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This is a classic puzzle which is discussed in some more detail by the excellent Wired article.  The puzzle is best represented by the picture below.  We have a hunter who whilst in the jungle stumbles across a monkey on a tree branch.  However he knows that the monkey, being clever, will drop from the branch as soon as he hears the shot being fired.  The question is therefore, at what angle should the hunter aim so that he still hits the monkey?

Screen Shot 2015-11-02 at 7.54.01 PM

(picture from the Wired article – originally from a UCLA physics textbook)

The surprising conclusion is that counter to what you would expect, you should actually still aim at the monkey on the branch – and in this way your bullet’s trajectory will still hit the monkey as it falls.  You can see a video of this experiment at the top of the page.

You can use tracking software (such as the free software tracker ) to show this working graphically.  Tracker provides a video demo with the falling monkey experiment:

Screen Shot 2015-11-02 at 7.52.44 PM

As you can see from the still frame, we have the gun in the bottom left corner, lined up with the origin, the red trace representing the bullet and the blue trace representing the falling monkey.

Screen Shot 2015-11-02 at 8.07.33 PM

We can then generate a graph to represent this data.  The red line is the height of the bullet with respect to time.  The faint blue line (with yellow dots) is the height of the monkey with respect to time.  We can see clearly that the red line can be modeled as a quadratic.  The blue line should in theory also be a quadratic (see below):

Screen Shot 2015-11-02 at 8.10.57 PM

but in our model, the blue line is so flat as to be better modeled as a linear approximation – which is shown in pink.  Now we can use regression technology to find the equation of both of these lines, to show not only that they do intersect, but also the time of that intersection.

We have the linear approximation as y = -18.5t + 14.5
and the quadratic approximation as y = -56t2+39t +0.1

So the 2 graphs will indeed intersect when -18.5t + 14.6 = -56t2+39t +0.1

which will be around 0.45 seconds after the gun is fired.

Screen Shot 2015-11-02 at 8.56.11 PM

(A more humane version, also from Wired – where we can throw the monkey a banana)

Newtonian Mathematics

The next question is can we prove this using some algebra?  Of course!  The key point is that the force of gravity will affect both the bullet and the falling monkey equally (it will not be affected by the different weights of the two – see the previous post here about throwing cannonballs from the Leaning Tower of Pisa).  So even thought the bullet deviates from the straight line path lined up in the gun sights, the distance the bullet deviates will be exactly the same distance that the monkey falls.  So they still collide.  Mathematically we have:

The vertical height of the bullet given by:

y = V0t – 0.5gt2

Where V0 is the initial vertical speed, t is the time, g is the gravitational force (9.8)

The vertical height of the monkey is given by:

y = h – 0.5gt2

where h is the initial vertical height of the monkey.

Therefore these will intersect when:

V0t – 0.5gt2 = h – 0.5gt2
V0t = h
V0/h = t

And for any given non-zero value of V0 we will have a t value – which represents the time of collision.

Well done – you have successfully shot the monkey!

If you like this you might also like:

Throwing cannonballs off the Leaning tower of Pisa – why weight doesn’t affect falling velocity

War Maths – how cannon operators used projectile motion to win wars

 

How to Win at Rock, Paper, Scissors

You might think that winning at rock, paper, scissors was purely a matter of chance – after all mathematically each outcome has the same probability. We can express the likelihood of winning in terms of a game theory grid:

rock paper5

It is clear that in theory you would expect to win, draw and lose with probability 1/3.  However you can actually exploit human psychology to give yourself a significant edge at this game.  Below is a report of a Chinese study on the psychology of game players:

Zhijian and co carried out their experiments with 360 students recruited from Zhejiang University and divided into 60 groups of six players. In each group, the players played 300 rounds of Rock-Paper-Scissors against each other with their actions carefully recorded.

As an incentive, the winners were paid in local currency in proportion to the number of their victories. To test how this incentive influenced the strategy, Zhijian and co varied the payout for different groups. If a loss is worth nothing and a tie worth 1, the winning payout varied from 1.1 to 100.

