Monday, 31 October 2011

Halloween

 

Just a quick update from me to bring you this wonderful short video that I and James Grime made together. With it being Halloween and all we decided to make a short video on the mathematics of zombie invasion. Enjoy :).

For more information of the maths of zombies see this page.  Thanks to Martin Berube for the zombie picture.

Monday, 3 October 2011

A short hiatus

Currently, I'm up to my eyeballs in my doctoral thesis. It is very close to being done, however, it's those last few hurdles that take all the effort. Thus, the Laughing Mathematician will be taking a short break as I try to get on top of things.

It is certainly not that I've run out of ideas! I will be back and hopefully, when I am, I will be a doctor :).

Keep smiling.
Yours,
Thomas

Monday, 26 September 2011

Mathematical biology. Create your own Turing patterns.

Last week I showed some of the possible patterns that were available as solutions of mathematical equations. This week we end morphogenesis month by showing you how to create such Turing patterns for yourself.

Freely available at:
is a simulator (written in java) of the types of reaction-diffusion patterns that we have been considering. The supplementary information section gives a really nice overview of how Turing patterns work and how to use the software. However, for those amongst you who are too impatient to read such things click on the picture below to show a brief manual of how to use the program.


The main joy of this program is that you are able to alter the parameter values of the reaction diffusion equations, seen in the big box at the bottom, very quickly. Thus, you are able to experience different patterns being created right in front of your very eyes! Below are a few that I created in 5 minutes that easily show you the great complexity available through this simple mechanism.

Each simulation starts with a smiling face initial condition like the one seen below.
Initial condition
The first pattern occurs when the program is run with the default initial conditions. Clearly, we see that we get multiple line patterns framing the happy face. Altering the parameters slightly produces spots and stripes as seen in the second picture, giving the face a much more surprised look. Pushing the parameter values even further we generate oscillating patterns as shown in the two snapshots on the right, thus, the face changes colour rapidly. Finally, for certain patterns it is difficult to make out the initial condition at all. In  the bottom image the solution oscillates but in waves across the domain.

Click on the picture to make the images clearer.

Hopefully, you'll have some fun exploring the many different patterns you can get. Whilst you are generating complex structures think about how difficult it must have been for Alan Turing to have postulated these patterns without our moden day computers. He never saw the patterns that now bare his name, yet, he managed to construct the framework by which we understand them.


Finally, if these have entertained you and you would like to see even more peculiar patterns then try Tim Hutton's youtube channel. Through combing different types of models he was able to produce oscillating Turing patterns (seen below). Now they are really trippy!


Hopefully, you have enjoyed morphogenesis month as much as I have had making these posts. I deal with Turing's theory of morphogenesis everyday and it is such a beautiful subject that I decided everyone must know about it.

Next week, back to the normal mathematical silliness.
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References:

Monday, 19 September 2011

Mathematical biology. Pattern formation.

Today we actually consider the types of patterns we can produce. This is a subject dear to my heart as my area of research is in pattern formation and in particular I have recently published a paper discussing the patterning found on stingrays.

The patterns found on the stingays can be produced through using a mathematical model called the BVAM equations. The BVAM system is quite a simple set of equations but they are able to lead to extremely complex behaviour and demonstrate numerous different types of patterns. For instance, not only are they are they able to give spots like those on the stingrays:
video


but, with minor alterations, they are also able to produce striped patterns that look like certain types of catfish.
video
These patterns are created through visualising the solutions that can be created through using so called "reaction-diffusion" equations. Normally, the mathematical models describe the dynamics of two chemical species that react with each other and diffuse through the domain, hence the name. By seeding a random initial state (seen at the very start of the videos) we can evolve reaction-diffusion systems through time and these simulations calculates the distribution of the chemical species throughout space, over time.

Of course the output of these systems is simply just a set of numbers which give the value of  concentration at each point in space and time. But, by associating a colour with various levels of concentrations, we are able to produce the fantastically colourful videos above. This idea of associating numbers with colours is illustrated in the videos, as on the right you see a colour bar which tells you the value of the system based on the colour. Negative numbers are nearer the blue end of the spectrum, whilst positive numbers are nearer the red. Thus, what you are seeing in both of these simulations is the response of one chemical population reacting with a complementary species and diffusing through the domain.

