Wait, though, there’s a problem. As Relinde Jurrius pointed out on Twitter, the formulation used in the paper abstract was not quite the same.

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $N$ greater than $5$ is the sum of three primes. The present paper proves this conjecture.

The version Christian used makes the assertion using odd primes, whereas the paper abstract only claims “the sum of three primes”. The latter version includes $7$ because $7$ can be written as the sum of three primes, but not odd ones ($7 = 3+2+2$). Certainly, you can see both statements of the weak Goldbach conjecture used (for example, here’s the $\gt 5$ version and here’s the $\gt 7$ version). Are they equivalent?

Now, clearly if you can prove “every odd integer greater than $7$ is the sum of three odd primes” and point out that “$7=3+2+2$”, then you have “every odd integer greater than $5$ is the sum of three primes”. That, however, isn’t what the abstract claims. The question is, does the implication go the other way?

There are some cases where an odd number can be expressed as three primes including two $2$s (e.g. $15=11+2+2$ or $104733=104729+2+2$), but we can write these as three odd primes without the use of a $2$ (e.g. $15=3+5+7$ or $104733=104723+3+7$). Can we always do this?

We had a useful conversation about this on Google+, and Evelyn Lamb mentioned this on her AMS blog. I also emailed Harald Helfgott and he was kind enough to take the time to reply and post a comment on the Google+ feed.

“People”, Harald implores, “please don’t worry”. It turns out we’d missed him addressing this in the original paper. He acknowledges, on page 114, that “some prefer to state the ternary Goldbach conjecture as follows: every odd number $\ge 9$ is the sum of three odd primes”, and takes account of this when making his conclusion. In doing so, he says in the comment on Google+, “I prove both versions of the conjecture, at the cost of about three lines of extra work”.

In his email he explains that indeed only one version implies the other, so they are not quite equivalent, but that this difference doesn’t matter in practice since this proof, and all previous work on the matter, works equally well for both versions of the conjecture.

So be happy people, both slightly different versions are proven by Helfgott’s work, and we needn’t publish the first ever Aperiodical correction!

I heard about Ox Blocks by following along with Maths Jam night remotely via the Twitter hashtag #MathsJam. Alison Kiddle (@ajk_44) tweeted that the Cambridge Maths Jam had been playing back in March.

The game is played as follows. One player plays as ‘O’ (noughts) and the other ‘X’ (crosses). Each turn, a player rolls a cube, or block, which has two faces marked ‘O’, two ‘X’ and two blank. Rolling a ‘O’ or a ‘X’, the player must place the block on the board. Clearly if the roll is in favour of the opponent it must be placed in a way that isn’t useful, if possible. Rolling a blank means the player must remove a block from the board. Play continues with the usual rule of three in a row winning the game.

Standard Noughts and Crosses (or Tic Tac Toe) is a solved game. As described by Alison’s NRICH, this means “if you play correctly you never lose and if your opponent plays correctly you cannot win”. So the introduction of a probabilistic element has the potential to make an uninteresting game slightly interesting.

My first thought was that it must be possible to play with pencil, paper and a normal die. Then I saw that the blanks are not just blank faces, but are hollow, and so supposed that the uneven weight distribution would affect the odds. Specifically, I imagined that the hollow sides would be less likely to roll to the bottom (and therefore the top), thus reducing the odds of a game going on forever. Thankfully, Ox Blocks is readily available via a popular auction site, and I was able to pick up a 1970 copy for £3.80 including delivery.

I took Ox Blocks to the Nottingham Trent University Maths Arcade. This is a lunchtime meeting themed around strategy games for students and staff, well explained elsewhere. Following the example of Noel-Ann Bradshaw at Greenwich, I set up a Maths Arcade at Nottingham Trent after applying for a grant from the IMA with student Kingsley Webster (look out for a piece in the next issue of Mathematics Today!). We have met six times in six weeks towards the end of this academic year to play a variety of games. Most of the games are pure strategy, for good reason, but I thought a bit of probability might be fun one week.