The results reveal a surprising pattern of behavior. On average, the players in all the groups chose each action about a third of the time, which is exactly as expected if their choices were random.

But a closer inspection of their behavior reveals something else. Zhijian and co say that players who win tend to stick with the same action while those who lose switch to the next action in a clockwise direction (where R → P → S is clockwise).

So, for example if person A chooses Rock and person B chooses Paper, then person B wins.  Human nature therefore seems to mean that person B is more likely to stick to a winning strategy and choose Paper again, whilst person A is more likely to copy that previous winning behaviour and also choose Paper.  A draw.

So you can exploit this by always moving anticlockwise i.e R → S → P.  To look at our example again, person A chooses Rock and person B chooses Paper, then person B wins. This time person A follows his previous pattern and still chooses Paper, but person B exploits this new knowledge to choose Rock.  Player B wins.

rock paper6

You can play against a Wolfram Alpha AI player here.  This program will track your win percentage, and will also adapt its behavior to exploit any non-random behavior that you exhibit.  Even though you may not be conscious of your biases, they probably will still be there – and the designers of this simulator are confident that the program will be beating you after about 50 games.  Have a go!

There are some additional tips for winning at rock paper scissors – if you are in a single game competition then choose paper.  This is because men are most likely to choose rock, and scissors are the least popular choice.  Also you should try some reverse psychology and announce what you will throw.  Most opponents will not believe you and modify their throw as a result.

Rock, Paper, Scissors, Lizard, Spock

rock paper3

You can of course make the game as complicated as you wish – the version above was popularised (though not invented by) The Big Bang Theory.  The grid below shows the possible outcomes for this game:

rock paper2

And of course, why stop there?  Below is a 15 throw version of the game

rock paper4

If you’ve honed your strategy then maybe you could compete in the a professional rock, paper, scissors tournament – here you can watch the final of a $50,000 Las Vegas competition.

If you liked this post you might also like:

Game Theory and Tic Tac Toe – Tic Tac Toe has already been solved using Game Theory – this topic also brings in an introduction to Group Theory.

Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour

puzzle

Can you find a sequence of consecutive integers that add up to 1000?

This puzzle is based on the excellent book A First Step to Mathematical Olympiad Problems – which is full of problems that could be extended to become exploration ideas.

Step 1 – arithmetic formula

Our first step is to write out what we want:

a + (a+1) + (a+2) + … (a +n) = 1000

next we notice that the LHS is an arithmetic series with first term a, last term a+n and n+1 terms.  Therefore we can use the sum of an arithmetic sequence formula:

Sn = 0.5n(u1 + un)

Sn = 0.5(n+1)(a + a+n) = 1000

Sn = (n+1)(2a+n) = 2000

Step 2 – logic

However, we currently have 2 unknowns, n and a, and only 1 equation – so we can’t solve this straight away.  However we do know that both a and n are integers – and n can be taken as positive.

The next step is to see that one of the brackets (n+1)(2a+n) must be odd and the other even (if  n is odd then 2a + n is odd.  If n is even then n+1 is odd).   Therefore we can look at the odd factors of 2000:

Step 3 – prime factorisation

Using prime factorisation: 2000 = 24 x 5³

Therefore any odd factors must solely come from the prime factor combinations of 5 – i.e 5, 25 and 125.

Step 4 – trial and error

So we now know that either (n+1) or (2a+n) must be 5, 25, 125.  And therefore the other bracket must be 400, 80 or 16 (as 5 x 400 = 2000 etc).  Next we can equate the (n+1) bracket to one of these 6 values, find the value of n and hence find a.  For example:

If one bracket is 5 then the other bracket is 400.

So if (n+1) = 5 and (2a+n) = 400 then n = 4 and a = 198.

This means that the sequence: 198+199+200+201+202 = 1000.

If (n+1) = 400 and (2a+n) = 5 then n = 399 and a = -197.

This means the sequence: -197 + -196+ -195 … + 201 + 202 = 1000.

