Saturday, February 25, 2012

The hoax of Hanoi

The Tower of Hanoi is one of the more popular puzzles amongst recreational mathematicians and in education.  Almost everyone is familiar with it and the story that usually accompanies it - often couched in terms something like this...

“There is a legend of a temple near Hanoi in Vietnam.  The temple has three golden poles. One one pole sits 64 golden discs of different sizes. The rings are arranged with the largest at the base progressing in order to the smallest disc which sits on top of the pole.  The monks who tend the temple have been charged with moving the rings from their current pole to the end pole - one disc at a time - but there is a special rule that they must follow. They must never place a larger disc on top of a smaller one. When they complete this task the world will end. How many moves will it take the monks to do this?”

We probably don’t need to work ourselves up into a sweat worrying about the monks finishing their task which will signify the end of the world.  It has been calculated that if the monks were able to move the one disk per second it would take them around 585 billion years to complete the task - 18,446,744,073,551,615 turns.    In classrooms the number of disks is usually drastically reduced to around five in the early stages - which is enough to prompt some interesting explorations.   In this age of the Internet there are several interactive versions - which have the advantage of being readily customised and also of making suggestions for optimal moves. Examples can be accessed here and here.

It is a great puzzle - a surefire hit in the classroom.  But what makes me love this puzzle so much is the simple fact that it is a hoax.  Traditionally there was no such legend in Vietnam - or in India either, which is another popular location for the site of the temple.  The whole thing is the invention of mathematician Edouard Lucas. When the puzzle was first published in 1883 it appeared complete with legend and was credited to  N. Claus de Siam (an anagram of Lucas d’Amiens - Amiens being the town of his birth).  

Lucas was a significant mathematician in his own right - so significant that he has his own number series named after him - the Lucas sequence (similar to the Fibonacci sequence but instead of starting 1,1,2,3,5,7...  Lucas’ starts at with 2,1,3,4,7,11,18...) and he developed a test for large prime numbers still  in use today.

Lucas’ death is also worthy of interest - he is possibly the only mathematician in history to die as a result of a piece of crockery. A clumsy waiter dropped a plate which broke and shard from the plate cut Lucas on the cheek. The cut became infected and he later died of erysipelas - a serious streptococcus infection.

The myth and the puzzle are well known … but the puzzle has a rather curious property that  is less so.  The movements of the pieces can be represented in a graph as depicted here.
This bears an uncanny resemblance to  to the famous  Sierpinski triangle- which is a fractal.   The fact that the movements of a game can be linked to fractals is, to say the least, intriguing. (More on this at the source.)

The Tower of Hanoi has recently become a movie star. A disguised version of it appears in the move “Rise of the Planet of the Apes” - disguised, fittingly given the manner in which it as first published, as the Lucas Tower which was used to test the intelligence of the apes.

What can we learn from this wondrous hoax? When it comes to education sometimes a touch of showmanship and playfulness can improve a strictly mathematical experience. It is important that the mathematics remains central - but the “power of story” is a significant tool to generate engagement and interest  - which may contribute to the “holy grail” of teaching;  a self-directed learner.  We can help foster this when we show our students that there is more to mathematics than just the numbers.


Undirected graph of movements:
Sierpinski triangle:

If you enjoyed this post you may enjoy my other maths related posts available via the maths page or by clicking here.

Saturday, February 18, 2012

Put the “ah” back in arithmetic…

Maths that makes you go "Hmmm"
Mathematics often puzzles me.  Sometimes it also makes me wonder.  We get accustomed to the Fibonacci sequence and the Golden Ratio appearing in nature so often that we cease to register just how remarkable this actually is and sometimes fail to ask ourselves why this should be.  

Sometimes it takes something new to bring us to a fresh realisation that the presence of some forms of mathematics in nature is not only unexpected but  possibly unexplainable.  Some unexpected occurrences of  mathematics are truly remarkable.  The following clip provides a worthy example.

It is taken from the BBC series “Why Maths doesn’t add up” and features
features Marcus du Sautoy,  Professor of Mathematics at Oxford and Simonyi Chair for the Public Understanding of Science with actor, comedian and maths fall guy  Alan Davies. Early in the clip we see a graph first devised by Bernhard Riemann showing the random distribution of prime numbers. In this excerpt  the pair see what happens to the vibrations of a sphere of quartz when the quartz is struck with a ball bearing.    Quite unexpectedly the two graphs are startlingly similar.

