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Exam papers (2)

The first post showed this gem from a mock GCSE maths paper on the BBC website:

Just in case you think it was a typo, here is the corresponding extract from the solutions/marking scheme:

The problem, of course, is that the illustration shows a dodecahedron, which has 12 sides not 10.

So what exactly is the student supposed to do? Well, he could ignore the diagram and assume he is given a 10-sided die, which has an equal chance of coming up 1, 2, …, 9 or 10. If he did that then he would get full marks.

Or he could assume that the first line had two typos (“ten” and “10″) and work on the basis of 12. In that case, the answers should be (a) 1/12, (b) 1/4, (c) 5/12, all of which will be marked wrong.

So is the error the first line or the diagram? Well, it has to be the first line, because a dodecahedron is indeed equally likely to end up with any of its 12 sides uppermost. But there is no regular 10-sided solid, so it is hard to see how the question makes sense with 10. Offhand, I suspect that there is no 10-sided die (whatever the shape and size of its faces) which is “fair” (so that any side has an equal chance of landing uppermost). But proving it is probably too tough for an Oxford finals question.

But then the question does not say that the die is fair. The question has no answer if it is not fair. Well, that is not quite true. The Bayesian answer would be that since you have no prior information on which number is written on which side, it does not matter whether it is fair. Even if it is guaranteed always to show one particular side, the chance of that side being 3 (or any other number 1-10) is still 3/10. That, however, is way outside the GCSE syllabus.

Part of me wants to ask what kind of a moron could set a question like this, but that is not fair. Everyone makes mistakes. That is well-known, so the standard procedure for exams is to make a small number of competent people (normally permanent staff) take them. That way they should pick up the errors. Cambridge, for example, used to (and presumably still does) take its Tripos exams seriously. A small committee is responsible for each paper. Each person has to vet the questions carefully and the committee spends many meetings arguing about the precise wording of each question. So what happened here? Did the BBC fail to get the questions checked? Or, much more alarming, did the checkers (presumably secondary school teachers) fail to pick up the error?

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Exam papers (1)

The BBC website has a substantial section devoted to the three “Key Stages” (KS1, KS2, KS3) and GCSE (the UK exam aimed at 16 year-olds). It is part of its Learning: Schools beta site. Confusingly, there is a link at the bottom left entitled “The BBC is not responsible for the content of external sites. Read more.”. That suggests that maybe this is an external site. But no, if you click you get an elaborate description of “Our approach to external linking” which seems to say that whilst the BBC chooses with care which hyperlinks to include, it is not responsible for other people’s websites.

Duh! Maybe the lawyers demand this kind of drivel. Who knows, maybe people think they can sue the BBC because they do not like a site they clicked to from the BBC site. I am never quite sure whether I blame the lawyers for a type of over-caution that conveniently creates work for themselves, or whether I blame a minority of people for pursuing unjustified claims. I am inclined to the former. The fact is that it is impossible to find an appeal court judgment which upholds ridiculous claims, and extremely hard to find such judgments in the High Court (outside of the Family Division). So why do we all drown in legal drivel pointing out the obvious? Is it because the judges/magistrates at the bottom end are hopeless and no one wants the aggravation of constantly appealing them?

But returning to the site, Bitesize offers a more or less complete course up to GCSE complete with plenty of practice tests and even mock exams with realistic answer schemes. Admirable.

But I was a little disappointed to discover this in the Mock exam paper “Non-calculator – Foundation” for GCSE maths:

Does anything strike you as odd about this question?

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Solution to group puzzles

The first puzzle was here. You were asked to show that if group is defined as having a left identity e and every element g has a left inverse h with hg = e, then in fact the structure satisfies the normal definition of a group, so that e is also a right identity and the left inverses are also right inverses.

Take any element g of G. Let h be a left inverse, so that hg=e. Let k be a left inverse of h, so that kh=e. Then gh = e(gh) = (kh)(gh) = k(h(gh)) = k((hg)h) = k(eh) = kh = e. So any left inverse of g is also a right inverse.

