IB Maths and GCSE Maths Resources. Theory of Knowledge (ToK). Real life maths. Maths careers, Maths videos, Maths puzzles and Maths lesson resources.

The site is based around the idea of maths ToK – which tries to get students thinking about more than just syllabus content and to start exploring philosophical ideas, historical connections, cutting edge research and real life applications in mathematics.

Some of the content in the site includes:

  • An IB ToK Maths syllabus plan with a huge amount of ideas for incorporating maths ToK into lessons.
  • A large “Flipping the classroom” videos section for IB students.  These cover pretty much the entire IB HL and SL syllabus – with each topic taught in a short 10 minute or so video.  This should help students both prepare for lessons and also should be invaluable for revision.
  • A new School Code Challenge activity which allows students to practice their code breaking skills – each code hides the password needed to access the next level.
  • Over 200 ideas to help with students’ Maths Explorations – many with links to additional information to research.
  • A large number of posts on everything from imagining extra dimensions to modelling asteroid impacts.

If you would like to contact me, you can do so here. There’s loads of content – so please explore!

euclidean

Non Euclidean Geometry IV – New Universes

The 19th century saw mathematicians finally throw off the shackles of Euclid’s 5th (parallel) postulate – and go on to discover a bewildering array of geometries which no longer took this assumption about parallel lines as an axiomatic fact.

1) A curved space model

 

euclid18

The surface of a sphere is a geometry where the parallel postulate does not hold.  This is because all straight lines in this geometry will meet.  We need to clarify what “straight” means in this geometry.  “Straight” lines are those lines defined to be of minimum distance from a to b on the surface of the sphere.  These lines therefore are defined by “great circles” which have the same radius as the sphere like those shown above.

A 2 dimensional being living on the surface of a 3D sphere would feel like he was travelling in a straight line from a to b when he was in fact travelling on the great circle containing both points.  He would not notice the curvature because the curvature would be occurring in the 3rd dimension – and as a 2 dimensional being he would not be able to experience this.

2) A field model –  Stereographic Projection for Riemann’s Sphere

 

Joh-RiemannSphere01

A field model can be thought of in reverse.  A curved space model is a curved surface where straight lines are parts of great circles.  A field model is a flat surface where “straight lines” are curved.

This may seem rather strange, however, the German mathematician Riemann devised a way of representing every point on the sphere as a point on the plane.  He did this by first centering the sphere on the origin – as shown in the diagram above.  Next he took a point on the complex plane (z = x + iy ) and joined up this point to the North pole of the sphere (marked W).  This created a straight line which intersected the sphere at a single point at the surface of the sphere (say at z’).  Therefore every point on the sphere (z’) can be represented as a unique point on the plane (z) – in mathematical language, there is a one-to-one mapping between the two.

The only point on the sphere which does not equate to a point on the complex plane is that of the North pole itself (point w).  This is because no line touching w and another point on the sphere surface can ever reach the complex plane.  Therefore Riemann assigned the value of infinity to the North pole, and therefore the the sphere is a 1-1 mapping of all the points in the complex plane (and infinity).

euclid19

On this field model (which is the flat complex plane), our straight lines are the stereographic projections of the great circles on the sphere.  As you can see from the sketch above, these projections will give us circles of varying sizes.  These are now our straight lines!

And this is where it starts to get really interesting – when we have two isometric spaces there is no way an inhabitant could actually know which one is his own reality.  A 2 dimensional being could be living in either the curved space model, or the field model and not know which was his true reality.

The difference between the 2 models is that in the first instance we accept an unexplained curvature of space that causes objects to travel in “straight” lines along great circles, and that in the second instance we accept an unexplained field which forces objects travelling in “straight” lines to follow curved paths.  Both of these ideas are fundamental to Einstein’s Theory of Relativity – where we must account for both the curvature of space-time and a gravitational force field.

Interestingly, our own 3 dimensional reality is isomorphic to the projection onto a 4 dimensional sphere (hypersphere) – and so our 3 dimensional universe is indistinguishable from a a curved 3D space which is the surface of a hypersphere.  A hypersphere may be a bit difficult to imagine, but the video above is about as close as we can get.

Such a scenario would allow for our space to be bounded rather than infinite, and for there to be an infinite number of 3D universes floating in the 4th dimension – each bounded by the surface of their own personal hypersphere.  Now that’s a bit more interesting than the Euclidean world of straight lines and circle theorems.