Anyway, my thoughts returned to what the odds are of being able to remove a block. So, in an attempt to settle it, I rolled one block 501 times. The results were as follows:

• ‘O’s: 161
• ‘X’s: 159
• blanks: 181

Surprisingly, this seems fairly even. Let’s make that a little more rigorous. Since I am interested in whether the block comes up blank or not, I can use a two-tailed binomial test with a null hypothesis that the block is a fair die. R gave me the following:

> binom.test(181,501,1/3,alternative="two.sided")

	Exact binomial test

data:  181 and 501 
number of successes = 181, number of trials = 501, p-value = 0.1848
alternative hypothesis: true probability of success is not equal to 0.3333333 
95 percent confidence interval:
 0.3191425 0.4050602 
sample estimates:
probability of success 
             0.3612774

This tests whether 181 blanks from 501 rolls is consistent with the probability of a blank being $\frac{1}{3}$. As the p-value is a giant $0.1848$, we don’t have evidence from my block rolling to reject the null hypothesis that blanks come up one third of the time. So, according to my experiment, there’s no reason not to play home grown Ox Blocks with pencil, paper and a normal die.

Of course, this doesn’t quite tell me that the blocks roll perfectly evenly, or if they do it doesn’t explain why the uneven shape doesn’t affect the roll. I suppose the unevenness might be so slight as to not matter, but then why not just make standard dice with blank faces? Any thoughts on the mechanics of the situation would be welcome.

From its first post (7 Feb 2008) to the most recent (2 April 2013), Travels in a Mathematical World was hosted by Blogger. This was fine, but last year I joined with Katie and Christian to form The Aperiodical and since then it has been an ambition to move the blog over here, for coherence and also because Christian’s custom WordPress setup here is far more capable than the Blogger system.

Anyway, after a lot of procrastination on the topic I have finally sorted out doing this. (Christian helped with this but the delay was mine.) There is a WordPress plugin that will import a Blogger blog. Christian set me up a throwaway WordPress site to have a dummy run and after that went well, and a couple of minor tweaks, we decided to push the button.

This morning the plugin imported 357 posts, comments and images and correct dates and all. So I think we call that a success. You can read the archive of posts in the Travels in a Mathematical World column. This is the first post posted here but not posted there. Brave new world.

I wrote a python script that fetched all the URLs and titles for posts in the ‘Travels in a Mathematical World’ category here. I did the same on Blogger previously, and I will use the two lists when I have a moment to program redirects from the old posts to the new, so that the old URLs still work (at the moment the blog is still mirrored in the old location), including the RSS and atom feeds. This isn’t completely necessary to move here but I am aware of a lot of legacy URLs sitting around the place, not least my database of automated “One from the archive” tweets.

So the moral of the story is: it was easy to move, and we’re on top of the process now so it will be even smoother if there is a next time. The importer deals with other blogging platforms than just Blogger. If you’d like to join us with a column at The Aperiodical and bring your old blog with you, get in touch.

All Squared (RSS, iTunes)

My Aperiodical co-conspirators Katie Steckles and Christian Perfect started All Squared, a maths magazine podcast, in February. The description for the first episode (or “number”, as Katie and Christian have it) overtly points out the “unusual paucity of maths podcasts at the moment” and promises “a half-hour podcast featuring maths, guests, puzzles and links from the internet”. The name is designed to be recognisable to mathematicians, who might find themselves reporting that an expression is “all squared”. As someone who named a podcast as overtly as it is possible to be, “Math/Maths”, this obfuscation amuses me. The three episodes so far have been enjoyable with a guest and main topic in each. As far as I’m concerned, this is far more the Aperiodical podcast that should exist than is The Aperiodcast with that third guy.

TES Maths Podcast (iTunes)

This one started just before Samuel Hansen and I went on our hiatus, but if you enjoyed the teaching aspects of what we did you can get a lot more on the theme from Craig Barton and his guests on the TES Maths Podcast. Craig promises “to share the latest news, resources and ideas that are relevant to secondary/high-school maths teachers and general number enthusiasts”.