As du Sautoy says in the clip, the similarity of the graphs is so striking that “it can not be a co-incidence”.

So if it isn’t a co-incidence...what is it?

Perhaps in pondering such “oddities” we will also remember that the “common” occurrence of the Fibonacci sequence and the Golden Ratio in nature are also worth of contemplation.  The unique Vi Hart has a series of her manic videos about the Fibonacci sequence which is well worth a diversion.

I believe it is this sense of wonderment and intrigue that we need to impart to our students as much as computational skills and procedural understanding.

Sierpinski Surprises
One of the wonderful things about mathematics is that it is frequently surprising.  Often relatively simple things reveal unexpected hidden depths - and do so with minimal probing.  One such aspect of mathematics is the Sierpinski triangle (aka Sierpinski gasket).  
The Sierpinski triangle was named after a Polish mathematician who explored the concept around 1915,  although it is not true that he “discovered” it as the basic shape apparently appears in art work dating from some centuries before.

The Sierpinski triangle is a self similar set - a pattern that can be made larger or smaller indefinitely while maintaining the same pattern,  in other words, it is a fractal.  Fractals are not just found on the pages of maths books - which is one of the reasons that they are so fascinating.  Fractals are found in many places in nature - from snowflakes to certain leaves, from ferns to forked lightning, even, most unexpectedly, in broccoli. Making Sierpinski triangles is something that even relatively young students can do. Start with an equilateral triangle. Inscribe that triangle with an inverted copy of itself (or, more simply, make an upside down triangle inside the first one). And keep repeating. (One set of instructions for use with young mathematicians can be found here.)  The pattern created, as depicted above, is surprisingly pleasing for what is essentially a mathematical process.

Creative types have added colour, and combined them to make them even more so as shown here.  So, the Sierpinski triangle is interesting and visually attractive in its own right...but there are some fascinating attributes that do not immediately meet the eye.  (Source)

The other famous triangle in mathematics is Pascal’s triangle. It has become customary to credit the discovery of this triangle to Blaise Pascal although the concept was known well before
him.  In Pascal’s triangle the two numbers above a cell are added to create the  number in the next row and the process repeats.  Now, this is where things get interesting.  If we shade only the odd numbers in Pascals triangle we get … something VERY similar to Sierpinski’s triangle - and the similarity strengthens as the larger the triangle continues.   

Why might this be?
Sierpinski and Chaos
There is a fascinating game invented by Michael Barnsley called “The Chaos Game” - which, ironically, proves that order can come where chaos is expected.
To “play” this game grab a sheet of paper and mark three points of an equilateral triangle.  Label the points A, B and C. Make a mark at any random point on the paper - call it X. Use a die - numbers 1 & 2 relate to point A, 3 & 4 to point B and 5 & 6 to point C. In our example imagine you roll a 1 (which therefore indicates point A).  Measure half the distance from P to A and make a new mark. This is the next point. Roll again and mark the point half way between the relevant point indicated by the die and the last point obtained. Repeat and repeat...and repeat.  The longer you “play” the game the more strongly a pattern emerges - and that pattern is the Sierpinski triangle!  
When I first read about this years ago I wasted hours “playing” the game in an attempt to test it. It proved to be true. These days actually playing the game is not strictly necessary as there are any number of interactive versions of the game available on the Internet.  One can be found here - I recommend that you try it for yourself.  Concentrate on your sense of wonderment as the patten unfolds. This is the feeling that we want our students to have when they explore mathematics.   Incidentally, this “game” has been called the “Creationist’s worse nightmare” - for those who like their maths flavoured with mysticism - and the site that uses the term explains why the pattern works well.
Counter-intuitive results such as that produced by the Chaos game can intrigue students - which leads to engagement...which leads to learning.
Tower of Sierpinski?
For me personally this is perhaps the most perplexing of the unexpected appearances of the Sierpinski triangle.
The Towers of Hanoi is a popular game / puzzle where the player is required to shift a number of disks from one of three “posts” to another of the three available and reassemble them in order with the largest disk at the bottom and the rest of the disk sitting on the bottom disks in order. It is a surprisingly simple yet engrossing game. 

