These are some STEP problems which can be stated very briefly and without technicalities, yet are very different from A level maths. Useful as “tasters”.

1. How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5? What is the average value of these integers? (I/1999/1 second part)

2. The Fibonacci numbers F(n) are defined by the conditions F(0) = 0, F(1) = 1 and F(n+1) = F(n) + F(n−1) for all n ≥ 1. Compute F(n+1)F(n−1)−F(n)^{2} for a few values of n; guess a general formula and prove it. (II/1996/3 first part, abridged)

3. A small goat is tethered by a rope to a point at ground level on a side of a square barn which stands in a large horizontal field of grass. The sides of the barn are of length 2a and the rope is of length 4a. Let A be the area of the grass that the goat can graze. Prove that A ≤ 14πa^{2} and determine the minimum value of A. (I/2006/2)

4. The points A, B, and C lie on the sides of a square of side 1 cm and no two points lie on the same side. Show that the length of at least one side of the triangle ABC must be less than or equal to √6−√2 cm. (I/2001/1)

5. Bar magnets are placed randomly end-to-end in a straight line. If adjacent magnets have ends of opposite polarities facing each other, they join together to form a single unit. If they have ends of the same polarity facing each other, they stand apart. Find the expected number of separate units in terms of the total number N of magnets. (I/1999/13, first part)

6. A regular octahedron is a polyhedron with eight faces each of which is an equilateral triangle. Show that the angle between any two faces of a regular octahedron is arccos (−1/3). (I/2007/5, first part)

**Follow-up**

1. How many integers greater than or equal to zero and less than 4179 are not divisible by 3 or 7? What is the average value of these integers? (I/1999/1 second part)

2. By induction on k, or otherwise, show that F(n+k) = F(k)F(n+1) + F(k−1)F(n) for all positive integers n and k [if F(.) are the Fibonacci numbers] (II/1996/3, second part)

3. In the bar-magnets problem above, find the variance of the number of separate units in terms of the total number N of magnets. (I/1999/13, second part)

4. Find the ratio of the volume of a regular octahedron to the volume of a cube whose vertices are the centres of the faces of the octahedron. (I/2007/5, second part)

5. I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given that my choice does not contain the digits 0, 6, 7, 8 or 9, show that the expected number of different digits in my choice is 3.3616. (I/2012/13)

6. A thin non-uniform bar AB of length 7d has centre of mass at a point G, where AG = 3d. A light inextensible string has one end attached to A and the other end attached to B. The string is hung over a smooth peg P and the bar hangs freely in equilibrium with B lower than A. Show that 3 sin ∠PAB = 4 sin ∠PBA. Given that cos ∠PBA = 4/5 and that ∠PAB is acute, find in terms of d the length of the string and show that the angle of inclination of the bar to the horizontal is arctan (1/7) (I/2011/11)

]]>The Norwegian Academy of Science and Letters has decided to award the Abel Prize in Mathematics for 2019 to Karen Keskulla Uhlenbeck of the University of Texas at Austin, USA, “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.”

When Karen Keskulla Uhlenbeck held a Plenary Lecture in Kyoto, Japan in 1990, at the world’s most important gathering of mathematicians, the ICM, or the International Congress of Mathematicians, she was only the second woman in history to have done so – the first being Emmy Noether in 1932…

She says: “you really need to… show students how imperfect people can be and still succeed. Everyone knows that if people are smart, funny, pretty, or well-dressed they will succeed. But it’s also possible to succeed with all of your imperfections”.

“Karen Uhlenbeck receives the Abel Prize 2019 for her fundamental work in geometric analysis and gauge theory, which has dramatically changed the mathematical landscape. Her theories have revolutionized our understanding of minimal surfaces, such as those formed by soap bubbles, and more general minimization problems in higher dimensions”, said Hans Munthe-Kaas, Chair of the Abel Committee…

Gauge theory is the mathematical language of theoretical physics, and Uhlenbeck’s fundamental work in this area is essential for the modern mathematical understanding of models in particle physics, string theory and general relativity.

• The Abel Prize recognizes contributions to the field of mathematics that are of extraordinary depth and influence. It is presented annually in Oslo by His Majesty King Harald V, and is administered by the Norwegian Academy of Science and Letters on behalf of the Norwegian Ministry of Education and Research.

