The trick for catching a liar in the preceding post is very easy to program (I will say a bit more about implementation below). Back in the good old days when my Psion held my diary and much more, I wrote a little program which posed the questions to the viewer. It was coupled with a Shakespearian insult generator which I got from Lewis Nowitz. (This consisted of three lists of about 50 words each: the first two lists were of adjectives and the third of nouns. If the player tells a lie, then before revealing his or her choices, the computer denounces him or her as “You X Y Z”.) Sad to say, the Psion no longer works, and I have lost the insult generator. But this little program was very attractive to children; sad to say, it seemed to have the effect of encouraging lying!

This particular Hamming code (or, strictly, the mnemonic I gave for using it) has a connection with the game of Nim. In this game, there are a number of heaps of matches; players take it in turn to pick up any positive number of matches from one pile. The player who takes the last match loses. The winning positions with at most three heaps each containing at most seven matches are of two types:

• two heaps of the same size; and
• three heaps whose sizes are the numbers on a line of the Fano plane given in the preceding post.

Thus, memorising them saves some mental effort if you are playing Nim.

Now I turn to Hamming codes in greater generality.

We start with the binary case, where F is the two-element field. Then a code C is a k-dimensional subspace of the n-dimensional vector space Fⁿ, hopefully with the property that any two codewords are quite far apart.

A code can be specified in either of two ways:

• C is the image of an injective map E: F^k→Fⁿ, the encoding map. Now E takes k bits of the message which is to be sent, and encodes them as the n bits of a word of C. So the rate at which information is transmitted is only k/n of what it would be without encoding; this is the price we pay for error correction.
• C is the kernel of a surjective map D: Fⁿ→F^n-k. This is not quite the decoding map, but it tells us if the word has been transmitted without error: this is the case if it mapped to zero by D.

Now encoding is simple arithmetic over the binary field, and can be done by lightweight and low-powered hardware. This is a great advantage in interplanetary exploration, where the space probe has weight and power limitations.

Decoding is more complicated; the easiest to describe is syndrome decoding. The received word can be written as t+e, where t is the transmitted word (which we want to find) and e the unknown error that occurred during transmission. For each coset of C in Fⁿ, choose a word of smallest weight in the coset, called the “coset leader”. Our assumption that the number of errors is not too great says precisely that the error pattern e will be a coset leader. Now D maps the coset leaders bijectively to F^n-k, so by a look-up table or other means we can calculate e from D(t+e) = D(e) [remember that D(t) = 0, since t is a codeword], and hence find t.

The Hamming code of length 2^d−1 and dimension 2^d−d−1 has the property that the matrix D (whose size is (2^d−1)×d) has as its rows all the non-zero binary strings of length d. We can regard these as being the base 2 representations of the integers from 1 to 2^d−1. If we write the rows so that the i-th row is the base 2 representation of i, then the syndrome of a received word (the result of multiplying it by this matrix) is 0 if no error occurred, and the base 2 representation of i if one error occurred in the i-th position. This is exactly the principle we used in the trick. This shows, in a very constructive way, that the Hamming code will correct one error. It is a perfect code, for each value of d.

Hamming codes exist over other finite fields. We take the matrix D to have as its rows, not all the non-zero strings of length d over the field, but a selection of them such that no row is a scalar multiple of any other. This is most conveniently done by arranging that the first non-zero entry in any row is 1. The number of rows is (q^d−1)/(q−1).

Here is an example with d=3 and q=3. I have transposed the matrix, to save space, and written the entries of the field as 0,+,−.

If one error occurs in transmission, the syndrome will be the i-th row or its negative according as 1 was added to or subtracted from the ith entry.

There is another use we can make of this: we can derive a solution of the “weighing pennies” problem. In this problem, we are given a number of pennies, one of which is known to be either lighter or heavier than all the others (but we don’t know which), and have to identify the odd penny with a given number of weighings on a balance. The traditional puzzle has twelve pennies and allows three weighings.