An interesting thing occurs when the moves leading to a solution of the Towers of Hanoi are graphed.
(source)The resemblance of this graph to the Sierpinski triangle is startling. It is worth pausing a moment to think about this. The Tower of Hanoi is a mental experiment, a “game”  devised by a human.  It is not a “naturally occurring”  phenomena such as a fractal snowflake or a symmetrical fern leaf - it is totally the product of human imagination.  Yet the solution to this totally invented  game, when graphed, has a strong resemblance to the Sierpinski triangle - which is a fractal.  It could be argued that Sierpinski triangle is also the product of the human mind - yet this does not diminish the sense of surprise when the link between the two concepts is established.  Why should this link exist? The sense that there is some intriguing connection between two such different things is tantalising.  In mathematics the enjoyment is in the exploration and the discovery.  Sometimes no answer is much more satisfying than a clinical definition that puts everything in its place.A slice of Biblical pi

Author Alex Bellos has been rightly praised for his fascinating book “Alex’s Adventures in Numberland”. It is a book full of mathematical gems presented in an accessible style not normally associated with mathematics books - which possibly explains its success.  One of the chapters is devoted to the study of pi and the people who have pushed back the boundaries of this number - a dry topic transformed into a engrossing read.  

For those who need a reminder - pi is the ratio of a circle’s circumference to its diameter.  As Bellos puts it ”...if you take the diameter of a circle and curve it around the circumference, you will find that it fits just over three times.”  It is that “just over” bit that makes pi interesting.  The “just over” bit is actually an irrational fraction - meaning it never ends  - it simply goes on forever.  The accuracy of pi has been calculated to a ridiculous degree - billions of digits. According to Bellos, manufacturers of precision instruments only need an accuracy of four decimal places so the quest for a more and more accurate figure for pi is no longer driven by any practical reasons.  However, early methods of calculation were ingenious and the chapter in Bellos’ book provides an entertaining overview of many of them.
The chapter contains this discussion of pi in, of all places, the Bible. 
“...A line in the Bible reveals a situation in which pi is taken a 3: ‘Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about’ (I Kings 7:23)”.
Thus the Bible is on shaky ground from a mathematical perspective.  After a very brief discussion on some squeamish “explanations” from devote believers on what the bible may have meant when giving that description Bellos throws in an absolute gem.
“A mystical explanation is much more enticing: due to the peculiarities of Hebrew pronunciation and spelling, the word ‘line’, or qwh, is pronounced qw. Totting up the numerological values of the letters gives 111 for qwh and 106 for qw. Multiplying three by 111/106 gives 3.1415, which is pi correct to five significant figures.
Today we would call this  numerology - but the practice has a long history. Traditionally the practice of ascribing numerical values to words was known as gematria and was wide-spread amongst the ancient Greeks.
I’m not sure what I find hardest to believe - that someone in Biblical times not only knew pi to five significant figures and could hide it via gematria in a passage to be decoded by the enlightened...or that purely random chance and an accident of linguistics has delivered pi to five significant figures - in a passage describing a circle.
Irrational numbers like pi can be tricky to understand.  A certain Dr Edward J. Goodwin came up with a … novel  … “solution”.  The good doctor decided that pi,  featuring all those uncooperative numerals after the decimal  point, was simply too complicated and decided that pi should be LEGISLATED to be 3.2.  He would grant to the state of Indiana a legal right to use his version of pi = but every one else would have to pay him a royalty when using his version of pi!  Legislation to introduce this actually made it to the Indiana legislature.  More of the story can be read here

What has this mathematical musing have to do with education? Two things. As teachers we don’t always have the answers - and we should admit this to our students - not necessarily in relation to the concepts discussed here but in general terms. The second is that the simple fascination of the unknown and the delight of exploration that is associated with mathematics should be shared with our students. Again, not necessarily these concepts, but with mathematics in general.  We should present mathematics as a subject full of fascination to be explored rather than allow it to be reduced to a collection of rote exercises, algorithms and meaningless formulae.
The worth of a word:
Gematria is an ancient belief. For most of us it is as valid as the notion of a flat earth or that the earth was a disk sitting on the back of a turtle floating through space.  However, if you want to calculate the numeric value of certain words or names then clicking here will take you to  a site that may be useful as well as amusing.  The site also provides words with the same numerical value as the text you input.

Have fun.

