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IF JOEY was Chloe鈥檚 age when he was twice as old as Zoe was, how many times older will Zoe be when Chloe is twice as old as Joey is now?

Or try this one for size. Two farmers inherit a square field containing a crop planted in a circle. Without knowing the exact size of the field or crop, or the crop鈥檚 position within the field, how can they draw a single line to divide both the crop and field equally?

You鈥檝e either fallen into a cold sweat or you鈥檙e sharpening your pencil (if you can鈥檛 wait for the answer, you can聽check the bottom of this page). Either way, although both problems count as 鈥渕aths鈥� 鈥� or 鈥渕ath鈥� if you insist 鈥� they are clearly very different. One is arithmetic, which deals with the properties of whole numbers: 1, 2, 3 and so on as far as you can count. It cares about how many separate things there are, but not what they look like or how they behave. The other is geometry, a discipline built on ideas of continuity: of lines, shapes and other objects that can be measured, and the spatial relationships between them.

Mathematicians have long sought to build bridges between these two ancient subjects, and construct something like a 鈥済rand unified theory鈥� of their discipline. Just recently, one brilliant young researcher might have brought them decisively closer. His radical new geometrical insights might not only unite mathematics, but also help solve one of the deepest number problems of them all: the riddle of the primes. With the biggest prizes in mathematics, the Fields medals, to be awarded this August, he is beginning to look like a shoo-in.

The ancient Greek philosopher and mathematician Aristotle once wrote, 鈥淲e cannot鈥� prove geometrical truths by arithmetic.鈥� He left little doubt he believed geometry couldn鈥檛 help with numbers, either. It was hardly a controversial thought for the time. The geometrical proofs of Aristotle鈥檚 near-contemporary Euclid, often called the father of geometry, relied not on numbers, but logical axioms extended into proofs by drawing lines and shapes. Numbers existed on an entirely different, more abstract plane, inaccessible to geometers鈥� tools.

And so it largely remained until, in the 1600s, the Frenchman Ren茅 Descartes used the techniques of algebra 鈥� of equation-solving and the manipulation of abstract symbols 鈥� to put Euclid鈥檚 geometry on a completely new footing. By introducing the notion that geometrical points, lines and shapes could all be described by numerical coordinates on an underlying grid, he allowed geometers to make use of arithmetic鈥檚 toolkit, and solve problems numerically.

This was a moonshot that let us, eventually, do things like send rockets into space or pinpoint positions to needle-sharp accuracy on Earth. But to a pure mathematician it is only a halfway house. A circle, for instance, can be perfectly encapsulated by an algebraic equation. But a circle drawn on graph paper, produced by plotting out the equation鈥檚 solutions, would only ever capture a fragment of that truth. Change the system of numbers you use, for example 鈥� as a pure mathematician might do 鈥� and the equation remains valid, while the drawing may no longer be helpful.

Wind forward to 1940 and another Frenchman was deeply exercised by the divide between geometry and numbers. Andr茅 Weil was being held as a conscientious objector in a prison just outside Rouen, having refused to enlist in the months preceding the German occupation of France 鈥� a lucky break, as it turned out. In a letter to his wife, he wrote: 鈥淚f it鈥檚 only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?鈥�

Weil hoped to find a Rosetta stone between algebra and geometry, a reference work that would allow truths in one field to be translated into the other. While behind bars, he found a fragment.

It had to do with the Riemann hypothesis, a notorious problem concerning how those most fascinating numbers, the primes, are distributed (see below). There had already been hints that the hypothesis might have geometrical parallels. Back in the 1930s, a variant had been proved for objects known as elliptic curves. Instead of trying to work out how prime numbers are distributed, says mathematician at Imperial College London, 鈥測ou can relate it to asking how many points a curve has鈥�.

Weil proved that this Riemann-hypothesis equivalent applied for a range of more complicated curves too. The wall that had stood between the two disciplines since Ancient Greek times finally seemed to be crumbling. 鈥淲eil鈥檚 proof marks the beginning of the science with the most un-Aristotelian name of arithmetic geometry,鈥� says of Columbia University in New York.

In the post-war years, in the more comfortable setting of the University of Chicago, Weil tried to apply his insight to the broader riddle of the primes, without success. The torch was taken up by Alexander Grothendieck, a mathematician ranked as one of the greatest of the 20th century. In the 1960s, he redefined arithmetic geometry.

Among other innovations, Grothendieck gave the set of whole numbers what he called a 鈥渟pectrum鈥�, for short Spec(Z). The points of this undrawable geometrical entity were intimately connected to the prime numbers. If you could ever work out its overall shape, you might gain insights into the prime numbers鈥� distribution. You would have built a bridge between arithmetic and geometry that ran straight through the Riemann hypothesis.