A mathematical solution is one thing but a physical experiment is quite another. As I have mentioned before, one of the key criticisms behind using this mathematical theory in biology is that we lack any definite evidence that morphogens can produce Turing patterns and other spatial complexity like that seen in the simulations above.

However, chemistry is there to save the day! Below is a demonstration of the Belousov-Zhabotinsky (BZ) reaction. Amazingly, it can show numerous different types of dynamics, all of which can be described through reaction-diffusion models.

The footage firstly demonstrates the colour oscillations that are hallmarks of such reactions. The footage then moves on to show the spatial patterns that the BZ reactions create. These phenomena are so peculiar that when they were first discovered by Boris Belousov in the 1950s no one believed his results and his findings were rejected by the chemistry community. This left him a broken man causing him to leave science.

Much like Turing he was a man ahead of his time. So much so that we are only really beginning to fully appreciate the importance of his work.

Next week, I bring morphogenesis month to a close as I demonstrate how to simulate mathematical patterns for yourself.

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References:
Catfish: http://de.wikipedia.org/w/index.php?title=Datei:Pseudoplatystoma_fasciatum.jpg

BZ reactions footage: http://www.youtube.com/watch?v=IBa4kgXI4Cg

To read more about the work on stingrays I direct you to the University of Oxford's science blog,
http://www.ox.ac.uk/media/science_blog/101126.html

Monday, 12 September 2011

Mathematical biology. Morphogens.

What creates a pattern? As yet we do not know. However, this has not stopped mathematicians from suggesting mechanisms by which they form.

Although we now have many mechanisms which can produce patterns through mathematics the best understood and most used is still that of Alan Turing's diffusion-driven instability. It is a testament to his true genius that an idea that he postulated in 1952 is still applicable to today's research.

To understand his work we first need to understand the concept of a morphogen and how they are used to create patterns. You may want to have a look back at the post on diffusion as we assume that this is how many morphogens move.

What is a morphogen?
A morphogen is any substance which is able to produce a pattern.

Normally, we think of morphogens as chemicals that are able to diffuse and interact with each other creating  complex forms. However, the concept of morphogen is much broader. For instance it could be a source of nutrients for a fungus that causes the fungues to grow into patterns like that seen in the left.

For now we will simply consider the case of chemical morphogens, which are able to diffuse through animal, or plant, cells. Finally, we assume that these cells can sense the morphogens and alter their behaviour because of them.

The French flag  pattern
Suppose a system of cells has a constant source of morphogen on its left side. This morphogen will then produce a concentration profile, or gradient, that is higher on the left than the right (see row A below). 

The cells on the left sense a higher concentration of morphogen and respond in some way e.g. they turn blue. The centre cells sense a middling concentration and the right-hand cells will sense a low concentration and, so, they produce different responses e.g. they turn white and red, respectively.

Hence, through simple diffusing morphogens we are able to produce the so called French flag pattern. If you are feeling more adventurous (or patriotic) you can couple multiple sources together to create the flag of the Netherlands, the Danish flag and even the Union Jack (although I've never seen this done :) ).






















In rows A and B we can see that simply through diffusion we are able to produce quite complex patterns. If we now let the morphogens react with each other much more complicated structures are able to form such as spots, stripes and labyrinthine patterns as seen in row C.

It is these Turing patterns that we will be considering next week.

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References:
The original idea that diffusion could create patterns instead of just wiping them out first proposed by Alan Turing in his landmark paper, The Chemical Basis of Morphogenesis, As such Turing can be thought of as the founder of mathematical biology.
Morphogen diagram: Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation by Shigeru Kondo and Takashi Miura. Art work by S. Miyazawa


Monday, 5 September 2011

Mathematical biology. Diffusion.

As noted in the previous post (and the zombie post), diffusion is a very important process, biologically as well as generally. Today, we take a closer look at this undervalued mechanism of motion.

To give an intuitive idea of the motion, consider a small blob of ink added to water that is not stirred or heated. Although the water is not being disturbed the ink will begin to spread out and, after enough time, it will colour all of the water. The ink particles are constantly and randomly bombarded from all sides by molecules of water, exactly like billiard balls. These random collisions cause the ink to diffuse through the water in a process called Brownian motion. 

