Constraints and Creativity in Mathematics and Fiction

I wrote earlier about Hannu Rajaniemi and his wonderful performance at the  The International Science Day. Here is a video from the scene. Mathematics and fiction are not very different.

Hannu Rajaniemi: Escape from the Castle of If (Constraints and Creativity in Mathematics and Fiction) from Suzi McEndoo on Vimeo.

Exploring the Intersection Between Science and Art

I had the most wonderful day Last Sunday. My hometown Turku is one of the European Capitals of Culture 2011 and there are all kinds of events throughout the year. I did not however expect anything very special for me, but it turned out I was wrong. In the local newspaper I found out about The International Science Day - Science Meets Art. I have watched Ted talks and wondered how nice it would be to attend one and now I got the chance to see many outstanding speakers talk about quantum physics, mathematics and how these two hars subjects can be presented in a fun and a creative way.

One of my favorites was Hannu Rajaniemi, the author of Quantum Thief, a new sci-fi book which I love! Hannu is Finnish but Lives in Scotland, has a Ph.D. in Mathematical Physics. He was talking about the differencies and similarities in Mathematics and Fiction. Mathematicians create imaginary objects and determine relations and rules between them and start playing with these imaginary things. That is just what I have tried to tell people about math. It is based on the creativity of the scientists! Other sciences use the results of this play, but it is not relevant for the mathematician. On the other hand Fiction has to have the plot, style, relations between characters and other rules. For me writing is like that; sometimes I am much more comfortable playing with my math than trying to figure out how to write logical, understandable sentences.

I also saw the premiere of the Documentary "Inside the Light", the story about dealing with single atoms and photons and making their presence visible.

Mathematics and quantum physics was used to describe what it looks like in a human's brain, what mathematics has to do with joggling and how music could be created from numbers and arithmetical operations. All the speakers made these hard scientific things so much more comprehensible even for people who don't know the theory so well.

I will get back to more details later, but here is a short video about the conference with a couple of interviews



 

I have never attended a cofenference or a seminar which would have fascinated me more!

Some More Math Fun

As an answer to George Hart's Möbius Bagel, Serious Eats has constructed a Möbius Doughnut

Then I saw a picture on Wired Science and thought that somebody had knitted a mathematical equation in 3-D.

It turned out that it was a Mandelbulb, a 3-D visualisation of a fractal called a Mandelbrot set. After seeing some videos like the next one, I still think that this is just like a cool knitting work...

Here is an example from a Finnish knitting blog nekkisneuleet. Using different patterns, someone could really knit a visualisation. Too bad I am not good in designing...

Finally Makezine led me to some really impressive polyhedra made of playing cards, again by George Hart

When Your Math Papers Come Alive

Last night I was listening The Drill Down, a podcast which is well known among the Diggers. I don't much listen to it nowadays live after they changed the time they send the podcast. Now it comes out in the middle of the night for me (1.00 am Monday morning) and if I have to work the next day, I don't want to to stay up that late. Anyway my TV was open on the background and it was on The Voice TV Finland, playing some music videos. Suddenly one of the videos caught my attention even though I had my headset on for listening to the podcast. The video is just so fascinating and the song is not bad either. My papers live also their own life but usually the other way around. Numbers and calculations start crawling among the other text and pictures :)

The Fray - Heartless:


The FRAY - Heartless from IE HAGY on Vimeo.

Inspired By Mathematics

As in honour of my Christmas holiday I found two mathematics inspired items yesterday.

This ring is inspired by the Fibonacci sequence 1,1,2,3,... The pattern of the beads is the beginning of the sequence.

holmescraft's shop (via Neatorama)

This is a mathematically correct breakfast, a bagel cut and linked to make a Möbius strip

You can find the instructions here (also via Neatorama)

Money Exchange Game

I have played a lot of flash games on PrimaryGames website. The game named Coinz! is disturbingly hard and addictive at the same time. The target of the game is to make money disappear by picking up the same amount of money using two currencies.

The first game involved exchanging Singapore Dollars to Russian Rubles. This was hard. There were lots of coins marked 1, but it is harder to determine if the coin is 1.oo or o.o1. Choosing Hint from the list on the right you can see the value when marking a coin.

After all the Singapore Dollars were exchanged the level was completed. Now the new currency was Lithuanian Litas. The exchange rate is shown above the box and below the box you can see how many Rubles you should find matching the Litas you have chosen.

Interestingly the third level seemed easier to me. Now Polish Zlotys were exchanged to Litas, and the value of the currencies being nearly the same made it easy.