Wrong, but Useful (RSS)

Wrong, but Useful is a new podcast featuring “a mathematical conversation” between Colin Beveridge and Dave Gale that sets out its stall as a response to the lack of Math/Maths episodes. The title is another nod to the mathematically minded without being overt, referring to a quote from George Box and Norman Draper who wrote “essentially, all models are wrong, but some are useful” (Empirical Model-Building and Response Surfaces, 1987). Episode 1 sees Colin and Dave finding their feet in a rambling, wide-ranging mathematically-themed discussion. There were a couple of awkward moments that gave me Math/Maths early episode flashbacks but I’m looking forward to Colin and Dave getting into the swing for the next episode.

Happy listening!

The number pi, written using the symbol π, is a mathematical constant that is the ratio of a circle’s circumference to its diameter, and has been claimed since antiquity to be an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, and that therefore its decimal expansion never ends or settles into a permanent repeating pattern. Here a proof is given that π can indeed be expressed as a ratio of two integers, 4/17, a fact that has unbelievably been overlooked until now. Moreover, this proof is understandable to anyone with a basic knowledge of algebra and calculus and arises from simply considering a standard integral at two values of x, x=1/4 and x=1. Of course I doubted the result at first, given that it has been overlooked for so many years, but I have checked the proof and verified it to be correct. This is a crucial and important revelation that will significantly alter all of mathematics.

Towards the end of the conversation, they speak a little about some of the ambiguity in spoken mathematics and how this might affect blind mathematicians. Particularly, they speak about timing and ambiguity, and Christian gives the example (where the comma is a pause): ‘x plus, y squared’ and ‘x plus y, squared’. The placement of the pause changes the meaning of the formula substantially.

A few minutes ago the conference ‘Supporting Disabled Students in STEM‘ began at the Royal Society. This is the spring conference of the STEM Disability Committee (an alphabet soup collaboration between IOP, RAEng, RSC, SoB, CaSE and RS). I am not attending this, though I was asked a couple of weeks ago for two examples for a session at this.

For the first example, I was asked for a formula that would be a bit difficult to read aloud. The aim is not for an unrealistic expression or to trick the audience, but just for something that might cause ambiguity and mean that the reader would stop and think. Here is the equation (apologies for the use of images throughout):

The exercise, later today, will use two volunteers. One reads the formula aloud and the other, with back turned to the screen, will write it down. Try it now with the person on the next desk or in the next office. I’ll wait.

This is the Taylor expansion of ln(x) around x=2. It is an example from a problem sheet in my first year methods course a few weeks ago, so it isn’t unrealistic. I chose it because there are two particular points of possible ambiguity. The first is in “minus one to the power i plus one”. The second is to have the top be “all over n times two to the n”. Is the “one” included in the power? Is the “two to the n” included in the bottom half of the fraction? There are also some conventions for non-mathematicians in the room to take note of, such as the pronunciation of “ln” and the way we read out “sum over i from one to infinity”. All of this means that to be read well requires a thoughtful, mathematically-trained reader.

If the person reading the equation in the session today does it badly then the other volunteer will write the equation wrong and the point will be made. If they do it well, it will be interesting to hear them speak to the group about what choices they made when reading the equation to reduce the ambiguity. 

The second example was a piece of mathematics that would need to be broken over two lines when enlarged, and that would be difficult to do so. Here the grey is supposed to indicate the edge of the page. Have a think about it: how would you break this equation to fit on the page (you can’t make it smaller!)?

Again this is an example from class, a partial fractions question. A naive solution might be to simply cut the equation at the page edge. You can hopefully see in the following image that this is unacceptable, in particular because the 2 on the next line looks like -2.

A more intelligent cut might be to take whole terms onto the next line. However, this may cause confusion because it looks like the (x+3) is part of a separate fraction. Is it one fraction plus another? One fraction multiplied by another? This perhaps isn’t unusable, provided you explain to the reader what you have done, but you are presenting the reader (your student?) with non-standard and, strictly speaking, incorrect mathematical notation. If part of being a mathematician is learning to speak and write mathematics so that other mathematicians can understand you, you are doing damage here.