We follow this same method for brackets 25, 80 and 125,16.  This gives the following other sequences:

28+29+30+…+51+52 = 1000

-54+-53+-52+…+69+70 = 1000

-27+-26+-25+…+51+52 = 1000

55+56+57+…+69+70 = 1000

So with a mixture of mathematical formulae, prime factorisation, logic and trial and error we have our solutions.  A good example of how mathematics is often solved in reality!

canonball

Tetrahedral Numbers – Stacking Cannonballs

This is one of those deceptively simple topics which actually contains a lot of mathematics – and it involves how spheres can be stacked, and how they can be stacked most efficiently.  Starting off with the basics we can explore the sequence:

1, 4, 10, 20, 35, 56….

These are the total number of cannons in a stack as the stack gets higher.  From the diagram we can see that this sequence is in fact a sum of the triangular numbers:

S1 = 1

S2 1+3

S3 1+3+6

S4 1+3+6+10

So we can sum the first n triangular numbers to get the general term of the tetrahedral numbers. Now, the general term of the triangular numbers is 0.5n2 + 0.5n therefore we can think of tetrahedral numbers as the summation:

\bf \sum_{k=1}^{n}0.5k+0.5k^2 = \sum_{k=1}^{n}0.5k+\sum_{k=1}^{n}0.5k^2

But we have known results for the 2 summations on the right hand side:

\bf \sum_{k=1}^{n}0.5k =\frac{n(n+1)}{4}

and

\bf \huge \sum_{k=1}^{n}0.5k^2 = \frac{n(n+1)(2n+1)}{12}

and when we add these two together (with a bit of algebraic manipulation!) we get:

\bf S_n= \frac{n(n+1)(n+2)}{6}

This is the general formula for the total number of cannonballs in a stack n rows high. We can notice that this is also the same as the binomial coefficient:

\bf S_n={n+2\choose3}

cannonball2

Therefore we also can find the tetrahedral numbers in Pascals’ triangle (4th diagonal column above).

The classic maths puzzle (called the cannonball problem), which asks which tetrahedral number is also a square number was proved in 1878. It turns out there are only 3 possible answers. The first square number (1) is also a tetrahedral number, as is the second square number (4), as is the 140th square number (19,600).

We can also look at something called the generating function of the sequence. This is a polynomial whose coefficients give the sequence terms. In this case the generating function is:

\bf \frac{x}{(x-1)^4} = x + 4x^2 + 10x^3 + 20x^4 ...

canonball

Having looked at some of the basic ideas behind the maths of stacking spheres we can look at a much more complicated mathematical problem. This is called Kepler’s Conjecture – and was posed 400 years ago. Kepler was a 17th century mathematician who in 1611 conjectured that there was no way to pack spheres to make better use of the given space than the stack above. The spheres pictured above fill about 74% of the given space. This was thought to be intuitively true – but unproven. It was chosen by Hilbert in the 18th century as one of his famous 23 unsolved problems. Despite much mathematical efforts it was only finally proved in 1998.

If you like this post you might also like:

The Poincare Conjecture – the search for a solution to one of mathematics greatest problems.

hailstone

Hailstone Numbers

This is a post inspired by the article on the same topic by the ever brilliant Plus Maths. Hailstone numbers are created by the following rules:

if n is even: divide by 2

if n is odd: times by 3 and add 1

We can then generate a sequence from any starting number.  For example, starting with 10:

10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

we can see that this sequence loops into an infinitely repeating 4,2,1 sequence.  Trying another number, say 58:

58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

and we see the same loop of 4,2,1.

In fact we can use the generator in the Plus Maths article to check any numbers we can think of, and we still get the pattern 4,2,1 looping.  The question is, does every number end in this loop?  Well, we don’t know.  Every number mathematicians have checked do indeed lead to this loop, but that is not a proof.  Perhaps there is a counter-example, we just haven’t found it yet.

Hailstone numbers are called as such because they fall, reach one (the ground) before bouncing up again.  The proper mathematical name for this investigation is the Collatz conjecture. This was made in 1937 by a German mathematian, Lothar Collatz.