A very good game in mathematics learning. I could imagine that kids are faster to find the right coins than I was.

Geometric Art

BibliOdyssey introduces scans of wonderful geometric art.

Here is an example of Lorenz Ströer's 'Geometria et Perspectiva' in 1567:

and this is an example from the large Munich University Library manuscript called: 'Geometria et Perspectiva: Corpora Regulata et Irregulata':

See all the images in The Geometric Landscape and also follow the links on the page.

Make Puzzles Not War

I have often imagined what would have happened if mathematics concepts could be copyrighted. Newton developed differentiation and integration to complete his gravitational theory. What if he had realized that the idea is very valuable and had copyrighted it? All the other scientists would have had to use other methods for various problems or had to pay Newton to use it. Science would not have developed. Pure mathematics is actually a theory of different methods and the applications are used by physicists, economists, chemists, biologists and so forth. All intellectual development had suffered. In this case Leibniz was working on the same problem at the same time and there would have been a huge argument who actually got the original idea.

Luckily ideas can not be copyrighted. In the scientific world mathematicians publish their work for the benefit of everybody. When someone creates just a problem he may get his name in history like Fermat in Fermat’s Last Theorem. Andrew Wiles has now proved the theorem but it just may happen that someone will create a much better proof and Wiles will be forgotten. Mathematics strives for the best solution and the person who provides that, gets his/her name in history.

KenKen, CalcuDoku, Square Wisdom or Mathdoku?

KenKen is a mathematical puzzle. The goal of each puzzle is to fill a grid with digits so that no digit appears more than once in any row or column. Grids range in size from 3x3 to 9x9. Additionally, KenKen grids are divided into heavily outlined groups of cells and the numbers in the cells of each group must produce a certain given number when combined using a specified mathematical operation (either addition, subtraction, multiplication or division).

The image above is not a KenKen puzzle, it is Calcudoku. KenKen is now a registered Trademark of Nextoy, LLC (as well as KenDoku) and the books containing KenKen’s are authored by Tetsuya Miyamoto, and Will Shortz; Kendoku books are authored by by David Levy and Robert Fuhrer; Dj Ape used originally the title Square Wisdom; The Puzzle Society uses the name Sukendo; The Kindle version is Mathdoku and the puzzle above is a Conceptis made Calcudoku, which name is also used by Dj Ape in the later books. The trademark does not prevent people from doing similar puzzles, it just prevents others calling them with the same name.

The abundance of different names is confusing for the general public and makes it more difficult to assess the overall popularity of the puzzle. When the Sudoku popularity was at the highest, we had many different Sudoku books and magazines available in Finland made by local puzzle makers. Now we don’t have a single KenKen (or Calcudoku) book available. If all those puzzles had the same name people would find them better and the mathematical challenge could find also people who are not puzzle addicts like me. I constantly meet mathematics teachers who never have heard of these puzzles. It will never reach the popularity of Sudoku.

Variants of Sudoku

Actually KenKen is also one form of sudoku. It is also based on a Latin Square which is an nxn table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.

Here are 2 examples of 5x5 Latin squares.

Sudoku is a Latin square with an additional rule: The objective is to fill a 9?9 grid so that each column, each row, and each of the nine 3?3 boxes (also called blocks or regions) contains the digits from 1 to 9 only one time each. The puzzle setter provides a partially completed grid.

This puzzle is known to most people, because it became very popular and newspapers publish them daily even though the biggest boom is over. The amount of different Sudoku answers is of course smaller than that of 9x9 Latin Squares because of the 3x3 box rule. The amount of puzzles is bigger and depends on the givens, the numbers provided on the puzzle.

(For those who are interested on the mathematical calculations about the amount of Sudoku’s or the minimal amount of givens, I recommend the Wikipedia article on Mathematics of Sudoku. It contains links to several mathematical articles about the subject.)

Colour Sudoku

The classical Sudoku has the 3x3 boxes. The third rule can however be obtained in many ways and these puzzles are called Sudoku variants. Different lists of them can be found all over the internet. Some of the variants have made it to publications containing these variants; some of them are published regularly on Sudoku variants magazines, some others have been created just for examples. The longest list I found is made by Uwe Wiedemann.

One not so know variant is Colour-Sudoku. This example is from Added Bytes.

Here the third rule comes from the colours; every colour represents the “region” which has to contain each number only once. Here the region is totally spread all over the grid. The size would not have to be 9x9. The same arrangement could be obtained by any nxn grid.