Time series evolution of ink in water illustrating diffusion.
This post is going to contain some serious maths, which is unusual for a "fun" blog. However, I urge you to stick with it. If you have never studied maths before the symbols will appear esoteric. However, don't worry. I am going to guide you through, step by step, building up the diffusion equation until there should be no mystery any more.
  Suppose we define the concentration of a chemical at a point x and at a time t be U(x,t). This means that where we have a lot of substance U will be a high number and, conversely, where U is a small number, we will have less chemical.

Through this simple definition we find that physics tells us that the random motion has to satisfy the diffusion equation,


The diffusion equation describes the motion of any substance which can be thought to be moving randomly e.g. proteins in your body, water through soil, heat through metal and smells in the air. But where does this equation come from and what does it mean? Let us decipher these curious hieroglyphics.

The left hand side is the "derivative of U(x,t)", with respect to time, t. Explicitly,
This means that if the derivative is positive then U(x,t) is increasing at that point in time and space. If  the derivative is negative then the concentration is decreasing. This term allows us to consider how U(x,t) evolves over time. The factor D is a positive constant that controls the speed of movement.

The term on the right-hand side is a little more complex than the left, but essentially it encapsulates the idea that the chemicals are moving from high to low densities. This is illustrated with the help of the figure below.


An example of how diffusion smooths out peaks and troughs.
Initially, the concentration is higher on the left than the right. Just before the peak in density the arrow, which is the tangent to the curve, or
at this point, is pointing upwards. This means that as x increases, so does the chemical density. Hence, at this point,
Just after the peak the arrow is pointing down and, so, at this point,
Hence, at the peak, the derivative of U(x,t) with respect to x is decreasing as x increases. 

By extending the definition of derivatives that we are using, we know that

 is the rate of change of
 
as x increases and, moreover, we have just deduced that this rate is decreasing at the peak. Thus, from the diffusion equation we see that, at the peak,
this means that the peak of chemical concentration is decreasing over time.

By a similar argument we can show that U(x,t) is increasing at the trough. Overall, we see that diffusion causes the chemical to move from regions of high density to low density.

It is impossible to overstate the importance of the diffusion equation. As indicated above, it can be used in many biological and physics contexts. This week we have given you the insight that diffusion is a smoothing process. As time increases diffusion will tend to produce a uniform districbution of chemical throughout the domain as in the ink and water experiment above. Next week we consider the counterintuitive result that Turing postulated:
DIFFUSION CAN CAUSE PATTERNS TO OCCUR.

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References:

All of this work is motivated by Alan Turing's landmark paper, The Chemical Basis of Morphogenesis. As this is quite a difficult read, I've taken it upon myself to try and explain it to everyone through this blog.

Monday, 29 August 2011

Mathematical biology. Motivation.


Firstly, I should probably answer the question of what mathematical biology actually is. As the name suggests, it is the application of mathematical techniques to biological problems. But why should we want to do such as thing, when experiments can be run? Why would be want to turn the beauty of nature into an ugly equation? Well, here are a few reasons off the top of my head:
  • Experiments can identify cause and effect relationships. Mathematics can suggest mechanisms which underpin these relationships.
  • By predicting the outcome of altered systems we reduce experimental waste.
  • Once a system has been successfully mathematically modelled (whatever that may mean). We maybe able to understand the cause of pathological cases. This leads to theories on how to correct them.
As mentioned above, my main interest is in pattern formation. Mathematically, we have a number of models that produce qualitatively the same patterns as animal skins. You can see this below; on top you have a the skins of a cheetah, a poison arrow frog and a giraffe; all of which have distinct patterns. Below this you have the mathematical patterns which can be produced using only one mathematical theory.




An important aspect of Turing's is that it suggests many types of animals depend on the exact same mechanism to produce their individual patterns. This supports the idea that evolution has simply picked a simples mechanism, whilst mutations and various types of selection will specify how the model behaves. Interestingly, it is not just skins that are thought to use these patterns, they even appear of animal shells, as seen at the top of the page. The left picture in each couple is a real shell, whilst the right picture is a computer simulation.