The Hamming code above would solve the problem if only the weights of the pennies were elements of the integers mod 3 rather than real numbers. Let v be the vector whose i-th coordinate is +1 if the i-th penny is heavy, −1 if it is light, and 0 if it is normal. Multiplying this vector by the transpose of the above matrix outputs plus or minus the ith row of the transposed matrix if the ith penny is heavy or light respectively.

To get round this, we have to arrange that any row of the original matrix has equally many + and − entries. This is not possible as it stands, but might be if we delete a column consisting of non-zero entries, since then the weight of each row would become even. In the existing matrix, the numbers of +/− entries in the three rows are 9/0, 6/3 and 5/4 respectively. A little thought shows that if we change signs of the columns numbered 8, 9, 10, 11, these become 5/4, 4/5 and 5/4; so we have to delete the column (+,−,+)^T (the 12th) to obtain balanced rows. The resulting matrix gives the weighing scheme. For the i-th weighing, look at the ith row, and place the pennies corresponding to entries − in the left-hand pan and those corresponding to + in the right. So the weighings are:

• 8,9,10,11 v 5,6,7,12
• 8,9,10,12 v 2,3,4,11
• 4,7,9,12 v 1,3,6,10

The results of the weighings tell us the odd penny and whether it is light or heavy.

I will return to this, but I am going to take a longish detour first. If you want to go straight to the website and register your support, fine.

The person who has had the greatest impact on mathematical publishing in the recent past is Donald Knuth. Trained as a mathematician (he wrote his thesis on projective planes under Marshall Hall), he became a computer scientist, where he may be best known for his magnum opus, The Art of Computer Programming. Unsatisfied with the typesetting of the books, he took a short time out to create his own typesetting system. The result is TeX, freely available to the worldwide mathematical community and, after not much more than thirty years, universally used by mathematicians.

This was an extraordinary achievement. As well as the typesetting program TeX, Knuth produced a font design program METAFONT, and the Computer Modern font family, designed so that the mathematical characters and symbols would go harmoniously with the body text. Furthermore, in the interest of encouraging good documentation, he produced his “Web” system, with two accompanying programs Tangle and Weave (I do not remember now which was which), one of which turned the file into Pascal source code, and the other produced a human-readable version incorporating the comments. The TeX and METAFONT programs were each published as a book to illustrate the idea that even a very large program could be structured and documented using these tools.

The manual for the program, The TeXbook, evokes mixed reactions, but in my opinion it stands head and shoulders above any other software manual I know. It goes from the most elementary to the most arcane aspects of the program, and is written with style and charm. It includes self-reference: each chapter ends with a couple of apposite quotes, and one of these is taken from the book itself. In addition, the introduction promises that the book contains both jokes and lies. (Jokes and lies? In a software manual?) The very last exercise in the book reads

EXERCISE 27.5 Final exercise: Find all of the lies in this manual, and all of the jokes.

The first appendix is entitled “Answers to All the Exercises”. Turning to the appropriate place, we find

27.5. If this exercise isn’t just a joke, the title of this appendix is a lie.

The reason I have diverted to discuss this is hidden away at the start of Chapter 16, “Typing Math Formulas”:

Notice that all mathematical formulas are enclosed in special math brackets; we are using $ as the math bracket in this manual … because mathematics is supposedly expensive.

Indeed, in the days of hot metal typesetting, mathematical typesetting was a specialist task for which printers could charge premium rates. Perhaps Knuth’s greatest achievement was to give us the tool to do our own mathematical typesetting, with no charge at all!

TeX, in the particular format of LaTeX, eventually became the universal default for mathematical typesetting. Not only did mathematicians take to the convenience of the TeX conventions for mathematics in emailing mathematical content to one another, but even the largest publishers produced style files and accepted submissions in LaTeX (often exclusively so).