Now Conceptispuzzles has published a new variation called Chain Sudoku. The release article got a very angry response from Strimko. Their puzzles have the same idea; the third rule defining the region is obtained by lines connecting the numbers. This variant has also different sizes. Here is an example of Conceptis Chain Sudoku.

Battle

Now Strimko says that the intellectual idea belongs to them. In some comments they also claim that the streams have not existed before and this idea is totally different from Irregular Sudoku.

Here is an irregular Sudoku, and this would make a Chain Sudoku very easily:

The Chain Sudoku contains one feature which irregular Sudoku doesn’t, the ability to link numbers diagonally.

However this is not a new idea even for Conceptis. In August 2006 Sterling published a book Snakes on a Sudoku – official Snakes on a Plane Puzzle Book and the puzzles were created by Conceptis. Here is one scan from the book:

As you can see there is also the diagonal snake. Somehow the word stream describes these better than the previous ones. To me they look chains. The logic I use in solving both types is the same!

Mathematically speaking

I have also seen a more advanced approach to connecting the numbers, a Toroidal Sudoku:

In mathematics toroid is the doughnut-shaped object. These puzzles are scanned from a magazine BBC Mind Games Christmas 2006. This could really be clued to make a doughnut, but I think it’s better to solve it first. Dr Gareth Moore publishes these regularly on his Puzzlemix website.

Mathematically speaking Strimko’s and Chain Sudoku’s are graphs, more precisely they consist of connected acyclic graphs which are called trees. An acyclic graph with multiple connected components is sometimes can be called a forest.

There is also another puzzle closely connected to graph theory: Hashi. The rules in Hashis are however totally different

As a conclusion to the above; any puzzle having the third rule in addition to the row/column –rule is just another variation in the Sudoku family. Conceptis has an algorithm to create Sudoku’s and Dave Green states in Conceptis Forums:

“To demonstrate how close these puzzles behave, it took us only several minutes to modify the algorithm and the graphical export of our Sudoku generator to enable it to produce Chain Sudoku puzzles.”

I have done some programming but my main interest is algorithms in general and I know that if the code is written using good programming, the changes are easy.

I think I answered this question.

Gareth Moore's Colour Sudoku

There is still the design, but even there Strimko is not totally original. Here is another Sudoku from Gareth Moore which uses colours instead of numbers:

My love of logical puzzles started originally from my main interest, mathematics. I think logical puzzles are very good in learning logical thinking which is a very important skill in mathematics. Puzzles are not as threatening to students as mathematics sometimes seems to be. To me it is a very sad thing that copyright issues get in the way of making these puzzles publicly known. Sudoku’s became popular because the Japanese puzzle company Nikoli, which created the name Sudoku, never trademarked it outside Japan. (Nikoli did not invent the puzzle).

Crediting Strimko

Some of the comments have been criticizing that Conceptis does not give the credit to Strimko. With all the puzzle types Conceptis has provided info about the history of the particular puzzle and I have no doubt they would have made it this time also but quoting Serhiy Grabarchuk in Conceptis Forums:

"Some months ago I warned Conceptis Puzzles that if they will use the idea of Strimko (under any name) with the same features as our Strimko, we will understand that as a direct and clear infringement of our intellectual property rights."

I hope there will be some solution to this. I wrote my opinion to my blog because I see no point in joining all the puzzling forums Strimko has left their accusations. Those who are interested in them can use Google to find them. If Conceptis decides to drop out this puzzle, I will not solve them. I want to find my puzzles on the same website. I have too many links to various social media sites and educational sites as it is!

Arthur Benjamin about Math Education

I found this video on my usual bookmarking sites. It got Front Page on Digg and lots of Stumbles also. The video is very interesting from my point of view as a mathematics teacher. Arthur Benjamin, a mathematics professor and also the mathemagician offers a bold proposal on how to make math education relevant in the digital age.

I think he is a really good performer but he cuts some corners and speaks like a politician convincing people of something they know very little about.

Before I utter my criticism, I have to remind you that my perspective is Finnish and our mathematics education. I probably don't know enough of the conditions in USA. I have learned that there are schools where the students don't advance very well, but on the other hand the most of the best universities are located there. Anyway calculus and statistics are universal.

The biggest problem in my opinion is that statistics is not a science we can put under the main category mathematics. It belongs to social sciences (in the University of Turku for one) and the interpretations of statistic data and their relevance has very little to do with math. Probability is a part of applied mathematics and also the calculations regarding statistics use math, but something as vague as predicting the future can not be a part of a science which gives only absolute solutions or shows that there are not any! I have a master's degree and my major is mathematics, I have even had some predoctoral studies, but I never had a single course in real statistics.

