Earlier this year, I canvassed readers on the best way to get the measure of potential PhD students (May column) . To get you thinking, I described the experiences of a Stony Brook colleague and myself in giving graduate-school applicants cleverly chosen physics problems – and then asking them not to solve each problem, but to explain the solution as if tutoring an undergraduate.

Demetris Charalambous, a former lecturer in mathematical physics at the University of Lancaster, UK, proposed one interesting challenge along these lines. Consider two identical metal spheres, one hanging from a piece of string and the other resting on a table. If you supply each with the same quantity of heat and ignore heat transfer to the table, string and so on, which sphere ends up hotter? Isn’t the student you want, he asked, the one who first spots the relevance of the centre of gravity?

Colin Pykett, now retired after having worked in several UK Ministry of Defence labs, remembered five challenges for prospective recruits. First, why is it that a mirror reverses your facial features but does not make you appear upside down? Second, why are there two tides a day? Third, if you are standing on a pavement flanked by a fence with vertical posts at regular intervals receding into the distance, what might happen if you clap your hands? Fourth, is the received power versus range law for a radar or sonar system inverse square and, if not, what is it and why? Finally, what are some differences between optical microscopy and X-ray crystallography?

Martin van Exter, from the Huygens Laboratory at the University of Leiden in the Netherlands, asks prospective students to explain Maxwell’s equations. He does not expect a treatise, just basic lines of reasoning, such as that the divergence of the electric (E) field is linked to the charge (or charge density), that the divergence of the magnetic (B) field is zero (because there can be no monopoles), that electric current acts as a generator of magnetic field and that the E and B fields are linked via time and space derivatives (thus containing the speed of light).

Peter Haynes, an admissions tutor at the doctoral-training centre in theory and simulation of materials at Imperial College London, says his colleagues prefer not to ambush applicants. Instead, in advance they send each candidate three straightforward problems – a 1D heat-diffusion equation, a 1D particle in a box and a cantilever bending under its own weight – to be explained at the interview. “We found that advance warning was vital to enable us to get to the stage where we could ask probing questions on the timescale of an interview,” says Haynes. He notes that some candidates treat the problems as purely algebraic exercises – struggling when asked how to interpret the final mathematical expression physically – while others write down equations they have memorized. The best students, in Haynes’ view, focus on the relevant physical principles and then express these mathematically. “They can readily sketch solutions and know how to interpret results,” he says.

Other means

Darryl Holm, also at Imperial, objected to the challenge-problem approach, noting that it was “exploitative” to treat PhD applicants as “cheap undergraduate instructors”. He has two alternative methods. One is to ask prospective PhD students what they love about science and maths. The answers, he says, make it easy to separate the passionate from the diffident. Another is to show them a rattleback – a kind of top that refuses to spin in one direction and can change its rotation to a preferred direction. “If, when they see it unexpectedly reverse spin, they start laughing, or say ‘Oh shit!’ or are otherwise energized, I know I am on the right track,” he says. The blasé or confused ones drop to the bottom of the list.

Harvey Buckmaster, an adjunct physics professor at the University of Victoria in Canada, says in three decades of experience with graduate students one good marker is their extra-curricular activities. “Those with significant interest and involvement in cultural activities have done more significant research and wrote better theses,” he says, adding that creativity in the arts is “no different from that in the sciences”.

The critical point

George Hart of the Museum of Mathematics, which is set to open next year in New York City, gave me a good challenge problem in person that might be asked of maths students: “Why does a negative times a negative equal a positive?” he asked.

I had no idea.

Hart, who is a former colleague of mine at Stony Brook, said the answer can be explained simply as a consequence of the distributive law. He hunted for pencil and paper, jotted down 1 + –1 = 0, multiplied both sides by –1, applied the distributive law a(b + c) = ab + ac, rearranged terms and ended with –1 × –1 = 1. He said another way was to use scaling, and drew a number line. Multiplication, he explained, “scales” numbers from the 0 point: multiplying by 2, for instance, doubles the distance from 0 in either direction, while multiplying by a negative number flips a distance from 0 by 180°. “A negative times a negative is thus a double 180° transformation,” concluded Hart.

I asked if a prospective maths PhD would know this. “Sure!” he said. “Mathematics isn’t a set of arbitrary definitions. Each definition is bound up with other parts in an architecture – it has consequences. Even an undergraduate, if really interested in mathematics, will have thought about this issue enough to have some way of describing how it fits in that architecture.”

Physics, of course, has a similar structure – its different laws and definitions, too, are bound up in a fabric. That is what makes the challenge-problem approach effective. A student’s promise is not measured best by the ability to memorize formulae, but by how well they understand this fabric, and know how to fit even simple problems into it.