Tom remembers the day he decided he wanted to be a theoretical astrophysicist. He was deep into research about black holes, and had amassed a box of papers on his theories. In one he speculated about the relationship between black holes and white holes, hypothetical celestial objects that emit colossal amounts of energy. Black holes, he thought, must be linked across space-time with white holes. “I put them together and I thought, oh wow, that works! That’s when I knew I wanted to do this as a job.” Tom didn’t know enough maths to prove his theory, but he had time to learn. He was only five.

Tom is now 11. At home, his favourite way to relax is to devise maths exam papers complete with marking sheets. Last year for Christmas he asked his parents for the £125 registration fee to sit maths GCSE , an exam most children in Britain take at 16. He is currently working towards his maths A -level. Tom is an only child, and at first Chrissie, his mother, thought his love of numbers was normal. Gradually she realised it wasn’t. She would take him to lectures about dark matter at the Royal Observatory in London and notice that there were no other children there. His teacher reported that instead of playing outside with other kids at breaks, he wanted to stay indoors and do sums.

One day his parents took him to Milton Keynes to have his intelligence assessed by an organisation called Potential Plus, formerly the National Association for Gifted Children. “We told him it was a day of puzzles,” Chrissie says. “It was my dream world,” Tom says. “Half a day of tests!” His mother waited while he applied his mind to solving problems. When they were shown the results, Tom’s intelligence put him in the top 0.1% in Britain.

Precocious children are often dismissed as the product of pushy, middle-class parents. Nurture and environment clearly do play an important role in any child’s intellectual development. Talk to your child about politics over the dinner table and he is likely to develop confident opinions about the way the world should be run. Suggest that your toddler think of slices of cake in terms of angles and she may well display an early aptitude for mathematics. Practice can make perfect. The child with a gift for playing the piano who practises five hours a day is more likely to end up performing at Carnegie Hall than the equally gifted one who plays for just 20 minutes a week.

But children like Tom are different. He was brought up in an underprivileged part of south London: 97% of pupils at his first school didn’t speak English as a first language. When it comes to numbers – or his other passions such as Latin and astrophysics – Tom’s parents have little idea what he’s talking about. His genius is not of their engineering.

Intelligence tests are marked “on a curve”, meaning that the results are transformed into a bell curve: what matters is how you do compared with others who take them. By definition, most scores bunch in the middle: the average result in a cohort becomes an intelligence quotient ( IQ ) of 100; the middle two-thirds of scores become IQ s of 85 to 115. The outliers are few. About two people in 100 have IQ s below 70, and another two have IQ s above 130. By the time you get 45 points away from the average of 100 in either direction, you’re down to about one person in 1,000. But since only a small percentage of any population takes IQ tests, identifying very exceptional children is hard. Most schools have none.

Society prizes intelligence. Geniuses are viewed with awe and assumed to be guaranteed prosperity and success. Yet there is a dark side to intelligence. Like many gifted children, Tom’s childhood has often been unhappy. Aged five, he talked about wanting to end his life: he said he planned to do this by banging his head repeatedly against a wall. “Life’s like a maze, only bigger,” Tom told his mum. “I feel I’m getting lost.” His GP said he was suffering from severe depression, and reckoned its roots lay in Tom’s “genius”, and the frustration and isolation this was causing him.

Tom finds it hard to relate to other children and has few friends. At school he has been shunted out on his own into corridors and offices. “They didn’t want him in the class because he’s doing different stuff,” Chrissie says. To distract his mind from “dark thoughts”, Tom turns to puzzles and calculations, often late at night. He has long suffered from insomnia. The strain affects the whole family: “I don’t understand parents who seek this,” says Chrissie. “I can’t cope with it. I just want to take it away.”

Many others echo the pain of Tom and his family. Mensa, an international organisation founded in Britain in 1946 to nurture the country’s most intelligent people, has 20,000 members (you must apply to join). When I put out a request via Mensa to hear from gifted children and their parents, my inbox fills with emails, many of them anguished. Those that I speak to say that, for fear of inspiring jealousy, they don’t dare talk to others about their children’s abilities. Given a sympathetic ear, they pour out their woes at such length that I nearly despair of getting them off the phone. Almost all are afraid of being identified, and insist on fake names.

Some countries value extremely high intelligence more than others and offer specific educational provision for such children. Yet even if your genius is prized, admired and cultivated, social and psychological issues that often accompany great ability may make it an unwelcome gift. From the inside – and for many families that I spoke to – genius can feel more like a curse than a blessing.