But now ge = g(hg) = (gh)g = eg = g. Since g was an arbitrary element, the left identity e is also a right identity and we are done.

The second puzzle was to take the weaker axiom that G has one or more left identities, and that for any element g we can find h such that hg is a left identity.

It soon becomes clear that this is much more intractable to deal with. In fact there are innumerable counter-examples. For example:

Here e and e′ are both left identities, but there is no right identity. Every element is its own left inverse and hence also its own right inverse. But every element actually has two inverses. For example a2 = e, but b is also an inverse of a. We have ba = e and ab = e′. Note that ab is not equal to ba!

Checking the associative law holds is slightly laborious, but not hard.

If you managed all that, you did exceptionally well! Finding better ways of checking the associative law and explaining how one might find this counter-example (other than by computer search) is a topic for a future post.

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Groups – a couple of little puzzles

Maths is a curious subject, which is reflected in its practitioners. Even by academic standards, mathematicians tend to come across as distinctly odd. For most of the world, maths means a school exam which they did not enjoy. A modest proportion can probably do some basic mental arithmetic and occasionally find that useful, although most prefer not to delve to closely into exactly where their last pay cheque went. If they really need to add something up, they will use a calculator, or maybe a spreadsheet. They might remember doing some simple algebra at school, but unless they are puzzle nuts who use it to solve occasional “brainteasers”, they are probably glad to remember almost nothing about it. If pressed, they will assume that engineers and boffins will use maths to help create all the technology around us.

That indeed is correct. There are thousands or millions of people using maths in science and engineering for every person who is interested in creating it or studying it for its own sake (the proper definition of a mathematician). The relationship between maths and the real world is deeply mysterious, and I do not even have time to touch on it here. But in practice there is a drastic difference in outlook between the mathematician and the scientist or engineer. The latter simply regard maths as a tool; the former wants to understand why results are true and to find new results.

When I got to Trinity in 1968, there were two famous old mathematicians occasionally to be seen doddering around. One was Abram Besicovitch, who is best known for his comment that you judge a mathematician by the number of bad proofs he has produced. This sounds paradoxical, but Besicovitch was using “bad” in a rather technical sense. He did not mean bogus proofs, he meant proofs which are correct but far too long. The empirical fact is that the first proof of any significant result is almost always bad. Usually a significant result will have been around for quite a while before anyone finds a proof.

Of course, the snag is that until a proof is found, no one is quite sure whether the result is really true. After struggling with it for a year (or more usually much less) and failing to prove it, most people will begin to have the awful sinking feeling that maybe it is not true and give up. Often people will suspect it is half-true, in the sense that it needs a rather stronger hypothesis. Indeed that is usually the case. Indeed someone may find a “pathological” counter-example, meaning a counter-example which is weird, not the kind of thing anyone was envisaging when they looked at the result. That usually leads to a stronger hypothesis on the basis of which someone will manage to prove the result.

But many results are not like that. Fermat’s last theorem (there are no integer solutions to xn + yn = zn with n>2) was proved for vast numbers of special cases, but no counter-examples were found and the original conjecture remained irritatingly unproved 357 years after Pierre Fermat had stated it in 1637. Finally, Wiles and Taylor did manage to prove it, and a truly awful proof it is. No mathematician who is not a specialist in that area will find it at all easy to follow.

Another rather mysterious feature of maths is the interplay between geometry and more abstract branches of maths, such as algebra and analysis. The charm of geometry is that it easier to bring visualisation to bear on it. Mental pictures are much harder to come by if you are just manipulating symbols, and that means that it is much harder to think in a concentrated way about the problem with your eyes closed – usually considered the only way to solve tough math problems.

With that background, we turn to groups. In a sense the group is the simplest possible structure. Sets, of course, are simpler, but they do not have any inherent structure. Indeed “set theory” only really gets interesting when you allow infinite sets, and that rapidly becomes hard and counter-intuitive. So it is probably not a good place to start if you are trying to find your feet.