If you enjoyed this you might also like:

Imagining the 4th Dimension. How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.

Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.

Geometry, Relativity and the Fourth Dimension is a fantastic (and very readable despite its daunting title!) book full of information about non-Euclidean geometry and extra dimensions.

euclidean

Non Euclidean Geometry – Spherical Geometry

This article follow on from Non Euclidean Geometry – An Introduction – read that first!

Most geometers up until the 19th century had focused on trying to prove that Euclid’s 5th (parallel) postulate was true.  The underlying assumption was that Euclidean geometry was true and therefore the 5th postulate must also be true.

The German mathematician Franz Taurinus made huge strides towards developing non-Euclidean geometries when in 1826 he published his work on spherical trigonometry.

euclid14

Spherical trigonometry is a method of working out the sides and angles of triangles which are drawn on the surface of spheres.

One of the fundamental formula for spherical trigonometry, for a sphere of radius k is:

cos(a/k) = cos(b/k).cos(c/k) + sin(b/k).sin(c/k).cosA

So, say for example we have a triangle as sketched above.  We know the radius of the sphere is 1, that the angle A = 60 degrees, the length b = 1, the length c =1, we can use this formula to find out what the length a is:

cos(a) = cos(1).cos(1) + sin(1).sin(1).cos60

a = 0.99996

We can note that for the same triangle sketched on a flat surface we would be able to use the formula:

a2 = b2 + c2 – 2bc.cosA

a2= 1 + 1 – 2cos60

a = 1

Taurinus however wanted to investigate what would happen if the sphere had an imaginary radius (i).  Without worrying too much about what a sphere with an imaginary radius would look like, let’s see what this does to the previous spherical trigonometric equations:

The sphere now has a radius of ik where i = √-1, so:

cos(a/ik) = cos(b/ik).cos(c/ik) + sin(b/ik).sin(c/ik).cosA

But cos(ix) = cosh(x) and sin(ix) = (-1/i)sinh(x)  – where cosh(x) and sinh(x) are the hyperbolic trig functions.   So we can convert the above equation into:

cosh(a/k) = cosh(b/k)cosh(c/k) – sinh(b/k).sinh(c/k).cosA

This equation will give us the relationship between angles and sides on a triangle drawn on a sphere with an imaginary radius.

Now, here’s the incredible part – this new geometry based on an imaginary sphere (which Taurinus called Log-Spherical Geometry) actually agreed with the hypothesis of the acute angle  (the idea that triangles could have an angle sum less than 180 degrees).

Even more incredible, if you take the limit as k approaches infinity of this new equation, you are left with:

a2 = b2 + c2 – 2bc.cosA

What does this mean?  Well, if we have a sphere of infinite imaginary radius it stretches and flattens to be indistinguishable from a flat plane – and this is where our normal Euclidean geometry works.  So, Taurinus had created a geometry for which our own Euclidean geometry is simply a special case.

So what other remarkable things happen in this new geometric world?  Well we have triangles that look like this:

euclid15

This triangle has angle A = 0, angle C = 90 and lines AB and AC are parallel, (they never meet).  This sketch introduces a whole new concept of parallelism far removed from anything Euclid had imagined. The angle  β is called the angle of parallelism – and measures the angle between a perpendicular and parallel line.  Unlike in Euclidean geometry this angle does not have to be 90 degrees.  Indeed the angle  β will now change as we move the perpendicular along AC – as it is dependent on the length of the line a.

So, we are now into some genuinely weird and wonderful realms where normal geometry no longer makes sense.  Be warned – it gets even stranger!  More on that in the next post.

If you enjoyed this post you might also like:

The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.

Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.

euclidean

Non-Euclidean Geometry – A New Universe

This post follows on from Non-Euclidean Geometry – An Introduction – read that one first! 

The Hungarian army officer and mathematician Johan Bolyai wrote to his father in 1823 in excitement at his mathematical breakthrough with regards to the parallel postulate.  “I have created a new universe from nothing.” Johan Bolyai was one of the forerunners of 19th century mathematicians who, after noting that mathematicians had spent over 2000 years trying to prove the parallel postulate, decided to see what geometry would look like if the constraint of the postulate was removed.  The result was indeed, a new universe from nothing.

