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…

The talk was on a very natural question which, remarkably, has not been considered before:

Given a finite group G, what is the probability κ(G) that two (independent uniform) random elements of G are conjugate?

The talk was in three parts, of which the most substantial part concerned groups for which this probability is “large”. It is at most 1/2 for any non-trivial group, since given the first element, the size of its conjugacy class is not more than half of the group order. On the other hand, the main theorem of BBW asserts that, if κ(G) ≥ 1/4, then either G is the semi-direct product of an abelian group A of odd order with a group of order 2 inverting A, or G is one of finitely many exceptions (the largest of which has order 60). It follows that 1/4 is the largest limit point of values of κ(G).

More generally, if H is a finite group which is a Frobenius complement, then the number κ(H)−1/|H|² is a limit point of the values of κ(G) over Frobenius groups G having Frobenius complement H.

This provoked some discussion. Charles Leedham-Green remarked that these limit points are values of κ(G) (suitably defined) for profinite groups G, and wondered if perhaps every limit point was the value of κ for some profinite group. This is a question for which a proof seems out of reach with present knowledge; but perhaps a counterexample would be easier.

The second topic concerned groups for which the probability is “small”. In these cases it is better to consider the parameter |G|κ(G) instead. The value of this parameter is at least 1, with equality if and only if G is abelian. They show that, if G is non-abelian, then the value of the parameter is at least 7/4, with equality if and only if the centre of G has index 4.

This is reminiscent of a result for the probability that two random elements commute, which is 1 for abelian groups, and for non-abelian groups is at most 5/8, with equality if and only if the centre has index 4. It is not coincidence that something like this happens; John remarked that both |G|κ(G) and the probability that two elements commute are invariant under isoclinism.

[In any group G, the commutator map induces a function from (G/Z(G))×(G/Z(G)) to the derived group G'; two groups G and H are isoclinic if there are isomorphisms G/Z(G)→H/Z(H) and G'→H' which respect the commutator map.]

The final part concerned an intermediate class of groups, the symmetric groups. Here, John admitted that they had been scooped by a French team consisting of Flajolet, Fusy, Gourdon, Panario and Pouyanne. (This doesn’t contradict the statement that nobody had looked at the problem before, since for the symmetric group it is purely combinatorial: what is the probability that two random permutations have the same cycle structure?) The result is that κ(S_n) is asymptotically c/n², where (in a nice piece of self-reference) the constant c is the sum, over all n, of the values of κ(S_n). (The constant is about 4.26.) The proof by FFGPP, as you would expect, uses methods of analytic combinatorics, whereas BBW have a more elementary, though computational, proof. Incidentally, the value of n²κ(S_n) is maximised when n=13, the maximum being about 5.48.

Ian Wanless and I needed to know that the following result is true:

Let S be the set of positive integers n with the property that the prime-power factors of n are congruent to 0 or 1 (mod 3). Then, for any positive integer n, there is a member of S between n and n+o(n).

The proof went like this. A theorem of Euler asserts that a prime number p can be written in the form x²+3y² if and only if it is congruent to 0 or 1 (mod 3). Using this and a multiplicative identity asserting that the product of two numbers of the form x²+3y² again has this form, it follows that our set S consists of all integers expressible in this form. Now these representatations correspond to integer lattice points lying on a certain ellipse. Using a geometric argument it is easy to see that, if we take the ellipse x²+3y²=n and expand it by a small factor, we catch a new lattice point, and so find a new member of S.

The proof should have gone like this. The set S contains the set of squares, which clearly has the required density property.

But the nice thing is that I don’t have to feel stupid about this. I learned some number theory on the way, and had some good practice in doing estimates using Euclidean geometry.

Even better, the simple argument generalises and proves much more!