As well as not having to provide expert mathematical typesetting, publishers now (as a result of the internet) have less need to provide printing, warehousing, and distribution. However, their charges to the mathematical community for their services do not seem to have been reduced to reflect this, and their marketing practices aim at maximising their revenue at the expense of wide dissemination of information. It is because Elsevier are felt to be the worst in this respect that they have attracted the protest. (A Dutch colleague once told me, “All Dutch publishers are crooks”; since then the other large Dutch academic publisher, Kluwer, has become part of the Springer empire.)

The mathematical protest seems not to have created many ripples in the wider world yet, but the internet protest against ill-judged lawmaking on piracy has done so; the Wikipedia blackout affected many people. Nature last week devoted their first editorial to the affair. They are trying to position themselves as on the side of the angels. They write,

No one disagrees that a publisher of review articles deserves to charge for access to them. After all, the publisher’s staff have contributed value in various ways, identifying the author and the article’s aim, assessing and editing the draft, selecting peer reviewers, working with the author to build on their advice, developing illustrations, rendering the article into print and online forms, maintaining it online and including links, citation statistics and other enhancements.

I can hear the hollow laughter from mathematicians reading this. I consider my review articles to be probably my most important contributions to mathematics. No publisher has ever approached me for a review article; it has often been a hard task for me to convince the publisher that they want to publish it. Choice of referees is done by the (unpaid academic) editor, not the publisher’s staff; and the referees (who do the real work) are also unpaid academics. Again, the editor mediates between author and referees, and the publisher is presented with the finished product ready to be put on the website. The article usually does not appear on the website for some time because the journal has a backlog. And finally, I think anyone reading this will probably know my views about citation statistics and the damage they cause.

So to the boycott. It sounds fine in principle, but what will it achieve? Said otherwise, what event would be regarded as a successful conclusion? A few mathematicians are hardly likely to bring Reed Elsevier to their knees. Also it is not entirely clear what freedom authors have. The Elsevier journal Discrete Mathematics has published selected papers from the British Combinatorial Conferences for many years. Such publication is valuable to young authors, but it does require a lot of work by the guest editors and quite a bit of goodwill on both sides. Moreover, there is no viable alternative place for such publication. If the protest is directed against the bundling of journals, the publisher can easily offer individual journals but increase the price to keep up their profits; the impact factors of some journals might take a hit, but the popular ones will be unaffected.

Publishers say, “Publish with us; look at our impact factor.” Mathematicians say, “I don’t care about impact factors,” but increasingly the bureaucrats say, “Oh yes you do.” We were recently required to produce a list of “aspirational journals” to which we would submit our papers. Our research director, quite correctly (in my view) wanting to deal with this quickly so as not to eat into his research time, took the Australian Research Council listings (which the Australians have now disowned, by the way), and invited us to suggest additions to the A* category to use as the basis of our list. People of the stature of Tim Gowers and Terry Tao, and people of my age, can tell the bureaucrats where to put these lists; but not all our colleagues have this option.

The Nature article recognises two viable modes of running journals: subscription, or author-pays open access. Pressure from national bodies in some areas is likely to push us towards the second. This will produce a two-tier academic system, where academics at rich universities, or those with grants including money for page charges, will be able to publish, and others will not. However, there are two further modes: free and unrefereed repositories like the arXiv or personal web pages, and free refereed journals run by volunteers. The first of these two is officially supported by the research councils and some universities in the UK. (Our institutional publications list takes a feed from the arXiv.) When I raised these issues in the very early days of this blog, Laci Babai eloquently and passionately defended the last solution, journals run by volunteers; his own journal, Theory of Computing, is a fine example.

The French essayist Michel de Montaigne said,

If, like the truth, falsehood had only one face, we should know better where we are, for we should then take the opposite of what a liar said to be the truth. But the opposite of the truth has a hundred thousand shapes and a limitless field.

In similar vein, the philosopher Anthony Kenny said in his doctoral thesis,

. . . all worthwhile philosophical statements express an insight; and the opposite of an insight is not a contradictory sentence, but a muddle . . .