To recap, Euclid’s fifth postulate was as follows:

If a line crossing two other lines makes the interior angles on the same side less than two right angles, then these two lines will meet on that side when extended far enough.

euclid3

It had been understood in a number of (non-equivalent) ways – that parallel lines remain equidistant from each other, that non-parallel lines intersect, that if the lines L1 and L2 in the diagram are parallel then A + B = 180 degrees, that there can only be one line through  a point parallel to any given line.

Collectively these assumptions lead to the basis of numerous geometric proofs – such as the fact that angles in a triangle add up to 180 degrees and that angles in a quadrilateral add up to 360 degrees.

Gerolamo Saccheri

A geometry not based on the parallel postulate could therefore contain 3 possibilities, as outlined by the Italian mathematician Gerolamo Saccheri in 1733:

euclid7

1) A quadrilateral with (say) 2 right angles A,B and two other angles C,D also both right angles.  This is the hypothesis of the right angle – the “normal” geometry of Euclid.

2) A quadrilateral with (say) 2 right angles A,B and two other angles C,D both obtuse.  This is the hypothesis of the obtuse angle – a geometry in which the angles in quadrilaterals add up to more than 360 degrees.

3) A quadrilateral with (say) 2 right angles A,B and two other angles C,D also both acute.  This is the hypothesis of the acute angle – a geometry in which the angles in quadrilaterals add up to less than 360 degrees.

Don’t be misled by the sketch above – the top line of the quadrilateral is still “straight” in this new geometry – even if it can’t be represented in flat 2 dimensions.

Adrien Legendre

Mathematicians now set about trying to prove that both the cases (2) and (3) were false – thus proving that the Euclidean system was the only valid geometry.  The French mathematician Adrien Legendre, who made significant contributions to Number Theory tried to prove that the hypothesis of the obtuse angle was impossible.  His argument went as follows:

euclid8

1) Take a straight line and divide it into n equal segments.  In the diagram these are the 4 lines A1A2, A2A3, A3A4, A4A5

2) Complete the diagram as shown above so that the lengths B1B2, B2B3, B3B4, B4B5 are all equal.  From the sketch we will have lines A1B1 and A2B2 (and subsequent lines) equal.

3) Now we see what will happen if angle β is greater than α.  We compare the two triangles A1B1A2 and A2B2A3.  These have 2 sides the same.  Therefore if β is greater than α then the length A1A2 must be larger than B1B2.

euclid12

4) Now we note that the distance A1B1 + B1B2 + B2B3 + … BnBn+1 + Bn+1An+1 is greater than A1A2 + A2A3 + …AnAn+1.   In other words, the distance starting at A1 then travelling around the shape missing out the bottom line (the yellow line) is longer than the bottom line (green line).

5) Therefore we can write this as

A1B1 + nB1B2 + An+1Bn+1 > nA1A2

(Here we have simplified the expression by noting that as all the distances B1B2, B2B3 etc are equal)

6) Therefore this gives

2A1B1 > n(A1A2 -B1B2)

(Here we simplify by noting that A1B1 = An+1Bn+1 and then rearranging)

7) But this then gives a contradiction – because we can make the RHS as large as we like by simply subdividing the line into more pieces (thus increasing n), but the LHS remains bounded (as it is a fixed value).  Therefore as n tends to infinity, this inequality must be broken.

8) This means that β is not greater than α, so we can write β ≤ α.  This will therefore mean that the angles in the triangle A1B1A2 will be ≤ 180.  To see this

euclid13

We can work out the angles in A1B1A2 by noting that c = (180-α)/2 .  Therefore

angles in A1B1A2 = (180-α)/2 + (180-α)/2 + β

angles in A1B1A2 = 180 + β – α

But we know that β ≤ α.  Therefore β – α ≤ 0

So angles in A1B1A2 = 180 + β – α ≤ 180

Adrien Legendre therefore concluded that the hypothesis of the obtuse angle was impossible.  In fact, it isn’t – and the flaw wasn’t in the logic of his proof but in the underlying assumptions contained within it.  This will be revealed in the next post!

If you enjoyed this you might also like:

The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.

Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.

euclidean

Non Euclidean Geometry – An Introduction

It wouldn’t be an exaggeration to describe the development of non-Euclidean geometry in the 19th Century as one of the most profound mathematical achievements of the last 2000 years.  Ever since Euclid (c. 330-275BC) included in his geometrical proofs an assumption (postulate) about parallel lines, mathematicians had been trying to prove that this assumption was true.  In the 1800s however, mathematicians including Gauss started to wonder what would happen if this assumption was false – and along the way they discovered a whole new branch of mathematics.  A mathematics where there is an absolute measure of distance, where straight lines can be curved and where angles in triangles don’t add up to 180 degrees.  They discovered non-Euclidean geometry.

Euclid’s parallel postulate (5th postulate)

Euclid was a Greek mathematician – and one of the most influential men ever to live.  Through his collection of books, Elements, he created the foundations of geometry as a mathematical subject.  Anyone who studies geometry at secondary school will still be using results that directly stem from Euclid’s Elements – that angles in triangles add up to 180 degrees, that alternate angles are equal, the circle theorems, how to construct line and angle bisectors.  Indeed you might find it slightly depressing that you were doing nothing more than re-learn mathematics well understood over 2000 years ago!

All of Euclid’s results were based on rigorous deductive mathematical proof – if A was true, and A implied B, then B was also true.  However Euclid did need to make use of a small number of definitions (such as the definition of a line, point, parallel, right angle) before he could begin his first book  He also needed a small number of postulates (assumptions given without proof) – such as:  “(It is possible) to draw a line between 2 points” and “All right angles are equal”

Now the first 4 of these postulates are relatively uncontroversial in being assumed as true.  The 5th however drew the attention of mathematicians for centuries – as they struggled in vain to prove it.  It is:

If a line crossing two other lines makes the interior angles on the same side less than two right angles, then these two lines will meet on that side when extended far enough. 

euclid3

This might look a little complicated, but is made a little easier with the help of the sketch above.  We have the line L crossing lines L1 and L2, and we have the angles A and B such that A + B is less than 180 degrees.  Therefore we have the lines L1 and L2 intersecting.  Lines which are not parallel will therefore intersect.

Euclid’s postulate can be restated in simpler (though not quite logically equivalent language) as:

At most one line can be drawn through any point not on a given line parallel to the given line in a plane.

euclid2

In other words, if you have a given line (l) and a point (P), then there is only 1 line you can draw which is parallel to the given line and through the point (m).

Both of these versions do seem pretty self-evident, but equally there seems no reason why they should simply be assumed to be true.  Surely they can actually be proved?  Well, mathematicians spent the best part of 2000 years trying without success to do so.

Why is the 5th postulate so important? 

Because Euclid’s proofs in Elements were deductive in nature, that means that if the 5th postulate was false, then all the subsequent “proofs” based on this assumption would have to be thrown out.  Most mathematicians working on the problem did in fact believe it was true – but were keen to actually prove it.

As an example, the 5th postulate can be used to prove that the angles in a triangle add up to 180 degrees.

euclid3

The sketch above shows that if A + B are less than 180 degrees the lines will intersect.  Therefore because of symmetry (if one pair is more than 180 degrees, then other side will have a pair less than 180 degrees), a pair of parallel lines will have A + B = 180.  This gives us:

euclid4

This is the familiar diagram you learn at school – with alternate and corresponding angles.   If we accept the diagram above as true, we can proceed with proving that the angles in a triangle add up to 180 degrees.

euclid5

Once, we know that the two red angles are equal and the two green angles are equal, then we can use the fact that angles on a straight line add to 180 degrees to conclude that the angles in a triangle add to 180 degrees.  But it needs the parallel postulate to be true!

In fact there are geometries in which the parallel postulate is not true  – and so we can indeed have triangles whose angles don’t add to 180 degrees.  More on this in the next post.

If you enjoyed this you might also like:

The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.

Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.

 

One of the main benefits of flipping the classroom is allowing IB maths students to self-teach IB content. There are currently a good number of videos on youtube which allow students to self teach syllabus content, but no real opportunity to watch videos going through IB Higher Level past paper questions. So, I’ve started to put some of these together:

Playlist, Worked Exam Solutions:

The videos above are all around 10 minutes long and consist of talking through the solutions to 2-3 IB HL maths questions. The best way to use these videos is to pause the video at the start of the question, attempt it, then watch the video to check the answer and make notes on the method. Click on the top left hand corner to change the video being shown in the playlist.