A group is a set of objects G with a binary operation. A binary operation is just something that takes two objects of G and gives an object of G. It is usual to represent the operation as a product, so that if the two objects are b and c, their product is written bc. Groups are partly an abstraction of ordinary integers with the operation +. But there is an important difference: the operation is not necessarily commutative. In other words, we do not require that bc = cb. The axioms are:

(1) the operation is associative, in other words for all g, h, k we have (gh)k = g(hk);

(2) there is an object 1 in G, such that:

(A) for any g in G we have 1g = g1 = g; and

(B) for any g in G we can find an object h in G with hg = gh = 1.

The object 1 is called the identity, and in (B) the object h is called the inverse of g, written g-1.

It is almost obvious that 1 is unique and that each object has a unique inverse. For suppose e is an identity. Then since eg = g for all g, we have in particular that e1 = 1. But since 1 is an identity, we also have that e1 = e. Hence e = 1. That establishes the uniqueness of the identity.

Suppose g has another inverse h. Then since g-1g = 1 and 1h = h, we have that h = (g-1g)h. But (g-1g)h = g-1(gh) = g-11 = g-1, so h = g-1. That establishes the uniqueness of the inverse.

Where G is finite, we can conveniently represent the group by a table, eg

We find the product of b and c by looking along the row labeled b and down the column labeled c. It is obvious that this group is commutative (the table is symmetrical about the main diagonal), 1 is the identity and every element is its own inverse. The only tricky part is showing that it is associative. There are better methods, but brute force works (there are only 43 = 64 cases to try).

Note also that we can regard it as “generated” by two elements a and b subject to:

ba = ab; and
a2 = b2 = 1.

From this point of view, c is just shorthand for ab. A more complicated example is:

This is not commutative. For example, ab = d, but ba = f. We can regard it as generated by a and d subject to:

da = ad2; and
a2 = d3 = 1.

In this case brute force is clearly an unappealing (but feasible) way to check associativity (63 = 216 cases to try).

Now a mathematician will immediately suspect that these axioms are somewhat redundant. In other words, we are probably assuming more than we need.

Suppose we replace (2) by:

(2′) there is an object e in G, such that:

(A) for any g in G we have eg = g; and

(B) for any g in G we can find an object h in G with hg = e.

So speaking loosely, we are saying that G has a “left identity” and each object in G has a “left inverse”. Is that enough? In other words, can we prove that (2) follows? Well that is your first puzzle. It is non-trivial. In other words, it is harder than anything you will get in today’s A-levels, and plenty of first year maths undergraduates would fail to get a correct proof in a reasonable time (say 15 minutes). But it is far from hard. The proof is just a few lines of manipulation similar to what we used above to show uniqueness.

Now suppose we replace (2) by:

(2′′) We call e a “left identity” if for any g in G we have eg = g.

(A) G has (at least one) left identity;

(B) for any g in G we can find h in G such that hg is a left identity.

Note that this is weaker than (2′) because the left inverses of different elements could be associated with different left identities.

But is it enough? In other words does (2) follow? That is your second puzzle. You have to produce either a proof that it does, or a counter-example – a particular example of G and an operation satisfying (1) and (2′′) but not (2).

This is significantly harder. It soon becomes clear that a proof requires more than a few lines of doodling. On the other hand, although it is easy to produce a small set G with a binary operation satisfying (2′′) but not (2), it is less clear how we find one which is also associative.

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More sham apologies

I first took an interest in this topic when banking chiefs started parading before the Treasury Select Committee three years ago. I returned to it when Richard Williamson apologised to the pope shortly afterwards.

Recently, claims Christina Patterson in today’s i, sham apologies have been coming thick and fast. Her main concern was l’affaire PIP.

Jean-Claude Mas was arrested yesterday. The police have 48 hours to decide whether to charge or release him. Actually, it is a little unfair to complain about the lack of apology at this point. He is presumably concerned to find out the prosecution case against him before making too many admissions of guilt.