Nevertheless, I will discuss a very artificial situation, where there is a series of yes-no questions, and the respondent is permitted to lie once in answering the questions. The purpose of this is to introduce coding theory, as I will discuss later. This is prompted by a post about the Hamming code of length 7 by John Bamberg on SymOmega recently. I am discussing the same code, and the same use of it. Unable to shed my teacher’s persona, I will attempt to pull back the screen and show what lies behind.

Unlike John’s version of the trick, mine does without the cards; you have to remember the questions, and you have to do a small amount of brainwork to find the lie. When I do this in public, I usually get it right, but sometimes fumble it.

The game works like this. Instruct your volunteer to think of a whole number between 0 and 15 (inclusive), and then to answer a few questions about it. He or she is permitted to lie to at most one question, but this is not compulsory; truth-telling is allowed. Here are the questions.

1. Is the number 8 or greater?
2. Is it in the set {4,5,6,7,12,13,14,15}?
3. Is it in the set {2,3,6,7,10,11,14,15}?
4. Is it odd?
5. Is it in the set {1,2,4,7,9,10,12,15}?
6. Is it in the set {1,2,5,6,8,11,12,15}?
7. Is it in the set {1,3,4,6,8,10,13,15}?

At the end, you announce both the number thought of, and the question lied to (if any).

In order to do the trick, you have to remember the following simple diagram, which is the Fano plane with a certain natural labelling. (Put the first three powers of 2, namely 1, 2, 4, at the vertices of the big triangle, and then label the third point of each line with the sum of the two points already labelled.)

Now here are the decoding rules. First we identify the lie. Record the answers to the questions in order as 1 for “yes” and 0 for “no”, obtaining a binary string of length 7. The weight of this string is the number of ones it contains.

• If the weight is 0, no lie was told.
• If the weight is 1, the lie is in the position of the 1 in the string.
• If the weight is 2, then the positions of the two 1s lie on a unique line; the third point on the line is the lie.
• If the weight is 3, look at the three positions of the three 1s. If they form a line, then no lie was told. If not, then the complementary set of positions of the four 0s contains exactly one line; the point not on this line gives the lie.
• If the weight is 4 or more, then apply the same rules as above to the positions of the zeros.

Having found the lie, you can now correct it; the first four digits of the corrected string give the base 2 representation of the number thought of.

For example, if the answers given yielded the string 0111000, you conclude that the answer to the fifth question was a lie, and the number thought of was 7.

Why does it work?

Let us get the easy part out of the way first. If we knew the correct answers to the questions, we can recover the number thought of. This is simply a matter of looking at the first four questions and noting that they ask for the four digits in the base 2 representation of the number. In coding theory terms, these are the “information digits”, encoding the information we are trying to transmit. The remaining questions yield “check digits”, enabling errors to be spotted and fixed.

In fact, if you examine the set H of 16 strings produced by correct answers to the questions with each possible input, you will find two remarkable things:

• The set is closed under addition (mod 2). If you look further, you find that, if we represent each number from 0 to 15 in base 2, and regard the resulting string of length 4 as a binary vector in F⁴, where F is the two-element field, then each question is a linear functional on this 4-dimensional space (simply observe that the set of “no” answers forms a subspace), and so the entire procedure gives a linear map from F⁴ to F⁷, so its image is a subspace.
• The set contains the all-0 and all-1 vectors, the seven vectors whose supports are the lines of the Fano plane, and the seven vectors whose supports are the complements of lines.

It follows from these properties that any two elements of H differ in at least three positions. (The number of positions in which v and w differ is equal to the number of ones in v+w; if v and w belong to H, then so does v+w, so if non-zero it has at least three 1s.) So if we take an element of H and change a single coordinate (corresponding to telling a lie to one question), the result is still closer to the starting element than to any other. (If any two villages are at least 3km apart, and I walk 1km from one village, I am still closer to that village than to any other.)

So in principle, the decoding is possible; all I have to do is to run through the 16 elements of H and find which one differs in at most one position from the sequence produced by the answers to the questions. The procedure I gave above is a relatively simple method of doing this.