The playlists below combine these worked solutions with the syllabus content videos, all grouped into the relevant syllabus strands:

Playlist 1, Algebra 1:

Sequences, Binomial, Logs, Induction, Permutations, Gaussian elimination:

Playlist 2, Complex numbers:

Converting from Cartesian to Polar, De Moivre’s Theorem, Roots of Unity:

Playlist 3: Functions:

Sketching graphs, Finding Inverses, Factor and Remainder Theorem, Sketching 1/f(x), sketching absolute f(x), translating f(x):

telephone2

The Telephone Numbers – Graph Theory

The telephone numbers are the following sequence:

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496…

(where we start from n=0).

This pattern describes the total number of ways which a telephone exchange with n telephones can place a connection between pairs of people.

To illustrate this idea, the graph below is for n=4.  This is when we have 10 telephones:

telephone

Each red line represents a connection.  So the first diagram is for when we have no connections (this is counted in our sequence).  The next five diagrams all show a single connection between a pair of phones.  The last three diagrams show how we could have 2 pairs of telephones connected at the same time.  Therefore the 4th telephone number is 10.   These numbers get very large, very quickly.

Finding a recursive formula

The formula is given by the recursive relationship:

T(n) = T(n-1) + (n-1)T(n-2)

This means that to find (say) the 5th telephone number we do the following:

T(5) = T(5-1) + (5-1)T(5-2)

T(5) = T(4) + (4)T(3)

T(5) = 10 + (4)4

T(5) = 26

This is a quick way to work out the next term, as long as we have already calculated the previous terms.

Finding an nth term formula

The telephone numbers can be calculated using the nth term formula:

 

telephone7

 This is going to be pretty hard to derive!  I suppose the first step would start by working out the total number of connections possible between n phones – and this will be the the same as the graphs below:

telephone3

These clearly follow the same pattern as the triangular numbers which is 0.5(n² +n) when we start with n = 1.  We can also think of this as n choose 2 – because this gives us all the ways of linking 2 telephones from n possibilities.  Therefore n choose 2 also generates the triangular numbers.

But then you would have to work out all the permutations which were allowed – not easy!

Anyway, as an example of how to use the formula to calculate the telephone numbers, say we wanted to find the 5th number:

We have n = 5.  The summation will be from k = 0 and k = 2 (as 5/2 is not an integer).

Therefore T(5) = 5!/(20(5-0)!0!) + 5!/(21(5-2)!1!) + 5!/(22(5-4)!2!)

T(5) = 1 + 10 + 15 = 26.

Finding telephone numbers through calculus

Interestingly we can also find the telephone numbers by using the function:

y = e0.5x2+x

 and the nth telephone number (starting from n = 1)  is given by the nth derivative when x = 0.

For example,

telephone5

So when x = 0, the third derivative is 4.  Therefore the 3rd telephone number is 4.

The fifth derivative of the function is:

telephone6

So, when x =0 the fifth derivative is 26.  Therefore the 5th telephone number is 26.

If you liked this post you might also like:

Fermat’s Theorem on the Sum of two Squares – A lesser known theorem from Fermat – but an excellent introduction to the idea of proof.

Unbelievable: 1+2+3+4…. = -1/12 ? A result that at first glance looks ridiculous – and yet can be shown to be correct.  How?

happy number

Happy Numbers

Happy numbers are defined by the rule that you start with any positive integer, square each of the digits then add them together.  Now do the same with the new number.  Happy numbers will eventually spiral down to a number of 1.  Numbers that don’t eventually reach 1 are called unhappy numbers.

As an example, say we start with the number 23.  Next we do 2²+3² = 13.  Now, 1²+3² = 10.  Now 1²+o² = 1.  23 is therefore a happy number.

There are many things to investigate.  What are the happy numbers less than 100?  Is there a rule which dictates which numbers are happy?  Are there consecutive happy numbers?  How about prime happy numbers?  Can you find the infinite cycle of sadness?

Nrich has a discussion on some of the maths behind happy numbers.  You can use an online tool to test if numbers are happy or sad.

perfectnumber Perfect Numbers

Perfect numbers are numbers whose proper factors (factors excluding the number itself) add to the number.  This is easier to see with an example.

6 is a perfect number because its proper factors are 1,2,3 and 1+2+3 = 6

8 is not a perfect number because its proper factors are 1,2,4 and 1+2+4 = 7

Perfect numbers have been known about for about 2000 years – however they are exceptionally rare.  The first 4 perfect numbers are 6, 28, 496, 8128.  These were all known to the Greeks.  The next perfect number wasn’t discovered until around 1500 years later – and not surprisingly as it’s 33,550,336.