• The Abel Prize was established in 2002 on the 200th anniversary of Niels Henrik Abel’s birth, and it has been awarded to 19 laureates.

• Niels Henrik Abel (1802–1829) was a Norwegian mathematician. Despite living in poverty and dying at the age of 26, he pioneered the proof that equations with powers like x^{5} or higher do *not* (unlike quadratics, cubics, and quartics) have an algebraic formula to solve them, and the theory of elliptic functions.

The centre of mass of three equal masses placed each at one vertex of the triangle is called the centroid. It is also the centre of mass of a uniform lamina in the shape of the triangle. If the position vectors of the vertices A, B, C are a, b, c, then the centroid is at ^{1}⁄_{3}(a+b+c). All three medial lines (lines connecting a vertex to the midpoint of the opposite side) meet at that point.

Proof:

Centre of mass of the three equal masses is the same as the centre of mass of the system formed by the mass at A and a double mass at the midpoint M_{A} of BC, which has position vector ½(b+c).

That centre of mass must be on the line AM_{A}, and two-thirds of the way down from A to M_{A}, and so at ^{1}⁄_{3}(a+b+c).

That is also the centre of mass of a uniform lamina in the shape of the triangle.

The centre of mass must be on the medial line AM_{A}, because along any other line passing through A, every slice of the triangle parallel to the base BC like the one shown in green would have greater mass on one side of that line than the other.

Since the expression ^{1}⁄_{3}(a+b+c) is symmetric, and one medial line goes through it, all three medial lines must go through it. So it must be the centre of mass of the lamina.

The centre of mass of a triangular *frame* is a different point, the Spieker centre.

STEP I/2007/5:

(i) Show that the angle between any two faces of a regular octahedron is arccos(−^{1}⁄_{3})

(ii) Find the ratio of the volume of a regular octahedron to the volume of a cube whose vertices are the centres of the faces of the octahedron.

]]>The “Google Doodle” shown on the Google search page on 7 March 2019 depicted Olga Aleksandrovna Ladyzhenskaya, a Russian mathematician, on the 97th anniversary of her death.

She was known for her work on partial differential equations (especially Hilbert’s nineteenth problem, about classes of partial differential equations which admit analytic functions as solutions) and fluid dynamics. She provided the first rigorous proofs of the convergence of a finite difference method for the Navier–Stokes equations.

She did all that despite her father being killed by the Stalinist secret police in 1937, and she herself being consequently blocked from entry into Leningrad University.

]]>By “range” the question means the x-value (horizontal distance travelled) required to reach height h.

]]>*Above: a meeting of Portuguese mathematicians*

Cordelia Fine’s book “Delusions of Gender” finds that a repeated theme in arguments about biologically-determined inequalities is that girls and women are worse at maths, and mathematically-structured fields like physics.

The “compensating” claim is that girls and women are, supposedly, biologically determined to be better at empathy and caring.

The nearest to a “proof” of this claim is the results for “mental rotation performance”, “the largest and most reliable gender difference in cognition”. Males are usually found to do better, on average, than females in telling rotations of three-dimensional shapes apart from mirror-reflections.

Fine reports research with Italian high school students who were told, before being tested, that “women usually perform better than men in this test”. The women then performed just as well as the men.

Even if the difference were real, why would that generate difference in maths more than in, say, visual arts, design, and architecture?

Aside from visual dexterity, the other qualities required for maths are, if you think about it, ones stereotypically identified as “feminine”: precision, attention to detail, neatness, imagination, ability to hold onto many ideas simultaneously, persistence and stamina, self-critical attitudes (most mathematicians get most things wrong most of the time), and sociability and cooperativeness (maths is the most social of sciences).

The quality cited as essential for maths and supposedly “masculine” is logical, analytic, systematic thinking.

In the first place, why would logical, analytic, systematic thinking be less important for law than for maths? Yet the photo which my younger daughter has just put on her Instagram account of herself with the other Judge’s Associates at the Supreme Court of Queensland (the top-end just-graduated law students there) shows 13 women and 5 men.

