Walking Randomly

In my previous post I asked how people felt about the future of the Carnival of Maths since it had become poorly organised lately.  Well, a group of us got together, rolled up our sleeves and sorted something out.  The result should see the Carnival going for a couple more years yet at least.  Here are the measures we have taken

• I have taken over the running of the main carnival submission page and have updated it.
• I have created a Twitter feed for the carnival.
• Rod Carvalho has created a google group for people who have contributed to the carnival over the years.  A group of us have been discussing the carnival’s future there and, between us, we will be able to ensure that it stays well organised in the future.  As far as I am concerned, this group runs the show and if they vote to have me replaced as carnival co-ordinator at any time for any reason then I will hand over the reigns.
• The Carnival of Math is now monthly and will be published on the first Friday of every month.  The next one will be on Friday 6th November and will be over at Jason Dyer’s Number Warrior.
• From November, The Math Teachers at Play Carnival will be running on the 3rd Friday of every month in order to co-ordinate with the new Carnival of Math schedule.  This means that you are never far away from your next fix of mathematics blog posts.

That’s pretty much it.  Thanks to everyone who has contributed to the disucssion and to those who helped set things up.  Here’s to the next carnival :)

The Carnival of maths has been around for a while now, recently reaching it’s 58th edition, but its organisation has become erratic of late. The first carnival was put together by Alon Levy who organised it ever since, continuing even after he officially hung up his blogging hat. The mathematics blogging community owes Alon a big debt of gratitude so I hope you all don’t mind if I say “Thank You” to Alon on all our behalves.

Recently, however, it seems that Alon’s other life commitments have caught up with him since the main Carnival page has not been updated since Februrary and the Carnival submission form indicates that the carnival has closed.  I’ve also tried to email him a couple of times (before I hosted the 58th carnival for example) but as yet have received no response.  Alon – if you are reading this then I hope you are doing OK.

So, onto the question posed in this post’s title.  What should the future hold for the Carnival of Maths?  I have thought of a few possible options as follows

• Let it end here.  The community has Math Teachers at Play and that may be enough carnival for us all.
• Continue the carnival in it’s current form but co-ordinated by someone else.
• Start a new carnival series with a title like ‘New Carnival of Maths’.
• None of the above?  Your suggestions would be welcome.

Personally, I favour option 2 – to continue the carnival in its present from but to reduce posting frequency to once a month.  As for co-ordinator – I would be happy to do it if there were no objections (and host a co-ordinating page here) – but if it is felt that this is inappropriate then I would happily step aside and let someone else do it.

It’s your carnival so what do you think?  Comments welcomed.

Welcome to the 58th Carnival of Maths – the late edition!  I should have published this a couple of days ago but work and life commitments got in the way and I have only now found time to put it together.  So, working on the assumption that late is better than never let’s get on with the show.

Whenever I discuss mathematics on this blog I often focus on using technology and so naturally I turned to technology to help me answer the traditional carnival question ‘What is interesting about the number n?‘ – where n is the edition number of the carnival being hosted.  So, let’s see what I can find out about the number 58 using the best technology the web can offer.

The Number Gossip website informs me that 58 is the smallest Smith number (in base 10) with a prime sum of digits. In addition (and more mundanely), 58 is also composite, deficient, even, evil and square free.  Wolfram Alpha also has a lot to say about the number 58 such as the fact that it is an idoneal number and that it can be expressed as the sum of two squares: 58= 3² + 7². Wolfram Alpha doesn’t finish there though since it also gives a wealth of other trivia about 58 such as how it can be written in binary, Roman Numerals, Greek, Babylonian and Mayan.  It tells us that its prime factorisation is 2*29, that 58 is the ASCII code for a colon along with the set of quadratic residues modulo 58.  Finally, it finishes off by telling us that

is a near integer.  A spot of googling helps me discover that  58 is the sum of the first seven prime numbers (2+3+5+7+11+13+17=58) and is also the sum of the totient function for the first thirteen integers .  Hmmm, fascinating!  However, enough of the number trivia and on with the show.

First up, we have a post from Jason Dyer of Number Warrior where he analyses one of his baby daughter’s favourite toys.  That’s right ladies and gentlemen, only 6 months old and already inspiring mathematics.