The next perfect numbers are:

8,589,869,056 (discovered by Italian mathematician Cataldi in 1588)

137,438,691,328 (also discovered by Cataldi)

2,305,843,008,139,952,128 (discovered by Euler in 1772).

and they keep getting bigger.  The next number to be discovered has 37 digits are was discovered over 100 years later.  Today, even with vast computational power, only a total of 48 perfect numbers are known.  The largest has 34,850,340 digits.

There are a number of outstanding questions about perfect numbers.  Are there an infinite number of perfect numbers?  Is there any odd perfect number?

Euclid in around 300BC proved that that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime.  Euler (a rival with Euclid for one of the greatest mathematicians of all time), working on the same problem about 2000 years later went further and proved that this formula will provide every even perfect number.

This links perfect numbers with the search for Mersenne Primes – which are primes in the form 2p−1.  These are themselves very rare, but every new Mersenne Prime will also yield a new perfect number.

The first Mersenne Primes are

(22−1) = 3

(23−1) = 7

(25−1) = 31

(27−1) = 127

Therefore the first even perfect numbers are:

21(22−1) = 6

22(23−1) = 28

24(25−1) = 496

26(27−1) = 8128

friendlynumber

Friendly Numbers

Friendly numbers are numbers which share a relationship with other numbers.  They require the use of σ(a) which is called the divisor function and means the addition of all the factors of a.  For example σ(7) = 1 + 7 = 8 and σ(10) = 1 +2 +5 + 10 = 18.

Friendly numbers therefore satisfy:

σ(a)/a = σ(b)/b

As an example (from Wikipedia)

σ(6) / 6 = (1+2+3+6) / 6 = 2,

σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2

σ(496)/496 = (1+2+4+8+16+31+62+124+248+496)/496 = 2

Therefore 28 and 6 are happy numbers because they share a common relationship.  In fact all perfect numbers share the same common relationship of 2.  This is because of the definition of perfect numbers above!

Numbers who share the same common relationship are said to be in the same club.  For example, 30,140, 2480, 6200 and 40640 are all in the same club – because they all share the same common relationship 12/5.

(eg. σ(30) /30  = (1+2+3+5+6+10+15+30) / 6 = 12/5 )

Are some clubs of numbers infinitely big?  Which clubs share common integer relationships?  There are still a number of unsolved problems for friendly numbers.

solitarynumber

Solitary Numbers

Solitary numbers are numbers which don’t share a common relationship with any other numbers.  All primes, and prime powers are solitary.

Additionally all number that satisfy the following relationship:

HCF of σ(a) and  a = 1.

are solitary.  All this equation means is that the highest common factor (HCF) of σ(a) and a is 1.  For example lets choose the number 9.

σ(9)= 1+3+9 = 13.  The HCF of 9 and 13 = 1.  So 9 is solitary.

However there are some numbers which are not prime, prime powers or satisfy HCF (σ(a) and  a) = 1, but which are still solitary.   These numbers are much harder to find!  For example it is believed that the following numbers are solitary:

10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99

But no-one has been able to prove it so far.  Maybe you can!

 

crypto4

This is inspired from the great site,  Practical Cryptography which is a really good resource for code making and code breaking.  One of their articles is about how we can use the Chi Squared test to crack a Caesar Shift code.  Indeed, if you use an online program to crack a Caesar shift, they are probably using this technique.

crypto

This is the formula that you will be using for Chi Squared.  It looks more complicated than it is.  Say we have the following message (also from Practical Cryptography):

AOLJHLZHYJPWOLYPZVULVMAOLLHYSPLZARUVDUHUKZPTWSLZAJPWOLY ZPAPZHAFWLVMZBIZAPABAPVUJPWOLYPUDOPJOLHJOSLAALYPUAOLWSH PUALEAPZZOPMALKHJLYAHPUUBTILYVMWSHJLZKVDUAOLHSWOHILA

We first work out the frequency of each letter which we do using the Counton site.

crypto2

We next need to work out the expected values for each letter.  To do this we first need the expected percentages for the English language:

crypto3

Then we can count the number of letters in the code we want to crack (162 – again we can use an online tool)

Now, to find the expected number of As in the code we simply do 162 x 0.082 = 13.284.

