In 1786, in a classroom in Braunschweig, near Hanover, a bored schoolmaster in need of a nap set his pupils the tedious task of adding up every number between 1 and 100. Before the master could even lean back in his chair, one boy strode forward and placed his slate on the front desk.

Ligget se, he casually declared. There it is. And there it was: 5050. Carl Friedrich Gauss, at the age of nine, had announced his mathematical genius to the world.

Marcus du Sautoy begins his book about shortcuts not with the story, but with the story of the story. Du Sautoy explains that he, like Gauss, was a schoolboy sitting in a maths class when his teacher told the tale (which has been heavily embellished over the years) and explained that mathematics was “the art of the shortcut”.

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

Once you see that there are 50 pairs of numbers, each summing to 101, the answer — 5050 — is not far away.

At this moment a new vision of mathematics opened up for du Sautoy, “the ability to see structure in our mind’s eye without physically encountering it”. Mathematical shortcuts offer easy ways to solve difficult puzzles, or the mass production of solutions so that a problem need only be solved once to unlock every similar problem.

Du Sautoy is a gifted and tireless mathematical communicator with considerable range — his Brief History of Mathematics was masterful radio, and I’ve seen him compère concerts of classical music, interspersing the pieces with discussions of the mathematical patterns in Mozart and Bach.

In Thinking Better, he pauses between each mathematical chapter to interview experts in many domains, hoping to gain the secret to shortcuts in learning a language (he once studied Russian) or mastering the cello (Du Sautoy is a keen trumpeter).

These interludes are pleasant enough but feel distant from the mathematical ideas in the book. Virtuoso Natalie Clein explains that the secret to mastering the cello is endless, focused practice; no surprise there. A champion memoriser opines that if du Sautoy wants to improve his Russian, he needs repetition and testing — or, more promisingly, a Russian lover.

The joy of du Sautoy’s book isn’t really the art of the real-world shortcut at all. It is the romp through mathematical ideas

The reader who desires plausible shortcuts to real-life mastery might pick up a copy of Robert Twigger’s Micromastery, which advocates focus (to learn to cook, first learn to make a truly superlative omelette). Algorithms to Live By, by Brian Christian and Tom Griffiths, pulls surprising practical insights out of mathematical techniques — using computer science to explain how to sort your bookshelves, search for a new apartment, or organise piles of paper on your desk.

In contrast, the joy of du Sautoy’s book isn’t really the art of the real-world shortcut at all. It is the romp through mathematical ideas, from place value to non-Euclidean geometry to probability theory. He finishes with “NP-complete” problems, an important field in modern computing. A pithy explanation of what they are eludes me, but du Sautoy’s own description is the best I’ve seen: they’re “needle in a haystack” problems, because while it is very hard to find the needle, if you suspect you have succeeded it is very easy to check.

He frames this as a chapter about problems for which there seem to be no shortcuts, and ranges from quantum computing to problem-solving by slime moulds. He even includes the surprisingly difficult challenge of calculating which teams are still in with a mathematical prospect of winning the Premier League. (It was much easier when only two points were awarded for a win, rather than three, because the total number of points awarded in each game did not change.)

At times the reader needs to concentrate hard, because du Sautoy doesn’t shy away from equations or challenging ideas. He is, however, always a model of clarity. There are vivid historical examples of scientists and others using mathematical ideas to solve problems, from Eratosthenes calculating the circumference of the Earth to Florence Nightingale deploying data visualisation to make a powerful argument about public health. Columbus, Galileo, Alan Turing and Samuel Pepys all make appearances.

Many of the stories and examples — beginning with Gauss’s alleged feat of calculation — will be familiar to people who enjoy popularisations of mathematics. Du Sautoy’s mash-ups of art, music and mathematics have sometimes pushed the boundaries of mathematical popularisation in unusual directions, but here he mostly plays it straight. This is a “greatest hits” of mathematical ideas presented with trademark clarity and energy.

*Written for and first published in the Financial Times on 3 August 2021.*

*The paperback of “The Next 50 Things That Made The Modern Economy” is FINALLY coming – 26 August 2021. Please think about pre-ordering, which is hugely helpful in stimulating bookshops to stock and display the book.*

*“Endlessly insightful and full of surprises — exactly what you would expect from Tim Harford.”- Bill Bryson*

*“Harford is a fine, perceptive writer, and an effortless explainer of tricky concepts. His book teems with good things, and will expand the mind of anyone lucky enough to read it.”- The Daily Mail*

*I’ve set up a storefront on Bookshop in the United States and the United Kingdom – have a look and see all my recommendations; Bookshop is set up to support local independent retailers.*