One way to investigate this conjecture is to look at the length of time it takes a number to reach the number 1.  Some numbers take longer than others.  If we could find a number that didn’t reach 1 even in an infinite length of time then the Collatz conjecture would be false.

hailstone2

The following graphic from wikipedia shows how different numbers (x axis) take a different number of iterations (y axis) to reach 1.  We can see that some numbers take much longer than others to reach one.  For example, the number 73 has the following pattern:

73, 220, 110, 55, 166, 83, 250, 125, 376, 188, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1…

so investigating what it is about certain numbers that leads to long chains is one possible approach to solving the conjecture. This conjecture has been checked by computers up to a staggering 5.8 x 1018 numbers. That would suggest that the conjecture could be true – but doesn’t prove it is. Despite looking deceptively simple, Paul Erdos – one of the great 20th century mathematicians stated in the 1980s that “mathematics is not yet ready for such problems” – and it has remained unsolved over the past few decades.  Maybe you could be the one to crack this problem!

If you liked this post you might also like:

Friendly Numbers, Solitary Numbers, Perfect Numbers – a look at some other number sequence problems.

stellar1

Stellar Numbers Investigation

This is an old IB internal assessment question and so can not be used for the new IB exploration – however it does give a good example of the sort of pattern investigation that is possible.

The task starts off with the fairly straightforward problem of trying to find the nth term formula for the triangular numbers:

stellar2

Method 1

There are a number of ways to do this, probably the easiest is to notice that the second differences are always constant (+1 each time).  Therefore we have a quadratic sequence in the form an2 + bn + c

We can now substitute the known values when n = 1, 2, 3 into this to find 3 equations:

a(1) + b(1) + c = 1

a(2)2 + b(2) + c = 3

a(3)2 + b(3) + c = 6

this gives us:

a + b + c = 1

4a + 2b + c = 3

9a + 3b + c = 6

We can then eliminate using simultaneous equations to find a, b, c.  In fact our job is made easier by knowing that if the second difference is a constant, then the a in our formula will be half that value.  Therefore as our second difference was 1, the value of a will be 1/2.  We then find that b = 1/2 and c = 0.  So the formula for the triangular numbers is:

0.5n2 + 0.5n

Method 2

We can also derive this formula by breaking down triangular numbers into the following series:

1

1+2

1+2+3

1+2+3+4

Therefore we have the sum of an arithmetic series, with first term 1, common difference 1 and last term n, and so we can use the sum of an arithmetic series formula:

Sn = 0.5n(a1 + an)

Sn = 0.5n(1 + n) = 0.5n2 + 0.5n

Once this is done, we are asked to find the nth term for the 6-stellar numbers (with 6 vertices) below:

stellar1

which give the pattern 1, 13, 37, 73

Method 1

Once again we can use the method for quadratic sequences.  The second difference is 12, giving us an2 + bn + c with a = 12/2 = 6. Substituting values gives us:

1 = 6(1)2 + b(1) + c
13 = 6(2)2 + b(2) + c

This simplifies to:

1 = 6 + b + c
13 = 24 + 2b + c

Therefore we can eliminate to find that b = -6 and c = 1.

which gives 6n2 – 6n + 1

Method 2

A more interesting method makes use of the triangular numbers.  We can first note a recurrence relationship in the stellar numbers – each subsequent pattern contains all the previous patterns inside.  In fact we can state the relationship as:

S1

S2 = S1 + outside star edge

S3 = S2 + outside star edge

S4 = S3 + outside star edge

stellar3

The outside star edge of S2 can be thought of as 6 copies of the 2nd triangular number

stellar4

The outside star edge of S3 can be thought of as 6 copies of the 3rd triangular number, but where we subtract 6×1 (the first triangular number) because we double count one of the internal points six times. We also subtract 6 as we double count each vertex.

stellar5

The outside star edge of S4 can be thought of as 6 copies of the 4th triangular number, but where we subtract 6 x 3 (the second triangular number) because we double count three of the internal points six times. We also subtract 6 as we double count each vertex.

The outside star edge of S5 can be thought of as 6 copies of the 5th triangular number, but where we subtract 6 x 6 (the third triangular number) because we double count six of the internal points six times. We also subtract 6 as we double count each vertex.

