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Misbehaving Prime Numbers


Two academics have shocked themselves and the world of mathematics by discovering a pattern in prime numbers. Primes – numbers greater than 1 that are divisible only by themselves and 1 – are considered the ‘building blocks’ of mathematics, because every number is either a prime or can be built by multiplying primes together – (84, for example, is 2x2x3x7). Their properties have baffled number theorists for centuries, but mathematicians have usually felt safe working on the assumption they could treat primes as if they occur randomly.

Now, however, Kannan Soundararajan and Robert Lemke Oliver of Stanford University in the US have discovered that when it comes to the last digit of prime numbers, there is a kind of pattern. Apart from 2 and 5, all prime numbers have to end in 1, 3, 7 or 9 so that they can’t be divided by 2 or 5. So if the numbers occurred randomly as expected, it wouldn’t matter what the last digit of the previous prime was. Each of the four possibilities – 1, 3, 7, or 9 – should have an equal 25 per cent (one in four) chance of appearing at the end of the next prime number.

But after devising a computer programme to search for the first 400 billion primes, the two mathematicians found prime numbers tend to avoid having the same last digit as their immediate predecessor – as if, in the words of Dr Lemke Oliver they “really hate to repeat themselves.”

A prime ending in 1 was followed by a prime ending in 1 only 18.5 per cent of the time, significantly less often than the expected 25 per cent. And, the pair found, primes ending in 3 tended be followed by primes ending in 9 more often than in 1 or 7.

The pattern – already being referred to as ‘the conspiracy among primes’ – has left mathematicians amazed that it could have remained undiscovered for so long. “I was floored,” Ken Ono, a number theorist at Emory University in Atlanta, told Quanta Magazine. “I have to rethink how I teach my class in analytic number theory now.” (source: The Independent)

Pure Gold in Math


In Mathematics there are numbers that are like pure gold – they glow with special properties. Like the magic ring in Lord of the Rings, they release powers of the imagination and can be used to inflict their magic on those unsuspecting students who look at them too closely. Yes, we are talking about The Golden Ratio. This is the ratio given by one side of a rectangle divided by another – in this way:
Make a square with side lengths of 10cm (or 4 inches). Mark the midpoint of one side (at 5cm or 2 inches). Join this midpoint to any corner. Use a ruler to measure this line and draw it along the bottom of your square. (Or you can use that line from the midpoint as the radius to draw an arc that defines the height of the rectangle). This new length from the midpoint divided by a side of your square is called the “Golden Ratio”. It is often symbolized using phi, after the 21st letter of the Greek alphabet, and is approx. 1.168 [H3 note: use a compass to be more accurate in your construction]

Even today, the golden ratio comes handy when we have to make visually appealing web pages, brochures, or even logos for your company or a client.

Notre Dame – Mathematics on Fire?


A very sad moment when we heard yesterday that Notre Dame cathedral was ablaze in Paris. H3 visited this amazing Gothic cathedral just a few years ago and the soaring roof detail seemed inexplicable – its heavy brick pattern seeming unsupported. Yes, this fire-proof lining may have saved the cathedral from absolute ruin. Notre Dame was build over a period of more than 200 years – amazing architecture which placed its achievement in the hands of mathematically-skilled craftsmen.

“The subtle grace of the Gothic cathedral touches us powerfully on so many levels. But that grace is vested in engineering design. Those barrel vaults, flying buttresses, Gothic arches, and spiral stone staircases had to be born of mathematics.

Mathematics means handling numerical quantities symbolically, not a subject medieval masons studied. In fact, some couldn’t even read. As we comb the rich medieval record, we find not only no mathematical basis for these glorious buildings, we don’t find architectural drawings. We find only the crudest sketches. Yet the medieval cathedral is geometry and proportion — from labyrinths in mosaic floor tiles to the criss-crossing ribs that hold the ceiling. It just doesn’t make sense. Then we realize:

The building is the geometry text. The master mason, with his fingers touching stone, used stone to express geometry. If mathematics is the symbolic expression of magnitude, that’s what the cathedral itself is. The balance of mass and space goes by square roots of 2 and 3, and the so-called Golden Section.

