of 60, which may have been used to allow ancient Babylonians to derive integers instead of fractions.

Norman Wildberger, explained that the research team reached their conclusions that the tablet was used for the study of triangles by findings based on ratios, not angles. In the top row of the tablet, said Wildberger, relatively equal ratios create a near equilateral triangle. Descending down the tablet, the ratios decrease the triangle’s inclination, creating narrower triangles.

It is a fascinating mathematical work that demonstrates undoubted genius,” said University of New South Wales researcher Daniel Mansfield in a press release.

The researchers speculate the tablet could have been used to survey fields or construct buildings. For example, knowing the height and width of a building, ancient builders would have been able to calculate the exact measurements need to build pyramid slopes. (source: National Geographic)

Watch more here…

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Sad loss to field of theoretical Mathematics


Maryam Mirzakhani, an Iranian-born mathematician who in 2014 became the first woman awarded the Fields Medal, often called the most prestigious prize in mathematics, died July 15. She was 40.

Stanford University, where she had been a professor since 2008, announced her death. The cause was breast cancer.

Dr Mirzakhani grew up in Tehran and came to the United States in 1999 for graduate study at Harvard University. Her mathematical interests included the theoretical study of complex geometric shapes and the movement of billiard balls across surfaces.

Her work was deeply theoretical, but other mathematicians considered it boldly original and of untold future importance. Her doctoral dissertation, which she completed in 2004, solved two long-standing mathematical problems and led to publications in three major mathematics journals. (source: stuff.co.nz)

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The Mathematical Genius Behind the oldest International Sport


Team New Zealand is the crew sailing against America for “The Old Mug”, a rather ugly trophy that represents the Formula 1 of yachting and is the oldest international sporting competition. It was initially sailed in large sloops, but is now hi-tech in catamarans than lift off the ocean on complex “foils” – thereby providing less hull resistance. This equals high speeds and the use of complex controls to gain speed advantage. Team New Zealand’s Technical Director, Dan Bernasconi, who holds a PhD in mathematical modelling and aerodynamics, backed up with a Masters from Cambridge University, has been credited as the key driving force behind Team NZ’s radical design. You can follow this exciting application of fluid Mathematics in action here.

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Oxford Exam Answer: According to Rebecca Cotton-Barratt, of Christ Church, this maths question tests abstract thinking”

“I’d initially ask the candidate what shape they think will be formed, and then ask them how they can test this hypothesis,” Cotton-Barratt says.

“They might initially try sketching the ladder at different stages – but ultimately what we want is something that we can generalise and that is accurate (you can’t be sure that your drawing is that accurate, particularly when you’re making a sketch on a whiteboard and don’t have a ruler). So eventually they will fall back on maths, and try to model the situation using equations.

“If they get stuck we would ask them what shape the ladder makes with the wall and floor, and they’ll eventually spot that at each stage the ladder is forming a right-angled triangle. Some might then immediately leap to Pythagoras’ Theorem and use that to find the answer (which is that it forms a quarter circle centred on the point where the floor meets the wall).Of course, Pythagoras could easily find the hypotenuse – it is the green line along the water! (Hint: the hypotenuse is always opposite the right angle!)

Frustratingly there is no definitive answer to the riddle, leaving guessers with no choice but to continue scratching their heads.

Dr Kevin Bowman, course leader for Mathematics at the University of Central Lancashire said: ‘You can interpret it in many ways; one way is no more correct than another.

“There’s no ambiguity in the first equation; 3 apples is 30, so one apple is worth 10.
The Fruit Puzzle…
This isn’t the first mind-bending puzzle to sweep the internet in recent months. Earlier this year, National Geographic’s puzzle asking you to identify which direction a bus is travelling in left thousands of adults scratching their heads (see earlier post). One person suggests that, “because all the bananas aren’t the same, you could say that they all represent different amounts. You might even say that the two coconut pieces in the third equation are different sizes, and therefore add up to three quarters or even seven eighths when put together. In that sense, there are an infinite amount of possible answers.”

Dr Kevin Bowman, course leader for Mathematics at the University of Central Lancashire said: ‘You can interpret it in many ways; one way is no more correct than another.

“There’s no ambiguity in the first equation; 3 apples is 30, so one apple is worth 10.”

Another said, “1 apple equals 10, coconut equals 6 and banana bunch equals 4 so your answer is 20.”

All exterior angles of one coin add up to 360 degrees. Since a coin has 12 sides, each exterior angle = 30 degrees. Two angles are formed between the two coins. Therefore, the angle formed is 60 degrees.

Quite an easy pattern in the Oct 10-11 Post. Subtract the first two numbers to get the first number in the right column; add the first two numbers in the left column to get the last two of the right column!

Parking Lot Puzzle: Turn your computer screen upside down (or stand on your head), then it becomes easy 🙂

In each row, adding gives the last 2 digits and subtracting gives the first.