Most experts reserve the term “gifted” for children who demonstrate three characteristics. First, gifted children begin to master a particular discipline – a language, maths or chess – much younger than most. They do so easily, so they also progress much faster than their peers.

Secondly, this mastery is achieved largely on their own, rather than as a result of parental prodding. A child’s surroundings and socio-economic background certainly affect their speed of development: there is a close correlation between the number of words a child’s parents have spoken to them by the time they’re three and the child’s academic success aged nine. Studies suggest that children born into professional families may have heard some 4m more words by then than the offspring of parents with lower educational backgrounds. Such families often have higher incomes to provide more educational opportunities too.

But Lyn Kendall, a consultant on gifted children at Mensa – who was herself a gifted child in a working-class family – insists that reading Nietzsche to your five-year-old, or forcing them to do three hours of extra homework, cannot “make” a genius.

Many children who have extremely high IQ s show signs of extraordinary ability even as tiny babies, before pushy parenting is able to have much impact. “From a very early age – pre-language – these children understand what is going on around them, understand what people say but cannot respond,” says Kendall. Most toddlers appear to explore the world as they encounter it, distracted by passing cars or the arrival of a new toy. By contrast, Kendall describes gifted children of that age as “driven”: “They never stop and they set themselves incredibly high standards.” We often associate the early years of childhood with taking joy in simple things, living in the present and an inability to think through the consequences of actions. Instead, says Kendall, watching gifted toddlers, “it’s almost as if someone has taken an 18-year-old and put them in a newborn body.”

A third characteristic of gifted children is that their interests often seem near-obsessive. They have what is sometimes called “a rage to master”. Jesse is five. When he was one and crawling, his father Richard tells me, he would do anything to avoid having his nappy changed. “We found that the only way we could keep him still was to give him things to take apart and put back together again. We had a yellow torch with a built-in bulb, and he would take the battery out, put it back in, and test whether it worked. If he’d put the battery in the wrong way round, he’d persevere until he got it right.”

The first IQ tests to measure intelligence were developed by Alfred Binet and Theodore Simon in the early 20th century. They evaluated short-term memory, analytical thinking and mathematical ability. Though the tests have changed since then, the basic skills they attempt to measure have remained the same. Within a few points either way, IQ is fixed throughout your life: the only way you’ll lose it is because of a brain injury.

So-called “intelligence” tests abound online. Many children take aptitude tests at school. Most of these can be gamed or, at least, children can be trained to excel at them. Mensa does its best to make its tests “culture fair” – in other words it aims to identify intelligence that is intrinsic rather than taught. “The original gifted children will have invented the wheel and discovered fire,” says Kendall. But even Kendall, who is in the business of evaluating children, admits that “testing IQ is not like measuring height”. No assessment is completely objective.

Most tests look only at particular types of intelligence, such as mathematical and verbal reasoning. That reflects how narrow society’s notions of giftedness are. Many other types of skill and characteristics are missed, such as voracious curiosity or the ability to make intellectual connections. The tests are unlikely to identify future novelists or poets, or children who may be exceptionally good at sports or music. We don’t yet have a way to measure creative, artistic or emotional intelligence. The sorts of children we rate as “geniuses” tend to be only those who fall into the standard categories.

Some people question the very notion of giftedness. The definition of a gifted child has fragmented over time, says Deborah Eyre, founder of High Performance Learning, an organisation that works with schools and teachers in Britain to try to help large numbers of children become “high performers”. She does not see aptitude as innate. Eyre says that no matter where you look in the world, the children of wealthy parents are over-represented in cohorts of gifted children. Those who come from minority backgrounds are under-represented: “Latinos don’t get selected [for programmes] in the US , Maoris don’t in New Zealand.”

She also says that what marks out brilliant and high-achieving children – and adults – is often determination. The difference between two equally talented physicists, one who goes on to win a Nobel prize and one who does not, is their will to succeed. Apparent genius, she argues, is a combination of some kind of potential, along with the right support and personal drive.

Eyre claims that a certain type of parent, usually a highly educated one, takes pride in having a “gifted child” to show off. But this view wasn’t borne out by the parents I spoke to, most of whom found their children’s gifts to be a source of anxiety, even distress.

Many of these parents face two main difficulties. One is how to cater to the advanced intellectual development of their child. The second dimension is more rarely voiced but may cause just as many problems: exceptionally intelligent children are often socially isolated, even disruptive. Gifts that are admired in the abstract often seem less welcome in person.