Medieval iconography regularly shows one mathematical instrument in the hands of the mason — a pair of dividers. When medieval artists show us God, He often appears as the Master Craftsman, holding a great pair of dividers. With dividers and a carpenter’s square alone, you can prove the Pythagorean theorem, and you can create any of those seemingly sacred proportions.

The cathedrals were not so much designed by mathematics as they are mathematics. They are mathematics that flowed straight from the mind’s eye to the fingers of masons who built them.”

From: John Lienhard, at the University of Houston

More on Gothic Design here

Update: Using lasers to peer inside ancient workmanship

(all photos taken by H3)

Pedestrian Mathematics


In a drizzling rain, a 63-year-old professor, George Nobl, stood on a stretch of Times Square sidewalk the other day beside an easel blithely daring passers-by to solve a math problem.

Fred can paint a room in three hours,” the problem, printed in block letters on paper pinned to the easel, said. ”Mary can paint the same room in two hours. How long will it take them to paint the room together?

Solve the problem, and win a Snickers bar,” the message added.

In an area known for bullhorn-wielding sidewalk evangelists, three-card monte games and even the occasional chess face-off, some thought Professor Nobl was a con artist. Some figured he was promoting a candy bar. Many walked away, but quite a few lingered to study the problem.

It turned out Professor Nobl had nothing up his sleeve except a quirky effort to promote the fun of math. Every Wednesday at noon since last summer, he has stood on West 42nd Street between Fifth and Sixth Avenues challenging passers-by in this eccentric fashion.

If people want to answer those trivial questions in ‘Who Wants to Be a Millionaire,’ then why not give them a math problem?” he asked.

After a few minutes, several people in the crowd whispered answers into his ears by turns, with most of them guessing that the answer was two and a half hours.

If it takes one person two hours to paint the room, how come it would take longer for them to do the task together?” Professor Nobl asked tartly.

Tom Mansley, a nearby office worker on his way to lunch, was one of the few who got the correct answer. ”You don’t need a lot of mathematical background for this problem, but it’s tricky,” he said after calculating on a notepad for five minutes. ”I would have stood here all day in the rain to solve it. I like to meet challenges.

David Williams, a social worker who graduated from public schools in the Bronx and went on to college, was less lucky. ”When I see a word problem, I don’t know which rule to use,” Mr. Williams said. ”I remember a lot of rules, but I don’t know which one is the right rule.” Full article here.
So, what was the correct answer that the professor was looking for? (see Post Support)

Math Prof wakes early to view black hole image


Astronomers have captured the first image of a black hole, proving the University of Canterbury’s distinguished Professor Roy Kerr’s 56-year-old theory correct.
Kerr said he set his alarm for 1am to see this exciting event.

“The event horizon telescope photo is just the beginning of a new phase in the understanding of our universe.

“The visual evidence will continue to get more and more sophisticated,” he said.

“I was surprised that the best image was not Sagittarius A* (compact astronomical radio source) but was a supermassive black hole 2000 times further away, and 2000 times larger.”

In 1963 before advanced computers existed Kerr achieved what had eluded others for nearly half a century with pen and paper – solving some of the most difficult equations of physics by hand. He found the exact solution of Albert Einstein’s equations that describe rotating black holes.

Kerr is an eminent mathematician known internationally for discovering the Kerr Vacuum, an exact solution to the Einstein field equation of general relativity. He began his long association with the University of Canterbury in 1951, earning a Bachelor of Science in 1954 and a Master of Science in 1955. He then went to Cambridge to research his PhD and was awarded his doctorate in 1959. From England he then moved to the United States where he worked with Professor Peter Bergmann, Albert Einstein’s collaborator.