Therefore we can form a formula for the outside star:

6(0.5n2 + 0.5n) – 6(0.5(n-2)2 + 0.5(n-2)) – 6

which simplifies to:

12(n -1)

We can now put this into our recurrence relationship:

S1 = 1

S2 = 1 + 12(n -1)

S3 = 1 + 12((n-1) -1) + 12(n -1)

S4 = 1 + 12((n-2) -1) + 12((n-1) -1) + 12(n -1)

Note that when we substituted the nth term formula for S2 into S3 we had to shift the n value to become n-1 as we were now on the 3rd term rather than 2nd term.

So:

S1 = 1

S2 = 1 + 12(n -1)

S3 = 1 + 12(n-1) + 12(n-2)

S4 = 1 + 12(n-1) + 12(n-2) + 12(n-3)

So:

S1 = 1 + 0

S2 = 1 + 12

S3 = 1 + 12+ 24

S4 = 1 + 12 + 24 + 36

So using the formula for the sum of an arithmetic Sn = 0.5n(a1 + an) we have

Sn = 1 + 0.5(n-1)(12 + 12(n-1))

Sn = 6n2 – 6n + 1

Quite a bit more convoluted – but also more interesting, and also more clearly demonstrating how the sequence is generated.

stella6

Generalising for p-stellar numbers

We can then generalise to find stellar number formulae for different numbers of vertices.  For example the 5-stellar numbers pictured above have the formula 5n2 – 5n + 1.  In fact the p-stellar numbers will have the formula pn2 – pn + 1.

We can prove this by using the same recurrence relationship before:

S1

S2 = S1 + outside star edge

S3 = S2 + outside star edge

S4 = S3 + outside star edge

and by noting that the outside star edge is found in the same way as before for a p-stellar shape – only this time we subtract p for the number of vertices counted twice.  This gives:

p(0.5n2 + 0.5n) – p(0.5(n-2)2 + 0.5(n-2)) – p

which simplifies to

2p(n-1)

and so substituting this into our recurrence formula:

S1 = 1

S2 = 1 + 2p(n-2)

S3 = 1 + 2p(n-2) + 2p(n-1)

S4 = 1 + 2p(n-3) + 2p(n-2) + 2p(n-1)

We have the same pattern as before – an arithmetic series in terms of 2p, and using Sn = 0.5n(a1 + an) we have:

Sn= 1 + 0.5(n-1)(2p + 2p(n-1) )

Sn = pn2 – pn + 1

Therefore, although our second method was slower, it allowed us to spot the pattern in the progression – and this then led very quickly to a general formula for the p-stellar numbers.

If you like this you might also like:

The Goldbach Conjecture – The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of 2 primes.  No one has ever managed to prove this.

Ib maths puzzles

Maths Puzzles

These should all be accessible for top sets in KS4 and post 16.  See if you can manage to get all 3 correct.

Puzzle Number 1

Why is xx undefined when x = 0 ?

Puzzle Number 2

I multiply 3 consecutive integers together. My answer is 8 times the middle of the 3 integers I multiplied. What 3 numbers could I have chosen?

Puzzle Number 3

You play a game as follows:

1 point for a prime number
2 points for an even number
-3 points for a square number

(note if you choose (say) the number 2 you get +1 for being a prime and +2 for being an even number giving a total of 3 points)

You have the numbers 1-9 to choose from. You need to choose 4 numbers such that their score adds to zero. How many different ways can you find to win this game?

Answers below in white text (highlight to reveal)

1) xx is undefined because using 2 different indices rules will give us contradictory results. 0 to any power will always be 0, however any number to the power 0 will always be 1. With 2 contradictory answers we leave it as undefined!

2) The equation we want is (x)(x+1)(x+2) = 8(x+1). This simplifies to x^3 + 3x^2 -6x – 8 = 0. We can solve this using the factor theorem, polynomial division or by plotting a graph to get 2 possible solutions – x = 2 or x = -4.