Here I try to keep the two halves of the fraction together by including in-line cuts. This is still weird, particularly the bottom half, but I could imagine writing something like this by hand (perhaps with a multiplication symbol on the bottom cut) so maybe it’s okay.

Finally, here is what I think I would do. The result is mathematically correct and using fairly standard notation (we wouldn’t ordinarily use multiplication symbols in g(x) but this is unusual rather than strictly wrong).

This is hopefully a good solution, but there are still two problems. Interpreting the two functions back into the partial fractions form is an extra cognitive load for this student (not related to the intended learning outcomes of the question), and producing this formatting is a giant amount of extra work compared with the original code. Here is the original LaTeX:

Find the following integral by first resolving the integrand into its partial fractions.
    $$ \int \frac{x^3+x^2+x+2}{(x+1)(x-2)(x+3)} \, dx $$

And here is the adapted LaTeX code that produced the last version.

Find the following integral by first resolving the integrand into its partial fractions.
    $$ \int \frac{f(x)}{g(x)} \, dx \text{,}$$
    where
    \begin{align*}
      f(x)= \, &x^3+x^2\\
      &+x+2
    \end{align*}
    and
    \begin{align*}
      &g(x)=(x+1)\times\\
      &(x-2)\times(x+3)
    \end{align*}

The person who adapts this page must be properly trained in mathematics, so they don’t introduce errors into the notation and adapt it sensibly, and in this case they must know LaTeX to be able to write the code. This is an unusual set of skills for a generalist disability support professional, and the maths department might be unable to commit personnel to do this. Sometimes the problems are that mathematics notation is difficult to adapt, and sometimes they are to do with the practicalities of who is able to do the work.

I will leave you with this quote from Christian in the podcast.

It is very rare for someone who is blind to get through to becoming a research mathematician, isn’t it? … So is that because the culture isn’t accommodating or because maths is a thing that really is much easier to understand visually?

Something for you to think about. Listen to the podcast to hear what Edmund thinks.

If you are interested in these issues, Emma Cliffe is running a workshop ‘Mathematical Study Without Pen and Paper: Experiences, Impacts and Options‘ for the Higher Education Academy on 20th March 2013 at Manchester Metropolitan University. Attendance is highly recommended.

1. It is an open question, as far as I know, whether the second episode will be number two or number four.

This journal is an online-only, diamond open-access*, peer-reviewed journal with an emphasis on “the aesthetic, cultural, historical, literary, pedagogical, philosophical, psychological, and sociological aspects as we look at mathematics as a human endeavor”. They publish “articles that focus mainly on the doing of mathematics, the teaching of mathematics, and the living of mathematics”. (Quotes from the Journal’s About page.)

My article’s synopsis is:

A reflective account is presented of experiences which took place alongside a research project and caused a change in approach to be more sceptical about implementation of learning technology. A critical evaluation is given of a previous e-assessment research project, undertaken from a position of naive enthusiasm for learning technology. Experiences of teaching classes and designing assessment tasks lead to doubts regarding the extent to which the previous project encouraged deep learning and contributed to graduate skills development. Investigations of the benefits of another technology—in-class response systems—lead to revelations about learning technology: its enthusiastic introduction in isolation cannot be expected to produce educational benefit; instead it must address some pedagogic need and should be evaluated against this. Overall, these experiences contribute to a shift away from a naive enthusiasm to an approach based on careful consideration of educational need before technology implementation.

Download the PDF of this article here.

P.S. Sorry the blog has become rather infrequent and quite education-focused. I am currently splitting my time between teaching and writing my thesis, so I have little time for anything else. My employment contract is only to teach until May and my thesis is due in July.

* Diamond open access means that you don’t have to pay to read it and I haven’t had to pay to publish it. It’s a kind of magic.