Brent Yorgey, of Math Less Travelled fame submits the post Challenge #12, Part II, where he discusses counting “hyperbinary” representations (like binary representations, but allowing up to two copies of each power of two instead of just one).  If you thought this carnival was late then you ain’t seen nothing yet ;)

Pat Ballew, author of the aptly titled Pat’s blog, sent in two related submissions for this edition of the carnival.  First up is a post about a problem and its solution in ‘problems from the land down under‘ (I don’t know about you but I can’t read that title without thinking of a certain song from the 80’s).   Pat worked on making the solution to this particular problem more understandable and, with a little help from his friends, he came up with a graphical solution.

The next submission comes from the blog Math with my kids where the writer asks some questions about sequences of rational numbers.  He knows the answers to four of the questions but the fifth question is open as far as he knows.

Moving over to The Endeavour, we have two related submissions from John D Cook all about The Mercator Projection and its inverse.

Finally, we have a submission from Paul Brabban who’s non-mathmo fiancee asked him ‘Why is a minus times a minus equal to a plus?’  How would you answer this question if asked?

And that’s pretty much it for this edition of the carnival – or at least I think it is.  I had to pull two of these submissions out of my spam bin so it seems that the filter has been a bit over-zealous recently.  I have published everything that was submitted to me so if you submitted something and it is not in this post then you have fallen foul of the spam filter gremlins. If you let me know via a comment to this post in the next few days then I’ll fix the situation.

Thanks to everyone who submitted.

Welcome to this – the last Carnival of Mathematics in 2008.  Sadly, submissions were on the low side this time around – probably due to the fact that many people were away from the internet over the festive season (and so they should be).  Rather than let this adversely affect the size of this carnival, I broke every rule in the carnival book and went searching for some submissions of my own.  I hope you enjoy the results…

Carnival tradition dictates that before I start the show I should entertain you all with some mathematical facts about the number 46 (since this is the 46th edition of the carnival – for those of you who have imbibed too much festive cheer) and I have managed to come up with:

• 46 is an evil number – but only for a certain definition of evil :)
• 46 is a Nonagonal (or enneagonal) number
• The prime factors of 46 are 2 and 23 which makes 46 an example of a semiprime number and a square-free number.
• 46 is a centred triangular number.

With tradition satisfied lets look at this edition’s submissions.

Rod Carvalho of Reasonable Deviations shows how to construct polynomials from their roots using a graphical approach.  This is a nice way of viewing the process of constructing polynomials, as all the cumbersome algebraic manipulation boils down to assigning values to the nodes of a graph.

Maria Andersen of TCMTB shows how to create an answer key to your calculus test (or any other test for that matter) using Windows Journal.  I have to confess that I have not used Windows Journal because it is part of Windows Vista – something I only ever use under duress.  Maria makes it look rather compelling though so maybe I need to convince a friend of mine to lend me their laptop!

John D. Cook of The Endeavour gives us a Constructive proof of the Chinese Remainder Theorem.  This comes at a good time for me because as part of my ‘fill in the huge gaps in my maths knowledge’ project, I have been learning about congruences from an Open University booklet I found in the library.  John also submitted his most popular post of 2008 – Jenga Mathematics which is more than worthy of your attention.

If I stuck to tradition then that would be it for this edition of the carnival since they were the only submissions I received.  It seems that the festive period really isn’t a good time for getting responses from math bloggers!   So, I threw tradition out of the window and went looking for some of my favourite maths blog posts from the last 12 months.  The only rules I tried to stick to were ‘one post per blog and one post per month’.

1 year, 12 months, 12 posts, 12 blogs – Happy new year to you all.

January:  Mathematics is logical, elegant and refined.  The real world isn’t! Just like a street fight, the real world is dirty, surprising and uncompromising and solving real-world mathematical problems in physics and engineering can often take phenomenal amounts of computational power. Enter street-fighting mathematics, an MIT course that teaches you some of the tricks and techniques you’ll need in order to get approximate answers to many real-world problems in just a few lines of mathematics.  I first learned of this course from a blog post by Michael Lugo of God Plays Dice back in January and yet I still haven’t found time to work through it.  With luck, I’ll manage it in 2009.

February: I like mathematics that’s pretty and so does Vlad Alexeev – author of both Mathpaint and Impossible World.  Back in February he highlighted a sculpture of a three dimensional Hilbert curve by Carlo H. Séquin.  This post also includes an image rendered by a piece of software called Maxwell Renderer which looks intruging – can anyone suggest the closest open source equivalent?

March: Thanks to Easter, a lot of people went egg-crazy back in March and Kathryn Cramer of Wolfram Research was one of them.  In response to her initial post a lot of us attempted to create Mathematical easter eggs using Mathematica.