The patterning systems we use tend to rely on diffusion as the key mechanism. In terms of evolution this is important as it suggests that no energy from the animal is needed to produce the pattern; only to create the reactive agents (called morphogens) which will naturally diffuse.

Next week I'll be demonstrating how we model diffusion mathematically.

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References:
The original idea that diffusion could create patterns instead of just wiping them out first proposed by Alan Turing in his landmark paper, The Chemical Basis of Morphogenesis, As such Turing can be thought of as the founder of mathematical biology.

The application of Turing's theory to shells can be seen in Han's Meinhardt's book, The Algorithmic Beauty of Sea Shells.







Monday, 22 August 2011

Mathematical biology. The creator.

2012 is the year of Alan Turing's centenary. Not only was he the father of computational logic as we know it today but he also had a lesser known interest in biology. Specifically, he was interested in such questions as:
  • Why is there something rather than nothing?
  • How are complex structures formed from simple components?
  • Why doesn't everything tend to a state of uniformity?
In 1952 he published a paper, The Chemical Basis of Morphogenesis, trying to answer these questions. His ideas were revolutionary and not fully appreciated at the time. However, his paper was the start of a whole new field of mathematics, which 60 years later still uses his ideas as their foundation.

Over the next few weeks I'll be giving you an insight into how mathematics can unravel the mysteries of biological development. Not only does this extend our understanding of the natural world but it also offers us a new way to appreciate the beauty, simplicity and diversity of the world around us.

We start next week, where I will illustrate what a mathematician interested in pattern formation can do.

Just a small selection of the patterns which Turing's theory can produce. Taken from http://www.rudyrucker.com/blog/2010/09/10/what-was-alan-turing-like/ which gives you the highlights of Turing's weirdness. 

Monday, 15 August 2011

Maths laugh 6. Up to date satire.

Ali G. was actually an excellent mathematical philosopher!

As usual, tweet me any maths jokes to illustrate @ThomasEWoolley.

Monday, 8 August 2011

What "The Code" could have been...

On Wednesday 27th July 2011 a new three part documentary hit the British television. It was called The Code, hosted by Professor Marcus du Sautoy and had the intention of demonstrating how mathematics can be used to unravel the mysterious code that surrounds the natural and man-made world. The reaction to the shows has been mainly positive and some of the footage has truly been breathtaking (see the footage below of the bubbles creating the platonic solids, incredible!).



Whatever your thoughts or feelings are about The Code, I wanted to share with you some information about what The Code could have been and let you make your own mind up as to whether they chose the right program.

At the beginning of February 2010 I had the amazing fortune to be asked if I would like to work as a researcher on a new mathematics program called The Code. Of course I jumped at the chance. So the things I explain here are my from my own experience. They are not hearsay or rumours, just plans that never came to fruition.

The competition aspect of The Code was there from the start and it was always the intention to have the programs weave with online content. However, the style of the show had originally been very different. The plan was to team Marcus up with a different celebrity each week and challenge them to do something amazing. The celebrity would admit that they could not complete the task and the program would track the celebrities progress as Marcus tries to teach them the various, simple mathematical principles that would allow the celebrity to achieve their goal.

My job was to develop the celebrity challenges. It certainly was not easy. I had to produce an exciting result using simple mathematics. My favourite example that I came up with was:
get the celebrity to try and measure the height of a cliff standing over water. Tape measures wouldn't be possible, so, instead, they would have to learn trigonometry.
Now you may think that is extremely mundane, however, the next part of the challenge was to:
get the celebrity to jump of the cliff using a bungee cord based on their calculation.
Not only do the have to work with heights but they have to take into account the extension of the chord as well. They would literally be placing their lives into the hands of mathematics.

I created five or six such activities like that mentioned above, wrote them up in a treatment, was paid for my work and left White City, having had an amazing experience. However, on my last day I was told what would then happen to my work. The pitch would go forward and either get funded or not. If the idea was not commissioned it would be recycled until they found a format that worked. If it got funded, the project would get sent to a production team who could COMPLETELY IGNORE all the development work!