3) The numbers will have the following values: 1 = -3, 2 = 3, 3 = 1, 4 = -1, 5 = 1, 6 = 2, 7 = 1, 8 = 2, 9 = -3. There are at least the following possible combinations:
1,2,3,4
1,2,5,4
1,2,7,4
9,2,3,4
9,2,5,4
9,2,7,4
6,8,4,9
6,8,4,1
Check to see I haven’t missed any!

If you like this post, you might also like:

A Maths Snooker Puzzle. A great little puzzle which tests logic skills.

Visualising Algebra Through Geometry.  How to use geometry to simplify puzzles

Analytic Continuation and the Riemann Zeta Function

Analytic Continuation is a very important mathematical technique which allows us to extend the domain of functions.  It is essential in higher level mathematics and physics and leads to some remarkable results. For example, by using analytic continuation we can prove that the sum of the natural numbers (1 + 2 + 3 + ….) is -1/12. Results don’t get more surprising than that!

Analytic continuation concerns functions of the form:

f(z) where z is a complex number and f(z) is (complex) differentiable.

Remember complex numbers are of the form a + bi and can be thought of as coordinate points in an x,y axis.  For the purposes of this post we will only look at real values of z (real numbers are still a subset of complex numbers).

The idea of analytic continuation is to take an original function with a restricted domain, then to find another function which is the same within that restricted domain, but also is valid outside that domain.  This sounds very complicated – but let’s look at a couple of examples:

f(z) = \frac{(z+1)(z+2)}{(z+1)}

This is a function which is defined for all values except for z = -1. When z = -1 we have zero on the denominator so the function doesn’t exist. However we can write a new function:

g(z) = (z+2)

Now, g(z) = f(z) for all z when z is not -1, but g(z) also exists when z = -1. Therefore we can regard g(z) as the analytic continuation of f(z), and we have extended the domain of f(z) from all values except -1, to all values of z.

A more interesting example is the following:

f(z) = \sum_{n=0}^\infty z^{n}

This is the infinite series:

1 + z + z^{2}+ ...

This function is analytic (complex differentiable) only when   -1 < z< 1.  (Don’t worry about how this is calculated – though it is related to the domain of convergence). Therefore this is our restricted domain.

But we can notice that the sum of a geometric sequence formula allows us to calculate f(z) in a different way:

\sum_{n=0}^\infty z^{n} = \frac{1}{(1-z)}

Here we have used the formula for summing a geometric, with the first term 1 and common ratio z.

Therefore we could write:

g(z) = \frac{1}{(1-z)}

f(z) = g(z) when -1 < z< 1 , but g(z) is complex differentiable for all values except for z = 1 (when the denominator is 0).  Therefore g(z) is the analytic continuation of f(z) from  -1 < z< 1 to all values of z except z = 1.

continue1

One example of analytic continuation that I’ve written about before is the Riemann Sphere. This extends by analytic continuation the complex plane into the complex plane plus infinity.

Another example is used in showing that the sum of natural numbers is -1/12. There are a few different methods to show this – some discussed previously here. I’m going to try and talk through another proof of this result. It’s a bit difficult, but try and understand the general method!

riemann2

The proof revolves around the Riemann Zeta function, (Riemann is pictured above). This is defined as:

\zeta(z)= \sum_{n=1}^{\infty}n^{-z}

This can also be written as:

\zeta(z)=\frac{1}{1^{z}} +\frac{1}{2^{z}} +\frac{1}{3^{z}}..

So, if we want to find the sum of 1 + 2 + 3 … then we need to substitute z = -1 into the above summation. However this formula for the zeta function is only valid for the domain z > 1, so we first need to extend the function through analytic continuation.

Through analytic continuation (where we extend the domain from z > 1 to all complex numbers apart from -1) we can rewrite the zeta function as:

\zeta(1-z)=2^{1-z}\pi^{-z}cos(0.5\pi z)\Gamma(z)\zeta(z)

and substituting z = 2 into this formula, so that we end up with zeta(-1) we get:

\zeta(-1)=2^{-1}\pi^{-2}cos(\pi)\Gamma(2)\zeta(2)

Now,

\zeta(2) = \frac{\pi^{2}}{6}

\Gamma(2) = 1

cos(\pi) = -1

Therefore

\zeta(-1)=-\frac{1}{12}

We have proved that 1 + 2 + 3 … = -1/12 !