The actual number of As in the code is 18.

Therefore we can do (13.284-18)2/18  following the formula at the top of the page.

We then do exactly the same for the Bs in the code.  The expected number is 162 x 0.015 = 2.43.  The actual number is 3.

Therefore we can do (3-2.43)2 /2.43

We do this same method for all the letters A-Z and then add all those numbers together.  This is our Chi Squared statistic.  The lower the value, the closer the 2 distributions are.  If the expected values and the observed values are the same then there will be a chi squared of zero.

If you add all the values together you get a Chi Squared value of ≈1634 – which is quite large!   This is what we would expect – because we already know that the code we have received has letter frequencies quite different to normal English sentences.  Now, what a Caesar Shift decoder can do is shift the received code through all the permutations and then for each one find out the Chi Squared value.  The permutation with the lowest Chi Squared will be the solution.

For example, if we shift every letter in our received code back by one – using the Counton tool (so A goes to Z etc) we get:

ZNKIGKYGXIOVNKXOYUTKULZNKKGXROKYZQTUCTGTJYOSVRKYZIOVNKX YOZOYGZEVKULYAHYZOZAZOUTIOVNKXOTCNOINKGINRKZZKXOTZNKVRG OTZKDZOYYNOLZKJGIKXZGOTTASHKXULVRGIKYJUCTZNKGRVNGHKZ

We can then do the same Chi Squared calculations as before.  This will give a Chi Squared of ≈3440 – which is an even worse fit than the last calculation.  If we carried this on so that A goes to T we would get:

THECAESARCIPHERISONEOFTHEEARLIESTKNOWNANDSIMPLESTCIPHER SITISATYPEOFSUBSTITUTIONCIPHERINWHICHEACHLETTERINTHEPLA INTEXTISSHIFTEDACERTAINNUMBEROFPLACESDOWNTHEALPHABET

and a Chi Squared on this would show that this has a Chi Squared of ≈33 – ie it is a very good fit.  (You will get closer to zero on very long code texts which follow standard English usage).  Now, obviously we could see that this is the correct decryption without even working out the Chi Squared value – but this method allows a computer to do it, without needing the ability to understand English.  Additionally a codebreaker who spoke no English would still be able to decipher this code, on mathematics alone.

The Practical Cryptography site have a tool for quickly working out Chi Squared values from texts – so you can experiment with your own codes.  Note that this is a slightly different use of Chi-Squared as here we are not comparing with a critical value, but instead comparing all Chi Squared to find the lowest value.

If you liked this post you might also like:

Code Breakers Wanted by the NSA – A look at some other code breaking techniques.

RSA Public Key Encryption – The Code that Secures the internet – How understanding RSA code is essential for all people involved in internet security.

penalties2

Statistics to win penalty shoot-outs

With the World Cup nearly upon us we can look forward to another heroic defeat on penalties by England. England are in fact the worst country of any of the major footballing nations at taking penalties, having won only 1 out of 7 shoot-outs at the Euros and World Cup. In fact of the 35 penalties taken in shoot-outs England have missed 12 – which is a miss rate of over 30%. Germany by comparison have won 5 out of 7 – and have a miss rate of only 15%.

With the stakes in penalty shoot-outs so high there have been a number of studies to look at optimum strategies for players.

Shoot left when ahead

One study published in Psychological Science looked at all the penalties taken in penalty shoot-outs in the World Cup since 1982. What they found was pretty incredible – goalkeepers have a subconscious bias for diving to the right when their team is behind.

penalties6

As is clear from the graphic, this is not a small bias towards the right, but a very strong one. When their team is behind the goalkeeper apparently favours his (likely) strong side 71% of the time. The strikers’ shot meanwhile continues to be placed either left or right with roughly the same likelihood as in the other situations. So, this built in bias makes the goalkeeper much less likely to help his team recover from a losing position in a shoot-out.

Shoot high

Analysis by Prozone looking at the data from the World Cups and European Championships between 1998 and 2010 compiled the following graphics:

penalties3

The first graphic above shows the part of the goal that scoring penalties were aimed at. With most strikers aiming bottom left and bottom right it’s no surprise to see that these were the most successful areas.

penalties4

The second graphic which shows where penalties were saved shows a more complete picture – goalkeepers made nearly all their saves low down. A striker who has the skill and control to lift the ball high makes it very unlikely that the goalkeeper will save his shot.

penalties5

The last graphic also shows the risk involved in shooting high. This data shows where all the missed penalties (which were off-target) were being aimed. Unsurprisingly strikers who were aiming down the middle of the goal managed to hit the target! Interestingly strikers aiming for the right corner (as the goalkeeper stands) were far more likely to drag their shot off target than those aiming for the left side. Perhaps this is to do with them being predominantly right footed and the angle of their shooting arc?