If you were to meet Ophelia Gregory, you’d think that the good fairies must have clustered around her cradle. Now 17, she is willowy and beautiful, with deep-green eyes. Her family – mother Kerry, father Tom and three younger brothers – is close and loving. At the age of 12, Ophelia clocked 162 in Mensa’s IQ test. It is the highest possible score for someone under 18, and on a level with Stephen Hawking, the ground-breaking cosmologist who died last year.

Yet so far, extraordinary intelligence has brought Ophelia little happiness. For her, being categorised as “gifted” is simply “more trouble than it’s worth”. She has been bullied and changed schools several times. I wonder what Kerry would say to a parent longing for a gifted child? “I’d say, ‘It should be a great thing, but it’s not. It never will be.’”

We have long known that some individuals have extraordinarily high intelligence. Only more recently have psychologists started to look at whether and how this affects other areas of these individuals’ lives. Gifted children often experience what psychologists call “asynchronous development”: exceptional abilities in some areas may be associated with, or come at the cost of other aspects of maturity. “The parts of the brain that control the learning of words, patterns and numbers develop extremely quickly in these children,” says Andrea Anguera of Potential Plus. “But the frontal lobe, which controls the regulation of emotions, doesn’t develop as fast.”

A gifted child may have an advanced ability to master something like maths, but more limited capacity to deal with their social environment which is another important part of growing up and fitting in over the course of their lives. “A gifted child might be prone to complete social meltdowns,” says Anguera. “They can’t understand how other children work, and they can’t control their emotions.” Being exceptionally able in some areas means they need “the right support” in others, she says.

In the early 20th century American psychologist Leta Hollingworth talked about “socially optimal intelligence”, which she associated with an IQ of between 125 and 155. Ratchet the score beyond that, and what Norman Geschwind, an American behavioural neurologist, termed a “pathology of superiority” can creep in: the dominance of one bit of the brain can affect the development of other parts.

We don’t yet know why this is, or whether it’s down to nature, nurture or both. One study shows that among members of Mensa in America, the rate of ADHD (attention deficit hyperactivity disorder) is almost twice that diagnosed in the general population. Others argue that because some gifted children are so different from their peers at school, and may interact little with them in the classroom, they may do so less in the playground too. Though in some ways their aptitudes are very adult, many find themselves unable to play games that we often refer to as “childish”: their social development is more restricted. If an exceptionally able five-year-old spends her free time doing algebra, says Anguera, she often doesn’t want to spend time with a peer who prefers to play with cars. Yet once a child is left out of some social situations, her opportunity to catch up or learn these skills diminishes.

Kendall identifies several characteristics common among gifted children who have no identified behavioural disorders. One trait is that many of them are deeply anxious, usually as a result of over-thinking everything. “Your brain has the capacity to work out all the variables,” she explains, “so it inevitably does.” Hilary emailed me about her son, Lorenzo: “I am finding it increasingly difficult to cope with his heightened emotion and anxiety.” Lorenzo, now 12, became a member of Mensa two years ago and so has opportunities to mix with other very bright kids both in person and online. Lorenzo scored 162 in his IQ test (“Same as Einstein,” Hilary tells me. I don’t have the heart to tell her that Einstein never had his IQ measured). He worries incessantly: “Waiting for a flight to Hong Kong recently, he asked so many questions about what might go wrong with the plane that the waiting hall cleared around us.”

The sleeping pattern of such children often differs from the norm: switching off their brains can be very difficult. The mother of one gifted child told me that he didn’t sleep for more than 90 minutes at a stretch until he was nearly five.

The emotional and physical health associations with genius don’t stop there. The American branch of Mensa, which has more than 50,000 members, refers to its affiliates as having “hyper brains”. A recent survey of its members suggested that people with exceptionally high intelligence very often have what Kazimierz Dabrowski, a Polish psychologist, dubs “over-excitabilities” or “super-sensibilities”, such as a heightened awareness of one of the five senses, experiencing extremely intense emotions or having very high levels of energy. Among these individuals, the incidence of depression, anxiety and ADHD is higher than in the average population.

Giftedness may even be linked with physiological conditions such as food allergies, asthma and autoimmune diseases, which sometimes go hand-in-hand with “sensory processing disorder”. For many exceptionally intelligent individuals, everyday stimuli such as a radio playing in the background, the colour or texture of food, a vibrant display on a classroom wall or a scratchy label in a piece of clothing can become almost unbearable. Because his brain function is so acute, Lorenzo’s senses are more than usually finely tuned, believes Hilary. “He can hear things that we can’t. He can find it impossible to do his homework in a room that would seem to most people completely silent.”