If you enjoyed this post you might also like:

The Riemann Hypothesis Explained. What is the Riemann Hypothesis – and how solving it can win you $1 million.

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

mmd1

Murder in the Maths Department

A murder has been committed in the maths department! A body has been discovered surrounded by mathematical objects and only the hardworking maths teachers were in school, doing long division sums for fun at the weekend. One of them must be the murderer.  (The wall of fame of successful detectives will be posted here)

Your task, should you choose to accept it, is to find:

1) The murderer
2) The room
3) The murder weapon

The murder suspects are:

1) Al Jabra – who was wearing a white, T-shirt with 2 stripes and ripped jeans on the day of the murder.
2) Polly Gon – who was wearing a knee-length green skirt, white blouse and gold watch.
3) Lisa Perbound- who was wearing a blue Adidas T-shirt with 3 stripes on the sleeves, Bermuda shorts and a baseball cap.
4) May Trix- who was wearing a black and white pin-stripe suit with shiny black shoes.
5) Ella Ment- who was wearing a blue knitted jumper with a picture of pi on the front, and brown cords.

The possible rooms are:

The possible rooms are:

1) The Canteen
2) The Tuck-shop
3) Room 20
4) Room 18
5) Room 17
6) Room 7

The possible murder weapons are:

1) A wooden metre ruler
2) A large metal stapler
3) A dusty trundle wheel
4) A sharp compass
5) A large maths textbook
6) An oversized calculator

Clue Number 1

mmd12

When you have solved this clue – click here, and enter the last word only of the decoded message (no capital letters).

The Poincare Conjecture and Grigori Perelman

In 2006 the Russian mathematician Grigori Perelman was awarded the mathematical equivalent of the mathematical Nobel prize (the Fields Medal).  He declined it.  In 2010 he was the first mathematician to be awarded $1 million – he turned it down.  What had Perelman done to achieve such (apparently unwanted) acclaim?  He had solved a puzzle that had frustrated mathematicians for over 100 years – the Poincare conjecture.

What is the Poincare Conjecture?

The Poincare Conjecture is that, “Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.”  At first glance that may look quite complicated – so looking at the definitions in turn:

poincare3

Simply connected means a shape without holes.  The two shapes on the left above are simply connected, the two on the right are not.  In 3 dimensions, a sphere and cube are simply connected, but a donut shape (torus) is not.

3D manifold means a 3 dimensional surface.  Imagine the surface of a sphere – that is a 2 dimensional surface.  So a 3 dimensional surface on a sphere would require a 4 dimensional sphere.  A 4 dimensional sphere is one which has a fixed radius in 4 dimensions (unlike in 3 dimensions for a sphere and 2 dimensions for a circle).

poincare5

Homeomorphic means it is mathematically equivalent in terms of the relationship between points.  Basically, if 2 shapes can be sqeezed or stretched to form another shape then they are homeomorphic.  In the above animation, the coffee mug and the donut (torus) are shown to be homeomorphic.

3-Sphere means a sphere in 4 dimensions (i.e with a 3 dimensional surface area).

poincare6

So, with those terms defined we can simplify the Poincare conjecture.  In regular 3 dimensions, conventional 3 dimensional shapes without a hole in them (cubes, cuboids etc) can all be squashed and squeezed to create a sphere.  Poincare conjectured that the same would be true in higher dimensions – i.e 4 dimensional cubes (a tesseract, as shown above) could be squashed and squeezed to make a 4 dimensional sphere.

poincare2

Grigori Perelman however was not interested in either the acclaim or the money on offer for solving one of the world’s most difficult mathematics problems.  In explaining why he turned down $1 million he said that the prize, “was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed.”

If you liked this post 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.

Non Euclidean Geometry V – The Shape of the Universe – Using mathematics to understand one of the most important questions of all.

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