So the next time you watch an episode of The Code (the last episode is to be aired Wednesday 10th August), think about what it could have turned into and ask yourself:
Did the BBC make the right choice?


Monday, 1 August 2011

Maths laugh 5. More food based maths.

Anyone else hungry?


For some more maths based food I recommend the following links from the Evil Mad Scientist Laboratories:

Here is a demonstration of how to cut a bagel into two connected pieces and, finally, turn your fruit into platonic a solid.

Monday, 25 July 2011

Family fun day at the Royal Institution

This week's blog post takes the form of a cheeky advertisement for the Royal Institution's family fun days.

Myself and other Mathemagicians have been there a few times now and we run mathematical puzzles and games that are all based on a specific theme given by the institute.

Previously, we have run mathematics puzzles based on food. Here, we considered the one-, two- and three-dimensional cutting problems. We also ran a load of weighing problems like the one I wrote about a few weeks ago.

Along with the Alchemists, we also attended the family fun day on forensics. We constructed a puzzle based on ideas from propositional logic. The puzzle was so successful that it even appeared in a Nature network blog. Have a go at cracking the mystery and, do not forget, there may be more than one possible suspect.

Me entertaining and educating the public in the ways of logic.

This coming Saturday, 30 July 2011, the day is all about waves and we have some great experiments lined up!

  • We will be showing how the current generation of 3D works using polarised light.
  • I have constructed a Doppler effect machine (I'm very proud of this).
  • What shape do water waves make in different containers?
  • Finally, (if we can get it to work) we will be demonstrating the interference phenomena of light using the Young's slit experiment.
And that is just what we will be doing! There will also be lectures throughout the day and a whole host of other experiments based on waves.

Come along and bring the family, there will surely be something to interest you. For more information see the Royal Institution's family fun day page.

Monday, 18 July 2011

Arts vs Science

A couple of years ago a debate took place in Oxford, the title of which was "Poetry is beautiful, but science is what matters". An evocative title I'm sure you'll agree. However, what I personally liked about the debate was how it was advertised, as shown in the above picture.

Upon a background of equations you have a smooth, regal, Einstein with an angelic glow to show support for science. On the left, in support of the "beauty" of poetry, is an elderly, wrinkled gentleman, smoking a cigarette with unkempt hair.

Now, this may have been done completely by accident, but I just love how biased this picture comes across in its advertising of the event.

[Edit based on the excellent suggestion of Christian Perfect.] Interestingly it was an internet debate so you can read the whole set of arguments here:


In summary, the motion was defeated 62% to 38%. As you might expect, the debate never really got around to stating which field matters more. Instead, the battle was over semantics. What does "matter" mean? What does science mean? Etc.

Personally, I don't subscribe to the whole arts vs science utility debate. As Richard Feynman once said,
"Physics is like sex. Sure, it may give some practical results, but that's not why we do it."


P.S. If you know who the poet is in the figure please say so in a comment below. I would love to know who it is.

P.P.S. I've just been informed that it is W. H. Auden. Isn't the internet wonderful?



Monday, 11 July 2011

Diffusing zombies.

As mentioned in a previous post, I  have written a mathematical article on zombiism as an infectious disease. Since Robert Smith? et al. had already done this already you maybe wondering what we added to the zombie theory.

Ask yourself the following question. What are zombies known for? You may come up with a long list of properties, but personally I can think of two defining characteristics:
  • they prey on humans;
  • the move slowly and unsteadily.
Although Robert et al. covered the first one very well they had ignored zombie movement. Now, although the speed of zombie movement is arguable, as recently they've started to run (and even ride motorbikes), we decided to stick with the slow moving variety as they are easier to understand.


So, how do we model zombie movement? Well, we based their movement on the idea of a "drunkard's walk"; the agent lurches to and fro in a random fashion with no bias in direction. Of course, you may argue that zombies head directly towards humans, which would be true. And, although we could mathematically model this, it is simpler to assume random movement. Thus, we can think of our equations portraying the earlier stages of a zombie infestation, i.e. when the zombies first arise, they will be very confused and will be moving around randomly.