“Neurologically, high IQ goes with increased efficiency in neural functioning,” says Sonja Falck, a psychotherapist in Britain who works almost exclusively with clients of “extreme intelligence”. “That’s measurable,” continues Falck. “If a person is getting a lot of stimulation and processing it very quickly, they are susceptible to being over-stimulated.”

Many gifted children struggle with failure. The trouble, Kendall explains, is that if you’re known for being a brainbox you don’t have to try, and so don’t build up resilience. She works with many bright children who “won’t put pen to paper”. At workshops she runs for gifted children, the kids sometimes play Twister, a game where players contort themselves over a mat covered with coloured dots. “They’re in hysterics,” Kendall says. “You can’t get it right so you’re teaching them to do something just for the joy of it.”

Rebecca’s daughter Lizzie is five. She was conceived with donor sperm and her biological father had three degrees. Ahead of her first birthday she was using whole sentences. She completed a puzzle with 48 pieces in which she had to match pictures to the corresponding words at 16 months. By her second birthday she could recite “The Gruffalo”, a 24-page children’s story written in rhyme; when Rebecca forgot her face-cloth at bath time, Lizzie chided, “Mummy, you are an abomination!” Aged three, she announced, “Mummy, I’m not pretty. It’s my chromosomes’ fault.” But like many gifted children, she can become distraught if she gets things wrong. “Some days I feel sorry for her,” says Rebecca. “I just want her to be as normal as possible.”

That is difficult. Ahead of play dates, Rebecca clears away Lizzie’s toys so that the other mothers can’t see how advanced she is. People look for gifted children to fail, says Rebecca, “I’ve learned to cover for Lizzie.” Rebecca teaches children with special needs, but says that for her daughter’s particular needs “there’s nothing”.

Sonja Falck is wary of the word “gifted” because “it connotes privilege”, in that the gifted person is seen as having an advantage over everyone else. But it’s not necessarily an advantage. “Someone who is gifted, but who grows up in an environment that is not supportive, can really suffer. This suffering is hugely under-acknowledged.” Falck tells me about a client of hers who had an abortion: she couldn’t bear the idea of giving birth to a child who might suffer for her “gifts” as she had.

Emily’s son Peter is nine. Since he was tiny he has preferred adult company to that of his peers: “At nursery, he used to sob all morning,” says Emily. Physically fragile and a loner, he has ended up in hospital three times after being beaten up at school. In common with many gifted children he has difficulty eating because he is hyper-sensitive to food textures. But for Peter, as for many other children, the greatest problem is that humdrum, day-to-day life is so hard to deal with. He finds school crushingly dull. His head teacher doesn’t see that this is a problem. “A bit of boredom is quite good for you,” he told Emily.

But boredom can be torture. A gifted student needs a fraction of the hours to master a GCSE subject that the school curriculum typically devotes to that subject, suggests Falck. She compares it to a seasoned runner being forced every day to trudge in step with people who walk extremely slowly.

How best to educate a gifted child? The challenges are complex and often competing. On the one hand they are able to master material sooner and more rapidly than their peers. On the other, because the social skills of many such children are poorly developed, it can be extremely difficult for them to be a child in the traditional sense, to fit in and to learn many of the non-verbal, non-testable skills that social activity teaches you in preparation for being an adult. And without meaning to, such children may come across as smart-arses who, even with the best of intentions, other kids and adults may simply not wish to be around. Adults, especially teachers, may find extremely clever children threatening: a small child talking to you as an equal can put you on the back foot. They literally know more than the adults around them and can’t help but tell them so.

After Tom’s assessment at Potential Plus, Chrissie sought advice on how best to educate him. It was obvious to her that his south-London primary school couldn’t cope. Apart from his first teacher at the school, whom Tom describes as “incredible” and who encouraged his interest in maths by sitting with him during break times to work through problems, his other teachers seemed to hate him. One appeared to enjoy belittling him, announcing to the class that “Tom found maths hard today,” while neglecting to mention that he was doing work meant for children ten years older than him.

Chrissie was told she had two options: she could either home-school Tom or send him to a private school that could give him more individual attention. Both ideas horrified her. She disagreed with home-schooling on principle – surely it would exacerbate his feeling of isolation. Private school was beyond the family’s financial means, but Tom received a bursary and now attends a respected, selective school in London, where the annual fees are £20,000. He still struggles to relate to other kids, and finds the economic disparity between him and his fellow pupils shocking. But he finds the teaching more stimulating. “I do like her, and she has given me harder work,” he says of his maths teacher.