The shape Grothendieck was seeking for Spec(Z) was entirely different from any geometrical form we might be familiar with: Euclid鈥檚 circles and triangles, or Descartes鈥檚 parabolas and ellipses drawn on graph paper. In a Euclidean or Cartesian plane, a point is just a dot on a flat surface, says Harris, 鈥渂ut a Grothendieck point is more like a way of thinking about the plane鈥�. It encompasses all the potential uses to which a plane could be put, such as the possibility of drawing a triangle or an ellipse on its surface, or even wrapping it map-like around a sphere.

If that leaves you lost, you are in good company. Even Grothendieck didn鈥檛 manage to work out the geometry of Spec(Z), let alone solve the Riemann hypothesis. That鈥檚 where Peter Scholze enters the story.

鈥淓ven the majority of mathematicians find most of the work unintelligible鈥�

Born in Dresden in what was then East Germany in 1987, Scholze is currently, at the age of 30, a . He laid the first bricks for his bridge linking arithmetic and geometry in his PhD dissertation, published in 2012 when he was 24. In it, he introduced an extension of Grothendieck-style geometry, which he termed perfectoid geometry. His construction is built on a system of numbers known as the p-adics that are intimately connected with the prime numbers (see 鈥�The p-adics: A different way of doing numbers鈥�). The key point is that in Scholze鈥檚 perfectoid geometry, a prime number, represented by its associated p-adics, can be made to behave like a variable in an equation, allowing geometrical methods to be applied in an arithmetical setting.

It鈥檚 not easy to explain much more. Scholze鈥檚 innovation represents 鈥渙ne of the most difficult notions ever introduced in arithmetic geometry, which has a long tradition of difficult notions鈥�, says Harris. Even the majority of working mathematicians find most of it unintelligible, he adds.

Be that as it may, in the past few years, Scholze and a few initiates have used the approach to solve or clarify many problems in arithmetic geometry, to great acclaim. 鈥淗e鈥檚 really unique as a mathematician,鈥� says Caraiani, who has been collaborating with him. 鈥淚t鈥檚 very exciting to be a mathematician working in the same field.鈥�

This August, the world鈥檚 mathematicians are set to gather in Rio de Janeiro, Brazil, for their latest international congress, a jamboree held every four years. A centrepiece of the event is the awarding of the Fields medals. Up to four of these awards are given each time to mathematicians under the age of 40, and this time round there is one name everyone expects to be on the list. 鈥淚 suspect the only way he can escape getting a Fields medal this year is if the committee decides he鈥檚 young enough to wait another four years,鈥� says Marcus du Sautoy at the University of Oxford.

With so many grand vistas opening up, the question of Spec(Z) and the Riemann hypothesis almost becomes a sideshow. But Scholze鈥檚 new methods have allowed him to study the geometry, in the sense Grothendieck pioneered, that you would see if you examined the curve Spec(Z) under a microscope around the point corresponding to a prime number p. That is still a long way from understanding the curve as a whole, or proving the Riemann hypothesis, but his work has given mathematicians hope that this distant goal might yet be reached. 鈥淓ven this is a huge breakthrough,鈥� says Caraiani.

Scholze鈥檚 perfectoid spaces have enabled bridges to be built in entirely different directions, too. A half-century ago, in 1967, the then 30-year-old Princeton mathematician Robert Langlands wrote a tentative letter to Weil outlining a grand new idea. 鈥淚f you are willing to read it as pure speculation I would appreciate that,鈥� he wrote. 鈥淚f not 鈥� I am sure you have a waste basket handy.鈥�

In his letter, Langlands suggested that two entirely distinct branches of mathematics, number theory and harmonic analysis, might be related. It contained the seeds of what became known as the Langlands program, a vastly influential series of conjectures some mathematicians have taken to calling a grand unified theory capable of linking the three core mathematical disciplines: arithmetic, geometry and analysis, a broad field that we encounter in school in the form of calculus. Hundreds of mathematicians around the world, including Scholze, are committed to its completion.

The full slate of Langlands conjectures is no more likely than the original Riemann hypothesis to be proved soon. But spectacular discoveries could lie in store: Fermat鈥檚 last theorem, which took 350 years to prove before the British mathematician Andrew Wiles finally did so in 1994, represents just one particular consequence of its conjectures. Recently, the French mathematician proposed a way to build on Scholze鈥檚 work to understand aspects of the Langlands program concerned with p-adics. It is rumoured that a partial solution could appear in time for the Rio meeting.

In March, Langlands won the other great mathematical award, the Abel prize, for his lifetime鈥檚 work. 鈥淚t took a long time for the importance of Langlands鈥檚 ideas to be recognised,鈥� says Caraiani, 鈥渁nd they were overdue for a major award.鈥� Scholze seems unlikely to have to wait so long.

• Answers: Zoe will be three times as old as she is now. The farmers should draw a line across the field that connects the centre points of the field and the crop.

This article appeared in print under the headline 鈥淭he shape of numbers鈥