I have been thinking a lot about assessment methods and their advantages and limitations for a chapter I am writing for my PhD thesis. For example, I could set a paper test and mark it by hand, as indeed I set one last week and will be marking it when I finish this post, and this allows me to give a personal touch and assess students’ written work but one downside is that I can’t return marks to students very quickly. I could return marks immediately if I used automated assessment, but then setting the assessment would be more difficult and I may be limited in the range of what I could assess. And so on.

I have been trying to classify these advantages and their paired limitations. My thinking is that by viewing different assessment methods as balanced sets of advantages and limitations we can justify different approaches in different circumstances and, particularly for my PhD, explore the advantage/limitation space for any untapped opportunities, which I won’t go into now (but ask me).

Here is my current list of potential advantages that assessment could access. These advantages are each something that I think that some assessment method can offer. My question is: what am I missing? I would be pleased to receive your thoughts on this in the comments.

• Immediate feedback. This is linked to learning from mistakes, confidence and motivation. It can also prioritise procedural learning over conceptual understanding. 
• Detailed, personalised feedback. Though there is much disagreement in what I have read whether a human, who can respond to individual student work, or a computer, which will tirelessly generate worked examples using the context of the question asked, will in practice provide this.
• Individualised assessment. This is achieved through randomisation of questions and is linked to repeated practice, deterring plagiarism, allowing students to discuss the method of a piece of work without the risk of copying or collusion.
• Assessing across the whole syllabus. For example, computers can’t mark every topic.
• Testing application of technique. Whether students can apply some procedure.
• Assessing deep or conceptual learning. For example, open-ended or project work may require a detailed manual review to mark. This is linked to graduate skills development, etc.
• Easy to write new questions. Assume it is easy for a lecturer to write questions that students can answer (it isn’t, but we’re talking principle here). Difficulty is introduced by having to second guess an automated system, or having to second guess students to program misconceptions. 
• Quick to set assessments. Assume that writing a test manually takes time. By quickly, I really mean choosing items from a question bank.
• Quick to mark assessments. Assume that marking by hand is not quick, perhaps unless the assessment is very short and student answer format very prescribed, in which case the assessment is limited. This is perhaps linked to problems of consistency and fairness when using multiple markers.
• Easy to monitor students. Clearly marking individual work from every student by hand will give great insight, but here I refer to the ability to gain a snapshot of how individuals and the cohort are doing as a whole with a concept, perhaps very soon after a lecture that introduced that concept has taken place. 
• Perception of anonymity. I’ve read that some students are happier to make their mistakes if only a computer knows. This can reduce stress.
• Testing mathematical writing. Clearly requires hand-written work.
• Testing computer skills. Clearly requires use of a computer.

For example, then it might be possible to offer ‘Easy to write new questions’, ‘Assessing deep or conceptual learning’ and ‘Testing mathematical writing’ through a traditional paper-based, hand-marked assessment, but this would preclude, for example, ‘Immediate feedback’.

Similarly, a multiple-choice question bank might offer ‘Quick to set assessments’ and ‘Quick to mark assessments’ at the expense of ‘Assessing across the whole syllabus’ and ‘Assessing deep or conceptual learning’.

And so on. I have loads of these for different assessment types.

My question really is, is there anything missing from my list that might be delivered by an assessment method?

There is something you can do to fill this mathematical podcasting gap, however. Samuel is trying to raise money through a Kickstarter to allow him upgrade his equipment and improve the quality, to pay for the travel to conduct face-to-face interviews and to make this his full-time job so he can concentrate on a regular release schedule, for his work in maths (math) and science communication over at ACMEScience.com.

At Kickstarter, Samuel says:

ACMEScience.com has spent the last four years trying to do something that very few others have ever attempted, create entertaining, insightful, and interesting content about mathematics and science. Started by Samuel Hansen in the beginning of 2009, ACMEScience has produced a pop-culture joke filled mathematical panel show, Combinations and Permutations, a show that interviews everyone from the CEO of a stats driven dating site to a stand up mathematician to Neil deGrasse Tyson, Strongly Connected Components, a show that tells the stories of the fights that behind DNA, dinosaurs, and the shape of the universe, Science Sparring Society, a video interview show that has featured predatory bacteria and crowdsourced questions, ACMEScience News Now, and a series of hour long journeys into the world of competitive AI checkers computers and stories of the most interesting 20th C mathematician and much more, Relatively Prime.