The blue cherry picker has an extension arm that can’t be seen very well. This has placed the workers closer to the camera and created a strong false sense of scale simply because your eye assumes that the workers should be on the same plane as the base of the cherry picker!

Yes, it was Major General Stanley in the “Pirates of Penzance!” Check out the link in the picture.

The extra rope needed is exactly 2 x pi or 6.28m!

Christmas Teaser: Today is the 1st of January. Bill’s 8th birthday was yesterday, so the day before (December 30) he was still 7 years old. This December he will turn 9 and, next year, will be 10!

What did the math mother feed her new baby? Formula Milk!

What is a bubble? It is a thin sphere of liquid enclosing air (in most cases) or another gas.

Number of toes = 5170

How many Mathematicians to change a light bulb? Why, n+1 of course (one to hold the light)!

Jan 24, 2014: Assuming a free fall rate of 9.8m/sec/sec it would take just 4.06sec to fall 81m.

= 1 (see first line in the post)

Yes, the TV show with hints of Mathematics and Physics (along with the usual tensions of flatmates?) – did you choose 79?

Leonhard Euler (1707-1783) was an incredibly productive mathematician who published almost 900 books! He took an interest in Latin Squares – grids where each row and column each contains a member of a set of numbers. This forms the basis for Sudoku!

Trig Ratios post: yes, the Sine and Cosine ratios are the same when their angles add up to 90 degrees! This relationship can be expressed as: Sine A = Cosine (90-A) or Cosine A = Sine (90-A)
Good work in identifying this trig pattern. Now, here is a follow up questions which we will address in the next post. Does this pattern suggest that there is a link between Sine and Cosine ratios? Come on, come on… be quick with your answer…Yes, well done – of course there must be!!

Yes, zero is an Integer (which keeps to negative and positive integers apart).

Sam had to position himself to make sure that he 8 the chocolate!

There are 7 days in a week (i.e. Modulo 7). 490 days will be the same day that you chose, so the 491st day will be tomorrow!

Yes, C is the missing section – giving the same difference between numbers in the rows and columns.

Other answers:
That’s a mean looking crocodile! Unless, of course, you knew that it measured just 40cm – yes, just over a foot long!! The camera’s wide angle lens has distorted the image and this makes tiny croc look menacing!

Yes, the 100m time for Bolt works out to be 37kms/hr or 22mls/hr. Of course that is just the average time, not the max speed he reached!

Category 3 climbs last approximately 5 kilometres (3.1 miles), have an average grade of 5 percent, and ascend 150 metres (500 feet).

Category 2 climbs are the same length or longer at an 8 percent grade and ascend 500 metres (1,600 feet).

Category 1 climbs last 20 kilometres (12.4 miles) with an average 6 percent grade and ascend 1,500 metres (4,900 feet).

Category H climbs are the hardest including an altitude difference of at least 1,000 metres (3,280 feet) from start to finish and have an average grade of at least 7 percent.
Finding missing numbers is great fun and many readers are regular users of Sudoku. In the recent post (July 13) we find that the sum of the numbers in each row and column is 6, 12, ? Therefore, we need to get 18 as the sum in the final row and column. So, 9 is the missing number in order to complete the puzzle.

Great to see some recent posts on Calculus and we hope that some of our junior students (Years 6+) have a close look at these and develop an interest in this (more advanced) Mathematics.

Trend lines are a practical way to analyse the patterns of data over time and are particularly helpful in population, commerce and environmental change, such as the arctic ice post. The best way to find the answer to the question posed in this post is to click on the original article, copy the graph and paste into (e.g.) Word, using the landscape format. Then, using a ruler, carefully draw the same lines that I have shown in the post. This will help arrive at a more accurate answer. When you have the answer, post a comment to the blog and we can check it out to see if you are right (or close). Good luck Junior Mathematician!

1 year = 31 556 926 seconds

1729 – A rather dull number?
The mathematician G. H. Hardy was visiting the Indian mathematician Ramanujan while he was ill in hospital. Hardy was making small talk and remarked that 1729, the number of the taxi that brought him to the hospital, was a rather dull number. “No Hardy!” repled Ramanujan, “It is a very interesting number. It is the smallest number which can be expressed as the sum of two cubes in two different ways!” You see, even “dull” numbers have special properties!

Blog Diary

Dear Blog Diary,

Our night sky has always fascinated H3, and there have been some recent releases of amazing images from our nearby galaxies. The size and sheer complexity of our solar system is staggering and, mathematically, quite difficult to describe because the numbers are simply so big!

The fireworks background gives readers some idea of how students feel when they suddenly get a mathematical concept and can apply it with success. This is what excites learners to do well in their math studies. This is also what inspires teachers to want to help students have these "aha" moments! As the famous Winston Churchill said, "Never, Never, Never, Never, Never give up on your maths!" (Well, he almost said that).

The "x" factor - it was intriguing to see the TED talk post that explained why we use x to indicate an unknown quantity in Algebra. Hope our readers also enjoyed this view on what we take for granted in our everyday Mathematics.