From this assumption we can model the zombie population as a diffusive substance. Now diffusion has two primary properties. Firstly, the agents move without directional bias and, secondly, the zombies move from places of high density to places of low density. This is shown in the figure above and the movie below which illustrates how a population of zombies all starting at the left hand side would move across the domain.

video

So what can this formulation tell us? Firstly, it implies that running away is better than trying to slow a zombie down. The reason behind this is that by doubling your distance between yourself and a zombie, you multiply the first meeting time by four. However, if you slow the zombie down by half, then you only multiply the first meeting time by two. Thus, here is my advice for today. If you see a zombie...
RUN!

Monday, 4 July 2011

Mathematics Genealogy Project.

Although, legitimately, I can only trace mine back to 1885 to Roland Weitzenbock, a polish mathematician, who worked on invariant theory and corresponded with Albert Einstein over the unified field theory. Roland did teach Otakar Boruvka, who can be traced right back to such great names as Gauss, Euler and Bernoulli.

Below is my mathematical genealogy. For an actual readable version, click on on the picture to go to the high resolution version on Picasa.



If you're interested in discovering your own mathematical genealogy. Go to http://genealogy.math.ndsu.nodak.edu/ for a fully searchable interface.

Monday, 27 June 2011

Answers to last week's questions.

Last week I posted some lateral thinking and trick questions. The idea being that, when I do outreach workshops to younger audiences, they will be less worried about being wrong if they realise that everyone will be as wrong as they are. Again, I would like to reiterate the point that this is nothing about making the students feel stupid, but rather empowered as they should not be worried about make "silly" suggestions.
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  1. The questions starts off with "You're driving a bus". Thus, the eye colour of the driver is whatever yours is.

    The idea behind this question is to emphasise the point that to produce an answer you first have to understand what information you need from the question.

  2. If you take 307 bananas from 429 bananas, how many bananas do you have? You have 307.

  3. All months have 29 days.

  4. Questions 2 and 3 deal with the idea of clarity of communication. When answering a question in mathematics you have to clearly define your terms and your assumptions.


  5. The probability that exactly five are in the right envelopes is zero. If five are in the right envelopes the sixth must be two.

    Here, again, we are dealing with the idea of taking the important information from the question. The most important word in the question is "exactly". Thus, any solution we generate should be weighed up against the requirement.

  6. Each dog takes five days to dig a hole. So ten dogs will take five days to dig ten holes.

  7. The full stop at the end of the sentence is the smallest circle.

  8. By now the students should realise that the questions are trick question, so questions 5 and 6 teaches them not to be too hasty with their answer and to think carefully even when the answer appears obvious.

  9. Tuesday, Thursday, today and tomorrow.

  10. This question asks them to find a solution to the seemingly impossible. The idea being that mathematicians need tenacity when working with a problem. Many times you it will seem like the question is intractable but eventually you will find the right path.

  11. Noah built the Ark not Moses.

  12. All the numbers are divisible by two. The question does not ask for integer answers.

  13. Questions 8 and 9, again, show the importance of reading questions carefully and fully understanding what is being asked. Trust me, as a mathematician, the hard part is not generating a solution, but rather, understanding what the question is asking.

  14. And the final puzzle. Did you spot it? The first instruction you are given is to write your name in the square. Next to this instruction is a rectangle. The square is at the bottom :).

Monday, 20 June 2011

Tricky.

Below are a few of my favourite lateral thinking and trick questions. Whenever I do outreach work shops on higher mathematics for secondary school I often open with this as a 5 minute ice breaker. Hopefully, the kids will not do very well. The reason I say hopefully is because I want to emphasise that they do not know everything. This is not meant in a bad way, but an encouraging way. Over the course of the workshop I touch on subjects such as logic and topology which they will have never seen. By getting the audience comfortable with being wrong and not knowing the answer I hope to encourage them to ask questions and provide solutions, even if they turn out to be wrong. 

Fear of being wrong or asking "silly questions" is a common barrier to over come in a class room situation. The participants are surrounded by their peers and the last thing they want it to do is to seem stupid. However, the workshops flow better if everyone is willing to suggest their insights on which to build. 