NB: This post is a combination of three earlier posts all with the theme of mathematics that makes you wonder. If you enjoyed this post you may find this link to my maths page which features other mathematical posts that go well with a cup of coffee.
Credits:Sierpinski triangle image link: 
Pi graphic:
Pascal/ Sierpinki triangle link:

Sunday, February 12, 2012

Classroom teachers - the real educational experts

Our staff had a professional development day recently.  The group was asked some “simple” questions such as “How do children learn” and “What do you do to facilitate learning”.  The thoughts of the group were then collected and presented as a word cloud.  Obviously this is a relatively crude way of summarising the ideas of the group - but it does capture the essence of what was discussed.

It seems to me that this is a solid representation of the “wisdom of the group” approach. There were no “outside” educational leaders or gurus present at this workshop, no-one was flown in from interstate to facilitate, no-one earned a cent in presenters fees.  Yet the result is a more than fair representation of the things that effective teachers do. In fact, if each item was to be expanded and clarified we would have the basis of a fairly satisfactory professional text.

Obviously knowing what to do and being able to do it are two different things. If our staff can enact these practices in the classroom our students will be the beneficiaries.  However, it strikes me that the knowledge base in our staffroom is a fairly significant one. There is an expression that an expert is someone who lives more than 100 miles away. Upon reflection it is obvious that we all must live 100 miles away from someone - and thus can all be classified as experts.

Whilst it must be acknowledged that effective teachers need to be effective learners themselves and always strive to develop their skills and knowledge base, perhaps it is time to place more value on the “home grown” wisdom within our schools.

Effective classroom teachers may well be the real educational experts.

Sunday, February 5, 2012

Those people

“Why did they built that new school there? There is no money for [my department] but they can build a school for them. Those people never appreciate anything they are given.”

This comment, paraphrased from a conversation reported to me recently, made my blood boil. The school in question is a brand new facility made to accommodate the students from a number of older schools in the area which were closed due to falling enrolments. The new “combined” school now has a critical mass of students which will enable it to provide a range of services that the component schools were struggling to provide previously. Thus,  in recurrent terms, it is more financially efficient than what it replaces while it should also allow for better educational outcomes too.  It should be acknowledged that the new school is state of the art and incorporates innovative design features that should allow educational approaches not possible in the schools it replaces. It should perhaps be added that the person who made the above comment lives in an area in which a similar infrastructure upgrades have taken place which will directly benefit her own school aged children. Her issue was not, apparently, spending on education per se  - but spending on those people.

Those people.

Presumably the fact that the new school services a lower socio-economic group motivates her comment.  “They never appreciate anything given to them.”

Given to them? Given?

Education is a right of every child - a right, not a privilege. The accident of birth which largely dictates a person’s postcode is irrelevant - every child deserves the best start that society can provide.  Even if people do not share this philosophical view point it is scarcely credible that anyone could argue against the value of improving the educational levels. History is full of high achievers that come from low socio-economic backgrounds.  The world’s first billionaire, John Rockerfeller, began life in poor circumstances as did fellow industrial giant Andrew Carnegie. The creator of Harry Potter, J.K. Rowling, was on welfare before her writing found fame and fortune. Global media personality Oprah Winfrey also came from poor circumstances.  The list of people who achieved success in their chosen field who came from humble circumstances is long indeed. And that is not to be sidetracked by the concept of “success” and the many ways in which it can be defined and achieved. Even at an economic level (and I do not endorse this as a gauge of a person’s value) it makes sense to provide the best education possible to the largest number of students.

Educators tend to be socially diplomatic - but that means we should allow such bigotry to go unchallenged.  Those people are MY students.

Thinking about this blog made me recall a thought exercise that runs something like this:
You are about to appoint a person to a responsible position.  You have some inside knowledge about their lifestyle.  

Applicant A
Is known to associate with crooked politicians and consult  astrologists. He is a chain smoker, drinks up to 10 martinis a day and has had two mistresses.
Applicant B
Lost his elected position twice, known to sleep until midday, used opium in his student days and drinks a quart of whiskey every evening.
Applicant  C
A decorated war hero who is a non-smoking vegetarian, whose alcohol consumption is limited to the occasional beer and has been faithful to his partner.

Which of these applicants would be your choice?

Applicant A is Franklin D. Roosevelt

Applicant  B is Winston Churchill

Applicant  C is Adolph Hitler

Thought exercise: - unable to trace the original source