Win the toss and go first

The Prozone data also showed the importance of winning the coin toss – 75% of the teams who went first went on to win. Equally, missing the first penalty is disastrous to a team’s chances – they went on to lose 81% of the time. The statistics also show a huge psychological role as well. Players who needed to score to keep their teams in the competition only scored a miserable 14% of the time. It would be interesting to see how these statistics are replicated over a larger data set.

Don’t dive

A different study which looked at 286 penalties from both domestic leagues and international competitions found that goalkeepers are actually best advised to stay in the centre of the goal rather than diving to one side. This had quite a significant affect on their ability to save the penalties – increasing the likelihood from around 13% to 33%. So, why don’t more goalkeepers stay still? Well, again this might come down to psychology – a diving save looks more dramatic and showcases the goalkeeper’s skill more than standing stationary in the centre.

penalties7

So, why do England always lose on penalties?

There are some interesting psychological studies which suggest that England suffer more than other teams because English players are inhibited by their high public status (in other words, there is more pressure on them to perform – and hence that pressure is harder to deal with).  One such study noted that the best penalty takers are the ones who compose themselves prior to the penalty.  England’s players start to run to the ball only 0.2 seconds after the referee has blown – making them much less composed than other teams.

However, I think you can put too much analysis on psychology – the answer is probably simpler – that other teams beat England because they have technically better players.  English footballing culture revolves much less around technical skill than elsewhere in Europe and South America – and when it comes to the penalty shoot-outs this has a dramatic effect.

As we can see from the statistics, players who are technically gifted enough to lift their shots into the top corners give the goalkeepers virtually no chance of saving them.  England’s less technically gifted players have to rely on hitting it hard and low to the corner – which gives the goalkeeper a much higher percentage chance of saving them.

Test yourself

You can test your penalty taking skills with this online game from the Open University – choose which players are best suited to the pressure, decide what advice they need and aim your shot in the best position.

If you liked this post you might also like:

Championship Wages Predict League Position? A look at how statistics can predict where teams finish in the league.

Premier League Wages Predict League Positions? A similar analysis of Premier League teams.

circular inversion2

Circular inversions II

There are some other interesting properties of circular inversions.  One of which is that they preserve the “angle” between intersecting circles.  Firstly, how can circles have an angle between them?  Well, we draw 2 tangents to both the circles at the point of intersection, and then measure the angle between the 2 tangents:

circular inversion17

Therefore we can see that the “angle” between these 2 circles is 59.85 degrees.  If we then carry out a circular inversion we see the following:

circular inversion16

The inversion has been done with regards to the black circle centred around the origin.  The red and blue circles are mapped from outside the the black circle onto circles inside the black circle.  Now if we do the same as before – by finding the 2 tangents at the point of intersection, we find that the angle has remained the same – it is still 59.85 degrees.

It is also possible to find circles which remain unchanged under the inversion.  This happens when a circle is orthogonal (at a 90 degree angle) to the circle with which the inversion is being carried out.

circular inversion 18

The small circle has an angle of 90 degrees with the large circle, and therefore when we invert with respect to the large circle, we map the small circle onto itself.

The question is, why is all this useful?  Well, an entire branch of mathematics (non-Euclidean geometry) is concerned with being able to map points in our traditional Euclidean worldview (the geometry of high school triangles, parallel lines and circle theorems) to different geometrical systems entirely.  Circular inversion is a good introduction to this concept.

Also, circular inversion can sometimes make studying mathematical shapes easier to understand and explain.  For example, (from Wolfram):

circular inversion21

It would be very difficult to explain mathematically how the shape above is generated – whilst there are patterns, it is not obvious how to explain them. However, if we invert this shape through a circular inversion (with the circle at centre of the image) then we get the following:

circular inversion20

This is the image inside the circle – and now we can clearly see the pattern behind the generated image.  So, inversion has a lot of potential for simplifying geometrical problems.

 

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