April: I have never taken a course in graph theory and so I don’t know much about it but I have attended talks on the subject and so have seen an old map of Königsberg several times.  Using googlemaps, the guys over at 360 showed that if we pose the Königsberg problem today then the result is completely different.

May:  If you read this blog for more than a couple of weeks then you will quickly realise that I like computer algebra systems and yet the free, open source pacakge Axiom is one that I haven’t played with much.  Alasdair of Alasdair’s Musings has though and he has also written a great 6-part introduction to the system which started in May.

June: No Carnival is complete without a puzzle to solve and Tanya Khovanova gave us a great one back in June.

July: Back in July, Eric Roland gave us details of his prime generating formula. 

August: Google isn’t just a search engine – it’s a calculator too but, like all calculators, it doesn’t always give the correct results.  Stephen Shankland gave us the details back in August.

September: Brian Hayes of bit-player invited us all to just shut up and program.  All too often I read articles from old-timers (such as myself – recently hitting 31) who lament about the loss of a supposedly golden age of computing.  You see, back in the 80s and early 90s we used computers such as the Sinclair Spectrum, Commodore 64, Acorn Archimedes and Amiga and all of them came with a programming language built in – usually some form of BASIC.  These old-timers argue that young-uns find it difficult to get into programming these days since computers no longer come with programming languages built in – or if they do then they are hidden from view in some way.

Of course this is a load of rubbish.  Hand me a computer with an internet connection and 60 seconds later I will hand it back to you with one or more programming environments installed that would be suitable for mathematical exploration (or mucking around as I prefer to think of it).  Brian’s article gives some ideas, both free and commercial, that might get you started.

October: Loren Shure of the Art of MATLAB explains how to create the Olympic rings using MATLAB.

November: What are p-adic numbers?  I have no idea – yet another subject that is on my list of subjects to study.  Dave Richeson of ‘Divison by Zero’ knows what they are though and gave a basic introduction to them back in November.  Thanks Dave – that post now represents the sum total of my knowledge on the subject.

December: Finally we reach December and a post from squareCircleZ who explains how Archimedes was doing calculus 2000 years before Newton and Leibniz.

So, that’s it.  The final carnival of mathematics for 2008.  I hope that no one minds the breaks from tradition and I hope you will join me in supporting the carnival throughout 2009.  Happy new year to you all

Hello and welcome to the 33rd edition of the Carnival of Mathematics. This carnival very nearly didn’t happen since I didn’t realise that no one had offered to host it until a couple of days ago! I toyed with the idea of letting this edition of the carnival lapse and write something in a fortnights time but then that would break the carnivals unbroken run of 33 publications (well..apart from that one time which we don’t talk about) and I simply couldn’t have that. So, with only two days to go I bent the standard carnival rules a little and started leaning on people I know in order to get submissions. After that I started leaning on people I didn’t know and I am glad to say that everyone came through and I have a nice selection of articles for you all.

Before I get onto the articles themselves, tradition dictates that I attempt to fascinate you with some interesting facts concerning the number 33. Well how about this one:

It is known that for all numbers N below 1000 that do not have the form it is possible to express N as a sum of three cubes. In other words

where a,b and c can be positive or negative. What does this have to do with the number 33? Well, 33 is the smallest such number for which a,b and c have not yet been found. If you fancy having a crack at solving this be aware that the solution for N=30 is

Anyway, enough with the trivia and on with the show!

As some of you know, I am a big fan of computer algebra systems (well most of them anyway) and so I thought I would start off with some submissions from three of the big names in the CAS world, Wolfram Research, The Mathworks and SAGE. I use the products of all three of these groups to one degree or another and so it is great to see submissions from them all. This is one of the areas where I bent the carnival rules slightly since I emailed the blog authors and said “Hi – please submit something to the carnival.” I thank them for humoring me and not consigning my email to the spam bin.

Loren from Loren on the Art of Matlab writes a regular blog on Matlab programming and her submission is a recent post entitled Acting on Specific Elements in a Matrix where she uses several methods to obtain the same result. This sort of article is very instructive when thinking about how to go about developing your code. Although she did not submit it, I thought that many carnival readers would also be interested in her post called Matlab Publishing for Teaching.

Next up from the Mathworks we have Doug whose submission is a coin tossing puzzle which he invites you to solve using Matlab. Some solutions can be found in the comments section so resist the urge to scroll down if you want to try and solve it yourself. Solving problems like this, using any system, can be a great way of learning how to use it – much more interesting than just reading through the manual; no matter how well written it is.