So, without further ado, try the questions yourself. Only give yourself 5 minutes to try out all the questions and, of course, don't cheat. I'll post the answers next Monday, so you can see how well you did. Remember
Research is what you do when you don't know what you are doing
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Write your name in the square:

  1. You're driving a bus that is leaving on a trip from A and ending in B. To start off with, there were 32 passengers on the bus. At the next bus stop, 11 people get off and 9 people get on. At the next bus stop, 2 people get off and 2 people get on. At the next bus stop, 12 people get on and 16 people get off. At the next bus stop, 5 people get on and 3 people get off.  Question:  What colour are the bus driver's eyes?

  2. If you take 307 bananas from 429 bananas, how many bananas do you have?

  3. In a leap year how many months have 29 days?

  4. A secretary prints out six different letters and is in a rush so she randomly stuffs the letters into six envelopes going to six different addresses. What is the probability that exactly 5 letters are in the right envelopes.

  5. If five dogs dig five holes in five days, how long will it take ten dogs to dig ten holes? 

  6. Highlight, in some way, the smallest circle.

  7. Name four days which start with the letter T.

  8. If animals enter the Ark in pairs at rate of 30 pairs per day, how many days would it take Moses to get 360 individual animals on board?

  9. How many numbers between zero and ten, inclusive, can be divided by two?
Score:





Monday, 13 June 2011

Mathematical poetry 2.

The following poem is about a certain Professor Felix Fiddlesticks:

F set the coins out in a row
And chalked on each a letter, so,
To form the words "F AM NOT LICKED"
(An idea in his brain had clicked).
And now his mother he'll enjoin:
MA DO LIKE
ME TO FIND
FAKE COIN
-- Cedric A.B. Smith

Now this may appear to not make much sense, but it is in fact a really clever solutions to the 12 coin problem:

Imagine you are given 12 coins, and a set of weighing scales. 11 of the coins have the same weight, but one has a different weight. To make matters worse, you do not know if it is heavier of lighter than the others. The problem is to find out which coin is different, and whether it is lighter or heavier, using at most three weighings on a pair of scales.

Before you see the answer and I explain the connection between the puzzle and the poem, have a go at solving the problem yourself. Here is a flash applet which allows you to play that game yourself:
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Did you solve it? Did the poem help? If you managed to do it, give yourself a pat on the back. Otherwise, let me explain. Firstly, do as the poem says; on the coins write one of the single letters F, A, M, N, O, T, L, I, C, K, E and D. Now, we weigh the coins
MADO against LIKE
METO against FIND
FAKE against COIN
 
in each of the weighings we have two possibilities. Either, the pans balance or they don't. If any of the weighings do balance, we immediately know that all 8 coins are genuine and the dodgy coin is in the four we haven't weighed. Conversely, if the pans do not balance then we know that the four coins not on the scales are genuine.

As a specific example, suppose that, in our weighings, the right pan is always lower. We can then logically deduce that the coin "I" is the dodgy one and that it is heavier, since coin "I" is the only coin that appears in the right balance in all three weighings. If you are interested in the entire solution to this problem look at the last post in this Dr. Math forum post.


Monday, 30 May 2011

Fractal fun.

As a Mathemagician (I didn't pick the name) I have been lucky enough to work with Prof. Marcus du Sautoy on a number of occasions. One of my favourite projects that I have worked on was when he asked me to do some illustrations for his most recent book, The Number Mysteries. Specifically, in Chapter 2, The Story of Illusive Shape, he wanted some pictures of fractals. So for those of you who have read the book you'll see some familiar figures below and for those of you who haven't then hopefully these will pique your interest and you will go and find out more.



Figure 1. Illustrations of how to measure the dimension of a fractal by covering it in boxes of different lengths.


Figure 2. The Koch snowflake is a very famous fractal. Normally, it is constructed using an equilateral triangle. Here, I alter the angle to make it isosceles. What do you think that might do to the dimension?


 Figure 3. The Koch snowflake is constructed by replacing the middle third of a triangle with a triangle a third of the size. Normally, all the triangles are taken to point outwards. However, above I randomise this process.
Figure 4. If you put three of the randomised Koch snowflakes together you get something that looks like a medieval map of Britain


If in the future I run out of things to say I may put up the codes so that you can also learn how to create such structures.