In Finland we have two different levels of mathematics the students can choose from, the short and the long course. Both levels have one course containing probability and basic statistics. The courses are very similar on both stages and I have noticed a certain trend with these courses: there are always students who get fascinated by the common-sense-math and perform much better than on the other courses BUT also some of the top students are finding this course very hard.

Calculus (we actually use the word Analysis) is taught about 2 courses on the shorter course (no integrals) and 5 on the longer course but also extra courses for those who want to choose them.

I have heard the statement: "I don't remember anything about derivatives" or: "I have not used derivatives after I left school" numerous times. I also know that most people don't, not even those who choose their career in engineering. However the basic calculus is needed in order to understand the further studies in engineering, physics, chemistry, mathematics and even statistics! A tool for the continuous probability distributions is integrals.

Teaching basic mathematical statistics is not as simple as it may sound. When I teach mean values and standard deviations the examples are either too simple to show anything about real statistics or too complicated to be handled without the calculator's built-in statistical functions. Mental mathematics skills are not very good even now, they would be worse if we take more calculators or computers in use. Regardless of the methods we use the examples take time. Typing all the input data is not so simple.

Arthur Benjamin mentions the digital age. I have a better suggestion, let's add Number Theory. It is simple, it is fun and it is a very relevant field of mathematics regarding computers, coding, cryptography and sharing secrets which will be needed in internet voting. These fields of math are hot at the moment, but the basics have been there for ages. I teach one extra course in Number theory (+ Math Logic) but there are many students who don't choose it.

It was actually Gauss (more known from the Normal Distribution) who said: "Mathematics is the queen of the sciences and number theory is the queen of mathematics."

Image credits: Wikipedia and zazzle

Teaching Mathematics in Finland With Japanese Puzzles

This is a repost from March for educational purposes with some pictures added here.



In the end of 2008 I had the chance to teach Japanese puzzles in my own school. Now my story has been published on Conceptis website :) I hope this gives ideas to other mathematics teachers as well. Some puzzles in the middle of normal work is refreshing and teaches logic reasoning. Logic is the foundation where mathematical thinking is built. I hope you enjoy the story.



read more | digg story

New Math Puzzles

Conceptis has added a new puzzle to their wide collection of weekly puzzles. The puzzles are named Calcudoku's. You can read more about them here

Here is the tutorial



It's all about math!

If you want to get a widget of your own, you can get the code here

Fractal Snowflake Cupcakes

I found these cool mathematical cupcakes. They go well with my Sierpinski cookies.

read more | digg story

Pi Day

Happy Pi Day for everyone celebrating it. Pi day is celebrated on March the 14th due to the American style of expressing dates, 3/14, and the fact that 3.14 is the approximation of Pi everyone usually knows. However we Europeans write the date 14.3.2009 and the idea of switching the day and month seems totally impossible. We could use 31.4 as Pi Day but April does not have 31 days...

The solution for us is Pi Approximation Day on July the 22th, due to pi being roughly equal to 22/7, an ancient approximation of pi.

I told my students about the Square Root Day 03.03.09 and they probably thought I was crazy. We don't actually have a culture for celebrations like this. Maybe I should start pushing them. We have numerous days dedicated to old Finnish culture figures. Let's add some mathematics to the culture!

Here is Lose Yourself (In The Digits) by Pi Diddy made for Pi Day last year.

Time and Mathematics

I have once blogged about a clock for math geeks but today I found a much cooler clock in Walyou, my favourite Tech website.

This really needs more advanced math than I am teaching.

Walyou has intoduced many math clocks earlier and I decided to check what else could be found in the net.

This is the first one I spotted in StumbleUpon almost 2 years ago:

What I found funny was the discussion in cnet news. The expression for 7 gives 6.999... and the argument about 0.999... being equal to 1 continues. Maybe I should blog about that next.


MathematiciansPictures.com has nice clocks featuring famous mathematicians. This one is my favourite, since it contains the most beautiful equation. The Euler Clock:

For a reminder, this is the one I introduced before. It also has some inaccuraties. 9 is only an estimate and 7 has two solutions.

I also love this Binary clock. I have one at home, a mother's day gift from my children. This one is from ThinkGeek

Kaboodle had this clock made of numbers

This is a unit circle clock and would be really helpful learning trigonometry. Found in Law9

This cool but confusing clock I found in WebUrbanist, which has altogether 24 different Modern Clock designs introduced.

Libraby of Math has a huge collection of math related clocks.

a different kind of binary clock

a trigonometry clock

a square root clock

This is actually a watch from Watchismo, but I think this would be much better on the classroom wall.