Moving over to the Wolfram Research Blog we have two posts in this edition of the carnival, the first of which is called Two Hundred Thousand New formulas on the Web which is a discussion of The Wolfram Functions Site. At the time of writing the site has over 307,00 formulas on it which is, quite frankly, astonishing! Pretty useful too!

Next up from Wolfram we have a blog post called Making Photo Mosaics. It never ceases to amaze me how much you can achieve with so little code – I will be having a play with this code using photos from my recent vacation :) Check out the video that Theodore has produced as part of this post as I think it’s fascinating.

Moving over to the world of open source we have a submission from William Stein – Can There be a Viable Free Open Source Alternative to Magma, Maple, Mathematica and Matlab? where he discusses the SAGE project. I have recently been looking at SAGE myself and have been very impressed with it.

This edition of the carnival isn’t just about computer algebra packages though – we also have lots of non-CAS submissions. The first of which is one from Maria over at the TCM Technology Blog where she writes about her talk, Exploring Online Calculus, at the Michigan MAA meeting. Gotta love those graphs :)

John of jd2718 asks Can we find the area of a quadrilateral from just it’s co-ordinates?, with some interesting answers in the comments section. I reckon a nice Wolfram Demonstration could be made from this idea.

Sam Shah thinks that algebraic manipulation is overrated – head over to his blog to see why. In another post, Sam also writes about some interesting calculus projects that he has assigned to his students. When I was at school I used to love open-ending projects as it used to give me a sense of ‘owning the material’. I distinctly remember doing a project on the Fibonacci sequence when I was 11 years old and spending ages on it. To this day I still have a fascination for the topic and probably always will. I wonder how often such projects can be done by school children in todays test-centric environment?

Moving on, we have Math for the Very Patient from Vlorbik on Math Ed. Vlorbik has already demonstrated his patience in the past since my blog looks horrible on his browser and yet he still reads what I have to say – thanks Vlorbik! I seem to have a problem with IE 6 that I have no idea how to fix. Just look at this blog in IE 6 compared to firefox to see what we mean. One hexadecimal pound (thats two pounds and fifty six pence) to the first person who can diagnose and fix the problem for me.

Over at blinkdagger (among other things, a great source of Matlab tutorials) they have a competition where you can win prizes from the people at the art of problem solving. There is still time to enter so take a look at BlinkDagger burgers and have a go.

If you like the level of your mathematics to be a bit higher and median graphs are your thing then you will be interested in David Eppstein’s submission Median graphs and binary majorization over at OxDE.

Denise of Let’s Play Math sent me the details of her latest post, The Function Machine Game. This is another one I remember doing when I was at school. As she suggests it’s probably best to limit the functions one can choose from – “Waddya mean you couldn’t get it – BesselJ(x) is simple!” I feel yet another Wolfram Demonstration coming on :)

Next we have a post from a blog that writes posts on the all time classic combination of subjects, cats and maths – Catsynth.com. The post is about how to calculate (that is the number of prime numbers below an integer x) without having to calculate all of the primes up to x. I wonder how the various CAS systems calculate this function? Anyone care to enlighten me?

Finally, in another bending of the rules, I’d like to present Five Open Problems Regarding Convex Polytopes from Gil Kalai’s blog, Combinatorics and more. He didn’t submit this post himself but it comes highly recommended and so I hope he will not mind having it included here.

And…that’s it for this 33rd edition of the carnival. Thank you to everyone who submitted something – without you the carnival would be..well..just me posting a load of links! Finally, would someone please volunteer to host the 34th edition of the carnival? I think it really is a lovely tradition that has been kept going by maths bloggers for almost 18 months now, which is like an eternity in internet years and it would be a shame to see it go. I think that it’s a great way of finding new math blogs and also of generating a sense of community in the maths blogsphere.

Enjoy!

Update: As it says in the comments, the next Carnival will be hosted over at 360 on May 30th so please head over there and submit a post. Making a submission is as easy as saying “Hi, what about this one…< insert link here>” 9 times out of 10 your post will be accepted so its an easy way to promote your blog.

Welcome everyone to this, the 25th edition of the Carnival of Mathematics. The 25th anniversary of many things is usually considered to be a little bit special and is often marked by a ‘Silver’ celebration of some sort. For example, in 1977, Queen Elizabeth II of the United Kingdom celebrated her 25th Year on the throne with a Silver Jubilee celebration and the British Royal Mail commemorated the event by releasing the postage stamps below

Technically speaking of course I should have waited until the Carnival had been around for 25 years rather than 25 posts but I thought I would exercise a little poetic license here – I hope the host of the real Silver Jubilee Edition in 24 years time will accept my heartfelt apologies.