In the second place, different people are good at maths in many different ways. Some can “see” mathematical relations quickly by visualisation; others can get little from diagrams. Some are brilliant calculators; others are quick at seeing possible approaches, but relatively slow at checking them out by calculation…

And some excel at painstaking logical analysis, while others excel at leaps of intuition and imagination. (See here for an example: Heaviside and Schwartz).

I see no evidence that women are systematically “good at maths” in different ways from men. But even if they were, they would still be “good at maths”.

The “mental rotation” test, as above, should identify women as worse at visual studies as much as at maths. Yet in British universities, 65% of students in creative arts and design subjects are women, 50% of architecture students (and 62% of law students). In all those fields, men do better at the best-paid senior end, but that can hardly be because of aptitudes biologically imprinted from babyhood.

Yet in the UK still only 37% of university maths students are women. (Only 21% of uni maths lecturers, and only 9% of professors).

Things are better elsewhere, and, oddly, not necessarily in countries where women’s equality is generally better.

Possibly best in the world is Portugal, where 46% of university maths and computer science students are women, 68% of maths Ph D students, 50% of maths lecturers, and 32% of maths professors.

Fine reports that in Armenia close to 50% of university students specialising in computer science are women (UK: 17%). The explanation which Fine got from Hasmik Gharibyan, an Armenian woman professor of computer science now working in the USA, is unexpected:

“In Armenia, ‘there is no cultural emphasis of having a job that one loves… The source of happiness for Armenians is their family and friendships, rather than their work… [So] there is a determination to have a profession that will guarantee a good living’.”

Fine explains how “stereotype threat” – the need to use mental energy fending off negative stereotypes – hurts in maths.

“Stereotype threat hits hardest those who actually care about their maths skills… and thus have the most to lose by doing badly, compared with women who don’t much identify with maths. Also, the more difficult and non-routine the work, the more vulnerable its performance will be to the sapping of working memory”.

She also quote findings from Shelley Correll in 1988 that among US high school students, boys tended to rate themselves as better at maths than girls would, at each level of actual achievement. There wasn’t the same male self-overrating, and female self-underrating, on verbal skills.

“Boys do not pursue mathematical activities at a higher rate than girls do because they are better at mathematics. They do so, at least partially, because they *think* they are better”.

Heather Mendick, who did a Ph D study on maths and gender in UK schools and came to talk with one of my classes about it in July 2018, found the same: girls are more reluctant to call themselves “good at maths”.

I think this may have shifted, or at least that more subtle mechanisms are operating.

Ten of my students were in class when Heather came, five girls, five boys. I asked them at the end: “Do you think you are good at maths?” Four of the girls said yes; one said she was just “ok”. Only one of the boys said “yes”; four said they were just “ok”. I’ve had similar responses elsewhere.

When in the course of the discussion Heather said something like: “Of course, maths isn’t easy for anyone”, the whole class immediately responded by laughing and indicating one of the girls: “Except for some!”

At least at some level, I think that girl knows she’s brilliant. In fact, most of my classes at that school have included girls recognised by me and by their classmates, and, at least eventually, by themselves, as stars.

Yet I’ve always found that the boys are more likely to attempt prize problems, outside the syllabus, than the girls. The boys are less worried by the risk of falling flat on their face with such problems.

My reading is that girls are more likely (on average! for socially-conditioned reasons!) to think that they’re good, even very good… but maybe not quite good enough. Boys are more likely to think that they’re maybe not brilliant… but good enough.

Fine’s argument about “stereotype threat” would fit into that picture.

Shortage of confidence (even at the level of thinking “I’m very good… but may still not good enough”) is a bigger difficulty in maths than in other subjects, I think. Asked to write an essay on the rise of Hitler, even the least confident student is likely to write something (and then can be told: “This is good!”) Asked to show, say, that in every right-angled triangle with whole-number sides, the three sides multiplied together must make a number divisible by 60, the less confident student is likely just to freeze and write nothing, while the more confident will make attempts, not be too worried if some attempts are way off the mark, and probably come up with at least some good ones.

The examples of Armenia and Portugal suggest that winning equality for girls and women in maths is not just a straightforward subsection of winning general equality. But it can be done.

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