Debate rages about the wisdom of accelerating children out of their age group. If they are moved up, they may struggle socially. If they stay down, they may switch off intellectually. Students need social and psychological support, says Leonie Kronborg of the University of Monash in Australia. She points to programmes for gifted adolescents like the Early Entrance Programme at the University of Washington in America: young teenagers can begin studying at university as part of a group of similarly advanced people their own age, so they are intellectually stimulated but keep socialising with their peers.

Faced with sons and daughters who are bored and miserable at school, many parents of gifted children opt to take things into their own hands. Chrissie’s fears aside, home-schooling is surprisingly common for gifted children of highly educated parents. In the mid-1980s a father and daughter, Harry and Ruth Lawrence, made a striking pair, travelling around Oxford on a tandem bicycle. Harry had given up his career in computing and home-educated Ruth since she was five; at 12 she won a place to study maths at Oxford University. Harry accompanied Ruth to all her lectures, making sure that she never “wasted” time by socialising with other young people. She now works as a respected – but not outstanding – mathematician. When she had her first child, she vowed not to push him to move any faster academically than he wanted to.

Some countries have cultivated an educational environment that is welcoming to gifted children. Singapore runs a highly selective programme designed to identify the most exceptionally intelligent students each year. At the age of eight or nine all children are assessed in maths, English and reasoning. The top 1% are transferred from “normal” classes to the Gifted Education Programme which is run in nine primary schools up to the age of 12. They can then choose whether to attend certain secondary schools that offer such classes. Selected children get “personalised education plans” that include teaching on particular topics in greater depth and breadth, access to additional self-taught online courses, placement in higher classes for specific subjects, and early admission to primary school for very young children. But emphasising educational attainment has proved controversial. Since 2007, there have been efforts to increase socialisation between children of different abilities.

Such an approach reflects a very traditional idea of intelligence – using certain types of tests to identify children with apparently innate intellectual abilities. Elsewhere educationalists are using a broader range of methods to spot highly intelligent children and increasing their focus on attitudes and personality traits often found in the most successful people – the drive, for instance, that Deborah Eyre talks about. In Project Bright Idea, a programme at Duke University in North Carolina, 10,000 ordinary nursery and primary-school children were taught using methods usually applied to the cleverest kids – fostering high expectations, encouraging complex problem-solving and developing meta-cognition (“thinking about thinking”). Nearly all of them went on to do much better in tests than their comparable peers.

What will become of Tom and Ophelia, Lizzie, Lorenzo and Peter? Raj Chetty, an American economist at Harvard University has calculated that those who score in the top 5% of standard tests at primary school are many times more likely than the other 95% to file patents as adults – and that probability is far higher among bright kids from rich families. Whatever their natural talents, children whose aptitudes are nurtured and given opportunities have a far better chance in life.

But gifted children do not necessarily shine later on. Some are what Chetty refers to as “lost Einsteins”: children who weren’t given an outlet for their intelligence or the encouragement to stretch their intellect, or who needed help to deal with the isolation of their experience. There are those whose abilities are missed by the limitations of IQ tests. And there are the many exceptional children who face barriers in later years because they never developed the interpersonal skills needed to succeed in the workplace or the wider world of social activity.

In the 1920s Lewis Terman, an American psychologist, studied 1,500 children with very high intelligence. Others followed up that group 70 years later. They found that they had accomplished no more than their socio-economic status would have predicted. One child Terman excluded as not bright enough, William Shockley, had co-invented the transistor and won the Nobel prize in physics.

And an unhappy childhood stays with you. Kim Ung-yong was a child prodigy in South Korea. Now a civil engineer in his 50s, he feels he was cheated of a childhood. He began speaking at six months and had mastered four languages by the age of two. He gained his first P h D aged eight, and was then headhunted to work for NASA . “I led my life like a machine,” he has said. “I woke up, solved the daily assigned equation, ate, slept…I was lonely and had no friends.” Even Albert Einstein, one of the most emblematic examples of genius, wrote in 1952: “It is strange to be known so universally and yet be so lonely.”

That’s a bleak message for the child geniuses of today. Looking to the future, Tom’s mum Chrissie doesn’t seem hopeful. “Show me a story of a child like this which ends well,” she says. “They don’t exist.” Then she turns to Tom reassuringly. “Maybe you will be the first.”■