Even though I love math, I think this Turing alarm clock is too much... My husband and daughter want to continue sleeping after the first alarm. I would go crazy when I had to listen to all the alarms several times! When I wake up, there is no chance to continue sleeping; I am awake.

Wild About Math!

During my 1.5-year journey in the blogosphere I have found amazingly many good resources. There are times when I find a place which leads to so many new resources that I spend days exploring them. One of these occasions was when I found Sol Lederman's blog Wild About Math!. Sol's blog is a serious math blog which mine is not; I have many times been more enthusiastic to write about puzzles or the Finnish lifestyle and of course Piitu.

Sol covers all aspects of math from the fear of it to the fun of it and takes care of all ages from kindergarten to university. Even his Blogroll is full of math bloggers.

Here are some posts I enjoyed the most:

8 really fun paper and pencil Math games
Interesting little Math problem
Five constants tie together multiple branches of mathematics
What patterns can you find in Pascal’s triangle?
Ti-Nspire inspires Math students

The last one is important to me at the moment. I have a trial version of the TI-Nspire program and I was planning to check it over the holidays. I guess I am running out of time...

Sol has also made great mathcasts (videos)

I still have lots to learn!

Math Anxiety

I am preparing for a two-day course in learning to use Moodle. Already a year ago I learned about WiZiQ which helps teachers to create online lessons. Browsing their site I found this great presentation about Math Anxiety. It is meant for both teachers and students. I think my conciense about handling the students is clear :)

The presentation is a bit long, but worth to watch.

Mathematics in Noodles

I found this cool video from kottke.org. It is a clip from Philip Morrison's 1987 PBS program "The Ring of Truth: Atoms" featuring chef Kin Jing Mark making noodles to demonstrate the principle of halving.

 

Math Trick

How to Read Someone's Mind With Math (Math Trick)

from wikiHow - The How to Manual That You Can Edit

Trick a friend with your "psychic powers" using only numbers. You don't have to be smart- just remember these steps.

Steps

1. Tell your partner to pick a number between 1- 10.
   
2. After they pick a #, tell them to multiply it by 2.
   
3. After that tell them to multiply it by 5.
   
4. After that ask them to divide their current number by their original number. (ex: in the 1st step if they pick 3, tell him/her to divide their current number (30) by 3)
   
5. Now tell your partner to subtract their number by 7.
   
6. If you did this right, it should be always be 3!
   

Tips

• How it works: so say in the first step they pick 5. In the 2nd and 3rd step, they times it 2 and 5, but actually, they multiply it by 10. so they get( in this example) 50. in the 4th step, they divide it by the original number, or 5. *so they always get 10. the 5th is optional, but it makes the trick less obvious.
  
• Try to be dramatic. After all, it is a 'magic' trick.
  
• give them some time on step 4. its kinda complicated.
  
• Don't forget the steps! This could be embarassing.
  
• Keep a straight face, but don't be too serious.
  
• If possible, give them a calculator. Step 3 is a tough one.
  

Warnings

• DON'T do this to 2 or more partners at a once.
  

Things You'll Need

• A partner( a friend, your mom, etc.)
  

Related wikiHows

• How to Perform a Card Trick Using Math
  
• How to Do a Math Magic Trick
  
• How to Learn Math
  
• How to Be a Great Prankster
  

Article provided by wikiHow, a collaborative writing project to build the world's largest, highest quality how-to manual. Please edit this article and find author credits at the original wikiHow article on How to Read Someone's Mind With Math (Math Trick). All content on wikiHow can be shared under a Creative Commons license.

Numberz!

I found a nice math online game. The aim is to clear the playfield making necessary sums of numbers in the neighbouring tiles. When a coloured tile is involved in the sum, it gets replaced by a white tile. You can play the game online or download the game on your computer.

The game is a Flash game and looks very nice. It was originally a bit too big for my laptop screen until I noticed that the size can be changed.

At first I thought that the game was too easy until I got stuck with a tile having number 8. It was a corner tile and all its neighbours were too large to sum up to 10. I had some difficulties deleting the neighbours and waiting for smaller tiles to show up :) This just shows the same thing I am trying to teach my students: consider the situation first before you start calculating. There may be a very quick solution if you think about it.

Mandelbrot - Mathematical Art

It seems that planning next years courses takes so much of my time and energy that the Madrid story gets delayed day after day.
In the meantime you can enjoy this video named Fractal Zoom Mandelbrot Corner

About Mandelbrot and Fractals in Wikipedia.