You may remember that Samuel raised money through a Kickstarter before, for the extremely well-received documentary series Relatively Prime. So you might judge this as evidence that he is capable of delivering this project. However, you may also remember that if he doesn’t raise the whole amount he needs then he gets nothing.

There are various pledge levels, with various rewards. Some of these are aimed at the individual who wants to own a piece of the project. Others are aimed at people who want to sponsor/advertise via the shows and get their message out there. Looking at the level of pledges so far, Samuel could really do with a few companies or individuals who want to get a message out to a mathematics or science audience coming forward and pledging some money. Relatively Prime was very well listened to, and you could get your message to a large, focused, engaged set of listeners.

There is not long to go (only four days at time of writing) and it doesn’t look good. So please pitch in and also tell everyone you know via your own blog/podcast/social networks/etc. so that others will support his effort.

Here is the video in which Samuel makes his case. It’s six minutes so at least watch that! The Kickstarter page is ACMEScience.com by Samuel Hansen. Donating is easy through Amazon payments.

A Celebration of Mind party is supposed to “celebrate the legacy of Martin Gardner on or around Sunday, October 21, 2012 through the enjoyment of [one or more of] Puzzles, Magic, Recreational Math, Lewis Carroll, Skepticism and Rationality”.

At Maths Jam I printed a bunch of flexagon material from the Flexagon Party page. I also had a plan: having finished two jobs in recent years with piles of business cards outstanding, I brought these to try some business card origami. In fact, we decided to make a business card Menger sponge. So we started folding.

Business card folding begins

Meanwhile, John Read had come equipped with some colourful designs to make hexahexaflexagons.

John Read’s first hexahexaflexagon of the evening. Designs from flexagon.net.

John Read’s second hexahexaflexagon. Designs from flexagon.net.

At the same time, Jon made a trihexagon.

Trying to make a triflexagon

Finally, after much business card folding,we had a Menger sponge* (*not a real one, it being a fractal after all!). Here it is, in Maths Jam-style, balanced on a pint of beer.

The completed business card Menger sponge

And here’s a shot through the Menger sponge, where a geometry puzzle is being attempted.

Through the completed business card Menger sponge, some geometry is happening

Here are a couple of the other puzzles that we tried, some from the #MathsJam tag on Twitter:

You have 100 coins, 10 of which are showing heads and 90 of which are showing tails (though these are indistinguishable by touch). Blindfolded, you must divide the coins into an even number of heads and tails. 

Which is bigger, 3^(21!) or 2^(31!)?

Then on Friday we made our way to Broadmarsh shopping centre for our afternoon at Nottingham’s STEM Pop Up Shop.

Nottingham STEM Pop Up Shop welcome notice

Here’s a picture of our stall, with Kathryn Taylor presiding, and in the foreground the posters about Martin Gardner, mathematical games and mathematicians that I had printed. Someone did ask me who Martin was and I explained a little; I think I also convinced him to come to Robin Wilson’s talk on Lewis Carroll next month in Derby.

Martin Gardner posters on our stall

Here’s the detail of our stall, which we called ‘Solving it like a mathematician’. You can get details of the set of puzzles on my website.

The STEM Pop Up Shop ‘Solving it like a mathematician’ stall

Looking around the shop, I requisitioned the Alan Turing postcards from the ‘My Favourite Scientist’ set for our stall!

Alan Turing postcards

Here’s a wider view of the stall, with Kathryn entertaining a customer.

Kathryn Taylor at our stall

And finally, here’s John Read bewitching a crowd with the loop on a chain trick.

John Read enthuses a crowd with a ring on a chain

Thank you for reading!

Thank you for visiting

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