Kerr returned to New Zealand and the University of Canterbury in 1971 where he became a Professor of Mathematics for 22 years until his retirement in 1993. Awarded the British Royal Society’s Hughes Medal in 1984 and the Rutherford Medal from the New Zealand Royal Society in 1993, he was also made a Companion of the New Zealand Order of Merit in 2011, and was awarded the 2013 Albert Einstein medal by the Albert Einstein Society in Switzerland.

The University of Canterbury awarded the rare honour of the title Canterbury Distinguished Professor to Emeritus Professor Roy Kerr, who also received the prestigious Crafoord Prize in Sweden in 2016.

Canterbury Distinguished Professor is the highest academic title that can be awarded by the University and has been conferred only twice before in the University’s history.

Title recipients are Nobel Prize winners or equivalent, such as the Crafoord Prize, which is worth over $NZ1 million. (source: NZ Herald here)

Do you want to host a visit by H3 Math?


Your blog host, H3, is visiting locations in Italy, France, Singapore, and the USA (West Coast) in August and September, 2019. If you are interested in hosting a school or class visit, please contact H3 in the comment section. The visit could include a brief outline of engaging math activities, an overview of this blog, and ‘why math really counts for school leavers’, etc. We look forward to hearing from you – our valued readers.

Average is anything but


A group of 5 students were comparing how much they earned over the weekend. Here are the results:

Jed $55 (mowing lawns)
Sarah $30 (pocket money)
Squid (not his real name) $0 (sad face)
Ginger $20 (thanks auntie)
Tim $225 (thanks eBay)

Squid (he has always loved math) said, “Wow, the average earnings was $66‘” which made Jed, Sarah and Ginger rather sad too. “But,” replied Tim, “the median amount was $30.” That made Jed and Sarah feel much better.
H3 Comment: What a big difference between the ‘average’ and ‘median’. The average is distorted by the large amount of money that Tim made, while the median (middle of ranked earning) is much closer to a measure of the middle.

Now, let’s apply this to a news article that came out of some real estate figures in Auckland, New Zealand this morning:

The Auckland property market has regained some of its momentum in March, says the city’s largest real agency, citing sales numbers that are almost doubled from February. The average sales price at $931,673 for March and the median price at $836,000 also showed a marked increase on those in February,” says the real estate agency’s managing director, Peter Thompson. (see full article here)

That is a whopping difference of over $99,000 between the average sales price and the median sales price. What happened? Just like the example involving the 5 students, in the case of Auckland house prices (now among the highest in the world) there were some large sales that skewed the data. If you were a buyer you could be misled if you were given the average figure as an indicator of the ‘middle’ value of properties being sold. The median is a better medium to use!

Numbers are Vital – Here’s the Proof


I love the “cover-up method” where I hide something and students are challenged to find the answer. For example, suggest some numbers that would make this new item make sense. Then check out the solutions in ‘Post Support’.

Job Purity …


Measurement Factoid


DYK (Did you know) – that a typical acre (0.40 hectares) of soil contains 400kg of earthworms, 1000kg of fungi, 680kg of bacteria and 400kg of insects.

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Post Support

George Nobl Problem:

Professor Nobl explained that the correct answer is one hour and 12 minutes. Here is how he solved it, he said. Fred paints a third of a room in one hour, and Mary a half. So together they can paint five-sixths of a room in an hour. To paint the entire room, they will need six-fifths of an hour, or an hour and 12 minutes.

Missing Numbers: ‘five-star’; ‘85000sq m floor area, $703m project with 1327 new underground carparks’; ‘33000 new international visitors,’ $90 million of ….. and 800 jobs. The 300-room, 5-star hotel will … further 150 jobs’; 3000 people or 4000 people for one-off events’.

Number of spare car-parking spaces. How about 8, or 9 with a tiny part of one on the lower right.

What is x? Try substituting 1,2,3,4. I think you have it solved now!


The pink triangle is one third of the area of the square. Use half base x height formula for area of the triangle, etc.

The coffee cup logic puzzle – Answer is Cup 5 as all the others have blocked pipes. 🙂


6×6 for the maximum dog pen area of 36 sq meters.


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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