We say that H is a 1-error-correcting code. It is the famous Hamming code of length 7 (which, arguably, was discovered in statistics by R. A. Fisher eight years before Hamming found it, but that is another story).

In fact, H has the additional property that any vector in F⁷ differs in at most one coordinate from an element of H. This is because H contains 16 vectors, and the 16×7=112 vectors obtained by changing one coordinate in a vector of H are all distinct; and 16+112 = 128 = 2⁷, so every vector is accounted for. We say that H is a perfect 1-error-correcting code.

The decoding method (the way to identify the lie) I gave above is known as syndrome decoding.

The underlying practical situation is that we are trying to send information through a noisy channel where some distortion will occur; during transmission of a binary string, it will occasionally happen that a 0 is changed into a 1 or vice versa. If we can assume that it is very unlikely that more than one of every seven digits transmitted will be received incorrectly, then the Hamming code allows us to recover from the errors in almost all cases.

an occasional series of interviews with female mathematical scientists. In this series we hope to showcase the achievements of inspirational women from all kinds of backgrounds and at all stages of their mathematical sciences careers. In addition, the Institute hopes that the Six questions with … series will help female mathematical scientists to share their experiences and will also encourage women to persist and excel at mathematical sciences research.

They make very interesting reading. Most encouragingly, they show (as I am not surprised to find) that female mathematicians are as diverse as mathematicians are, both in their approach to their subject and in their advice to young women starting their careers. Moreover, some value the opportunities for networking provided by organisations like European Women in Mathematics, whereas others feel that such a label is not in the long-term interest of female mathematicians.

Here is a selection of very brief excepts on “what keeps your interest fresh?” I don’t believe any mathematician could fail to be uplifted by reading these.

• “There are always new things to learn and discover.”
• “In every step of my mathematical life I discover beautiful things.”
• “I stay interested in maths by being interested in an wide range of real world problems.”
• “I participate in events and conferences on various mathematical themes that are not limited to my teaching area.”
• “The beauty of stochastic processes and its multiple calculi completely seduced me, and it still does.”
• “As I get older I see more and more questions left open by my work and the work of others.”
• “My research interests are fuelled by contacts and interactions with other statisticians in my field.”
• “I never felt my interest fading out – I always have some mathematics book on my bedside!”
• “If you solve one problem in mathematics it is like opening a door, behind which you find another unsolved problem (or more than one) already waiting for you.”
• “Maybe it is pertinent that so many of my friends are mathematicians … I talk maths to my friends. If I am stuck myself, my brain can be fired again by their enthusiasm in explaining their own work to me.”

Wordpress provide their bloggers with a raft of statistics about visitors to the site: how many, where they come from, where they go to. So, for example, there are on average nearly 400 times as many site visits as there are posts. This doesn’t mean that 400 people read each post, since several things distort the figures:

• Someone who comments on a post may come back to see replies to her comment.
• Most visitors come to the front page; they can read the ten most recent posts without another mouse click.
• There are about 50 “followers” who presumably get each post delivered and don’t need to visit the site at all. They all count in the statistics, even if they ignore the post.

Nevertheless, the figure gives some estimate of how many people read what I write. I do not know how many people read a mathematics paper I publish, on average, but it is probably not more than this.

It is also easy to find from the statistics page which are the most popular posts of all time. Of course, the front page wins by a large margin. But of individual posts, the top five (with numbers of views) are:

• Geomagic squares (1,965)
• Campaign for real mathematics (1,213)
• A fair coin (1,014)
• Lewis Carroll and algebra (1,002)
• The symmetric group, 2 (999)

The post on geomagic squares got a lot of hits because, about the time I posted this, there were articles in The Observer and New Scientist about geomagic squares, by two of my favourite mathematical journalists, Alex Bellos and Jacob Aron. People looking for pretty patterns would not have found them in my post, but I do think that there is a theory here waiting to be developed for arbitrary group actions, and I hope to return to this some day.

Other popular posts are the series on the symmetric group, and the post on using different fonts in LaTeX documents. I hope that these continue to attract readers because they are useful. (There hasn’t been another post on the symmetric group for a while. There will be more, but I want to talk next about representations, and this will require a bit of background.)