So what else is interesting about the number 25? Obviously it is a square number but did you know that it is also the smallest square that can be expressed as the sum of two squares . It is also a Cullen Number and is the atomic number of the element Manganese. Twenty-Five is also the name of a card game that is sometimes referred to as the national card game of Ireland.

Enough of the random meanderings and on with the show…

The first two submissions come from Mathmom over at Ramblings of a Math Mom who asks the question “Should gifted math students tutor others?” This is something I had personal experience of when I was at school (both good and bad) so I found her arguments interesting – feel free to head over there and add to the discussion, I am sure you will be made very welcome. Her second submission concerns probability-fallacies.

Next up we have Arvind Narayanan from the randomwalker’s journal (we have no connection other than both of our blogs have cool names!) who explains the mathematics behind part of Arthur Benjamin’s act in “Mathemagics” explained. I love this sort of stuff and may well be trying it out on some of my (long suffering) friends.

Over at Reasonable Deviations (I always think of Richard Feynman when I see that blog name), Rod presents an interesting problem in thinking about permutations. It has already sparked an interesting discussion in his comments section so why not head over and see if you can have a go at solving it? I tried, and failed, but you might have more luck.

Maria Miller highlights a link to Classic Math Mistakes over at Homeschool Math Blog which includes posters for classic howlers like “3.1hrs = 3 hours 10 minutes.” My favorite is “Finishing an exam early and then sitting doing nothing” which is clearly an elementary error when every true math geek knows that the correct procedure is to clear your throat loudly as you stand up to leave. This ensures that all of your classmates know that you have finished early and so must be better at maths than they are – just make sure that they never find out that your actual score on the exam was only 6% as it ruins the illusion.

The next submission comes from the blog of Mr Kruopatwa’s AP Calculus AB (2007-2008) class and concerns a favourite topic of mine – namely the evaluation of integrals. One of his students, Mr Siwwy (AKA Chris), asks the question ‘How “approximate” can approximate can be’ and discusses some of the elementary methods of numerical integration. In an ideal world we would always be able to come up with exact answers for our definite integrals but, as we all know, the world is far from ideal and so we often must make do with numerical approximations. Chris’ post discusses how you might start to go about making such approximations. I had not discovered this blog before now and, if all of the posts are going to be this good, then I look forward to reading more.

Over at Goods and Chattels, Amanda has been reminiscing about one of the problems from her student days in An interesting mathematics puzzle. Some maths problems seem trivial when you first read them and so you mutter “All too easy!” as you start working on them, expecting it to all be over after a few minutes. Several hours (and pieces of paper) later you give up in frustration, try to forget about it and get on with your life…but then another idea strikes you….another way of attacking it….this one might just work you know…just one more go….and it has you again. This is one of those problems. Have fun – but no peeking at the solution!

Sol’s Fun Math Blog has only been around for four months and yet it is one of the most read in the blath-sphere. Building up a Technorati rating of 88 in such a short amount of time says it all really – Sol writes stuff that the rest of us like to read and link to. His submission, “five constants tie together multiple branches of mathematics”, discusses some of the mathematics behind the equation that Feynman once called “The most remarkable formula in math”. I remember the first time I discovered this equation – my response was pretty similar to this one (don’t click if swearing offends you).

Denise discusses a quotation from Ralph P. Boas about what it takes to learn math over at her blog, Let’s Play Math. The phenomenon mentioned is something that I am sure we are all familiar with from our student (and teaching) days and her article is well worth a read. Any blog article that mentions a paper with the title “A Contribution to the Mathematical Theory of Big Game Hunting” is just begging to be read in my opinion.

What sort of calculations can do perform using nothing but your fingers and thumbs? Until I read Heathers’ article – Three finger tricks for multiplying – the best I could do was count to ten on them but now they quite a bit more versatile Head over to 360 if you want to upgrade your digits.

And now for something completely different…Rick from Big Ideas submitted an article called
Mathematical Beauty and the K₄ Crystal. Check out that gorgeous looking bit of perl – If only I could write stuff like that :)

Finally, we have a last minute submission from Brent, the author of The Math Less Traveled, who has written the third installment of his “Recounting the Rationals” series.

And – with that – I’m done. I hope you have enjoyed reading this carnival as much as I enjoyed writing it. Thanks to everybody who submitted articles – I loved reading through them all. The next carnival is over at 360 so start thinking about what your submissions might be,

Mike