Lewis and Clark explored routes to the American west...all the way to Oregon City where, today, there is a great museum to herald this famous migration period (see link in the post). So, the header image show canoes heading in which direction? East? How do you know? Should mathematicians expect every picture or drawing to point north? NO, of course not! So, to answer the post question - the canoes could be heading in ANY direction!

I had a discussion with a fellow teacher the other day that was along the lines of how sad it was that students today have lost a sense of fine craftsmanship when it comes to products and services. For example, old cameras were beautifully crafted and lasted, with regular servicing, for up to one or two generations. Today, with our "instant society" we are surrounding with products that have little permanency. The revival of fine architecture in the Art Deco movement is a recent highlighted post. In the same way, important mathematical proofs are timeless and give us all a better sense of something solid and permanent in our fragile world. I do hope that students who engage in Mathematics at any level also share this passion for numbers, patterns and proofs that are fixed and reliable signposts in a sea of turbulent ideas and rapid change.

Thanks to the positive feedback from Warren in Perth who wrote, "Congrats and good luck in your crusade to bring the joy and beauty of maths back to schools." See the Welcome page for the full comment. It is always great to have helpful ideas and feedback from blog readers. Again, thanks so much for taking the time to read H3 Maths.

It was in the news recently that Apple was looking to spend some $97 billion - that's 97,000,000,000. At the rate of $1000 a day, it would take an incredible 265,780 years to spend. That's an insane amount of money and it would be a good exercise to work out how this amount could help fix some of the big issues in the world today, such as the debt crisis in Europe, or Global Warming.

Being able to "roughly" work out an answer in Mathematics is called "Approximation". A good example of using this is in the little test post from the New York Times - looking at the rise in median house prices across a period of time. The answer is lower down in this column... :-)

Above is an algebraic expression with two sets of brackets, -
(x+1)(y-2). The brackets mean "multiply" so each bracket is a factor of an expanded algebraic expression. There are four parts to the bracketed factors, hence the term "quadratic" which comes up often in Year 9 and Year 10 (Freshman and Sophomore) grades. As a growing mathematician you will need to become competent with factorising and expanding algebraic terms.

Great to see so many visitors from 17 different countries - a Prime Number as well! Of course, there are more countries in our Visitor list but they did not show up on the new clustr map.

The blog about maths being all about language is really not entirely true...was just waiting for someone to comment! You see, Mathematics is also very much about shapes, patterns and trends, which were left of the list. In fact, maths is really about everything!! (Answer to median house prices = B)

Welcome to our first visitor from South Africa!

Numbers - they are the DNA of Mathematics and some recent posts will focus on the way that different number groups (called Number Sets) behave - very much like the different groups of people that you mix with (or not) at a party!

Making visual connections is an often forgotten focus in Mathematics yet is integral to most maths testing. I hope you enjoy the challenge of finding the right location for the van on Lombard Street! Your need a sense of orientation and scale but it is really not that difficult.

Welcome to our visitor from San Francisco, just after the San Fran posting! This is a great city, with so much architectural and cultural diversity as well as such a wonderful location.

Patterns - now here's a great subject to get your maths juices boiling! Show me a keen math student and I will guarantee that he or she is into patterns! Of course, the true-blue mathematician is also into random patterns - which we call "chaos" - and that is another great math topic to look at at some other (random) time! Do Zebra stripes count as random patterns? ;-)

The importance of a good breakfast is our focus for the weekmix!

Great to see a recent blog visitor from Gresham, Oregon. Great scenery around the Columbia River Gorge including the second highest waterfall in the USA. Home to some good mathematicians too!

A good friend and wonderful Mathematics teacher (now retired but used to live in Gresham too) send through this kind comment from the USA recently; ".. spent some time on your math blog and was very impressed. I am hoping that students are taking advantage of it. I was particularly impressed with your process of getting students to think mathematically and not just look at math as a hallway that is filled with hurdles called classroom exercises. The most exciting part of math is when you open a side door and explore other rooms that may lead to a maze of interrelated opportunities in math explorations." Many thanks!

A visitor reads our blog from the I-95 (see post). Is this a space-time warp from our Dr Who files or a wonky GPS?

Dear Blog,
Over 100 visitors for January. 100 visitors reminds me of the famous story regarding the great mathematician, Carl Friedrich Gauss. He started primary (elementary) school at age 7 and his genius became apparent when his teacher asked the class to add up (the sum) of all (integer) numbers from 1 through 100. Gauss did this almost instantly by noticing 1+100 = 101; 2+99 = 101, 3 + 98 = 101 for a total of 50 pairs. Therefore the total was 50 x 101 = 5050. He may have reached this mentally by doing 50x100=5000 + 50 = 5050? Whatever method, what a quick mathematical mind at such a young age! Yes, Gauss had a keen interest in how numbers worked and this is a key to doing well in Mathematics.

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