But something else has happened as the result of this blog. When I posted about a problem of Dennis Lin on what I called “hot” and “cold” matrices, Will Orrick and Gordon Royle made contributions which took our understanding of the problem further. I also got useful information from Christian Elsholtz about the number theory behind my brief recent post on the density of numbers of the form x²+3y².

From things like this, I learn a lot, and this helps to make the enterprise worthwhile from my point of view: it produces new mathematical insights.

So the rants keep me sane, the expositions are possibly useful to others, and the discussions of mathematical problems advance the subject. Good enough reason to continue? I think so.

I had a feeling of familiarity when I came across an account of the collaboration between Bob Dylan and Jacques Levy which led to the writing of most of the songs on Dylan’s Desire album.

Here is Dylan’s account of the writing of the first song, “Isis”:

I had bits and pieces of some songs I was working on and I played them for him on the piano, and asked him if they meant anything to him, and he took it someplace else and then I took it someplace else, then he went further, then I went further and it wound up that we had this song.

And here is Levy’s account of the same incident:

He had the general feeling of the song but hadn’t got further … so now what? … So the two of us started working on that together. I started writing words, then he would say: `Well, no, how about this, how about that?’ – a totally co-operative venture. It was just extraordinary, the two of us started to get hot together. And we began to work on this thing and we just kept going with it, and we’d stop and we didn’t know where the story was going to go next … we were just having a great time and coming up with one verse after another … and we kept on going until five in the morning and we finished the song … It’s impossible to remember now who did what, it’s like we’d push each other in the sense that he’d have an idea then I’d have an idea until we’d finally got to a point where we both recognized what the right idea was and what the right words were and whether it came from him or me it doesn’t make a difference.

This, incidentally, is from All Across the Telegraph, a collection edited by Michael Gray and John Bauldie, and maybe the best book on Bob Dylan I know.

I won’t go into the debate here: see the Wikipedia statement, or comments on the blogs by Paul Goldberg and Tim Gowers.

Photos of the authors are by the redoubtable Marc Atkins.

As the name implies, it does two jobs: it has papers (which can be substantial and presumably pass through some kind of editorial process), and blog posts (which so far are typically reports of things which have appeared elsewhere: some politician makes a speech about maths or IT in schools, or someone compares maths teaching in the USA and Finland, or whatever).

The very first paper is a reprint of a short article by the eponymous Augustus De Morgan, who was also founder and first president of the London Mathematical Society, about teaching induction – this first appeared in the Penny Cyclopedia in 1838. Two further papers have appeared: Roger Howe on “Three pillars of first grade mathematics”, and David Tall on “Perceptions, operations and proof in undergraduate mathematics”. Neither have attracted any comments as yet.

Some of the posts are more controversial. One reports a study in the USA which shows that

Elementary- and middle-school teachers who help raise their students’ standardized-test scores seem to have a wide-ranging, lasting positive effect on those students’ lives beyond academics, including lower teenage-pregnancy rates and greater college matriculation and adult earnings.

It is not clear whether these teachers raise their students’ test scores by the excellence of their teaching, or by encouraging cheating or feeding the students the test questions in advance. But even this post has not attracted a reply.

This journal and blog is clearly intended to be a forum. I am sure that people who read my blog will have opinions on some of the topics discussed there. Why not take a look, and make a comment, make a posting, or even submit a paper?

If a cluttered desk is a sign of a cluttered mind, of what is an empty desk a sign?

Unfortunately the administration here, in the belief that they are wiser than Einstein, have decided to target two professors in the School of Mathematical Sciences who have cluttered minds.

One of these professors has been threatened with the cancellation of permission for a research visit. The other has been told that the Vice-Principal himself will come and supervise the tidying of his office. (This is the same Vice-Principal that, as I mentioned earlier, says with pride that he acts first and thinks later.)

It seems to me that there is something a bit amiss with the priorities here…