With hindsight it may look crazy to have used a low grade of silicon for breast implants, but the question is what evidence was available to him at the time about the demerits of such an action. Certainly, there have been allegations that cast his behaviour in a poor light:

When I heard … that [he] … knew they hadn’t been authorised for medical treatment, and had ordered employees to hide them when safety inspectors visited the factory, and had carried on doing this for 13 years … [from Christina Patterson's article]

But one can imagine that, with the company being vilified around the world, a number of those involved may have an incentive to cast all the blame on Mas, whether or not he deserves it.

Ali Abdullah Saleh became president of North Yemen in 1978 and president of Yemen after the unification in 1990. He was badly injured in a rocket-propelled-grenade attack in June 2011 and was taken to a military hospital in Saudi Arabia where he had two operations. The vice-president was appointed acting-president.

Saleh returned to Yemen in late September and broadcast on 8 October that he would step down shortly. On 23 November he signed an agreement in Saudi Arabia to step down permanently. Elections are to be held next month. Last Sunday, he made a televised farewell address, and then flew to New York for further treatment, whilst the Yemeni parliament granted him immunity from prosecution. The point that annoyed Patterson was that in the speech he

asked the nation’s “pardon for any failure that occurred during my tenure.”

Formally, she is, of course, correct. If I apologise for “any mistakes I made”, or worse, “any mistakes I may have made”, or “any mistakes you think I have made”, then I am not actually admitting to any mistakes. To avoid the sham charge, an apology has to be absolutely clear and unconditional, and unqualified by any pleas in mitigation.

On the other hand, I find it hard to see that Saleh is a particularly egregious case. Certainly, the Arab dictators have not behaved well and have done little for their peoples, but that is not wholly their fault. To realise that your culture has serious shortcomings (such as grossly unequal treatment for women) and that, as leader, you need to institute major reforms requires someone of fairly heroic stature and considerable political skill. Changing practices hallowed by the centuries is far from easy.

Patterson’s third example was Diane Abbott who grudgingly apologised for “any offence caused” after a racist tweet about whites. But again, I find it hard to see Abbott as a bad case. Everyone took delight in her discomfiture because of her “hair-trigger sensitivity when white people make the same sorts of generalisations” about blacks (Rod Liddle’s comment in the Sunday Times), but her actual offence was fairly trivial and barely needed an apology.

I had more sympathy with her final example: Newt Gingrich’s apology when his second wife, Marianne (m 1981-99) recalled how he had asked her for an open marriage so that he could carry on more conveniently his affair with Callista Bisek, who later married him:

There’s no question at times of my life, partially driven by how passionately I felt about this country, that I worked too hard and things happened in my life that were not appropriate.

What was particularly outrageous about this was that his affair with Bisek (23 years younger than him) was in full flood just as he was pressing for the impeachment of Bill Clinton on the basis of his relationship with Monica Lewinsky.

It also does not help that Bisek is a cradle Catholic. She married Gingrich in August 2000, and later persuaded him to become a Catholic. He seems to have been formally received on 29 March 2009. So how exactly did that work? Surely it was wrong for her to get involved with him? How on earth did the Catholic Church come to marry them?

[The Free Lance-Star, Fredericksburg, VA for 12 May 2002]

Apparently in 2002 he got an annulment. It is usually reported – as above – to be an annulment of his marriage with Marianne on the basis that he was already married when she married Gingrich! But the Church would regard him as married to his first wife Jackie Battley, who is still alive.

The US bishops are notorious for dishing out annulments which are bogus under canon law*, or maybe the relevant bishop failed to grasp the full facts. Also if the annulment came after his marriage to Bisek in 2000, then that marriage would have been void, so they would need to be remarried subsequently. Maybe that happened at the time of the 2002 annulment. So far Gingrich has failed to give any but the sketchiest of answers to these questions and bizarrely he has not been much pressed on them.

[*An annulment is not a divorce, but a declaration that the marriage was never validly entered into. A typical argument, for example, is that Southern Baptists do not require that marriage should be until "death do us part". So a Southern Baptist entering into marriage might not have not have been intending to enter into marriage, because he was intending to enter into a partnership which he could terminate at will. So no marriage existed because he did not intend to enter into one. The fact that he split up twice would strengthen this argument. I do not find this particularly convincing.]

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