If I do anything, I want to do it well,” says Sarah Flannery, 1999‘s Irish Young Scientist of the Year and European Young Scientist of the Year. Already a media sensation for trying to develop a fast algorithm for secure encryption on the Internet, she is now co-author with her father of a book, In Code (Profile pound;14.99). Soon to give a prestigious Last Word lecture at the Royal Geographical Society, about to have her book serialised in the Daily Telegraph as a 12-part maths course in this Maths Year 2000, a guest at last year’s Nobel ceremony dinner: is this a girl for whom all doors fly open?

A year after her first interview for The TES at her school, Schol Mhuire Gan Sm l in Blarney, County Cork, she is still the same level-headed, thoughtful and modest person she was then.

With teenagers constantly blasted as lazy, destructive, anti-social and a problem, Sarah’s achievements are an inspiring reminder of the nurturing power of education. The promise, the potential and the energy are hers, but her teachers - particularly science teacher Sean Foley - and her mathematician father David Flannery, provided the guidance, encouragement and knowledge she has used to develop her own mathematical ideas. And others along the way, particularly William Whyte and Michael Purser at Internet cryptography company Baltimore Technologies - relished helping the budding talent unfold. Still, as Sarah says: “I’m not one of those people who likes not doing things. I want to learn.”

Sarah is 18 and lives in a small village outside Cork with her four younger brothers. Her mother, a microbiologist, works, cooks, sings and plays guitar with her sons, makes up limericks and casts a cool, calming eye on her family. Her father, a lecturer in mathematics at Cork Institute of Technology (CIT) is a teacher and thinker who’s always had a blackboard in the kitchen for the children to work out problems, and who’s always teasing them with puzzles. It was he who enticed Sarah to Tuesday evening Mathematical Excursions classes at CIT when she was 15, who hooked her on the joys of number theory, who worked with her when she hit snags and whose genial, playful method of teaching maths shines throughout In Code.

Beginning with a family history and going on to a series of puzzles, In Code is a paean to intellectual adventure. If you want to find out how to get from Euclid’s theory of prime numbers via Eratosthenes’ sieve, Mersenne’s numbers, Fermat’s little theorem and Euler’s phi function to modern number theory, you can. If you want an introduction to cryptography, from the old Caesar’s wheel (advancing letters against each other) to modern “double key” encryption, which uses very large numbers and non-commutative multiplications - called trap-door encoding - to create safe ways for you to use your credit card on the Internet, it is in here. And it’s all been thoroughly tested on Sarah’s mum.

As well as offering a quick-fire number theory course, the book tells the story of Sarah’s intellectual journey from bright schoolgirl to international sensation. Shortly after starting the CIT evening classes, Sarah embarked on her transition year. This uniquely Irish invention gives students 12 months between the two-year exam grinds of intermediate and leaving certificates to explore extended projects, creative abilities, community service and subsidiary academic interests. It often culminates with work experience.

Sean Foley, science teacher at Schol Mhuire Gan Sm l, always encourages his students to enter the Irish Young Scientist of the Year competitions. The chance of glory (Mr Foley’s son, Vince, has twice won Intel science awards) concentrates students’ minds and develops their study skills. The idea for Sarah to “do something with cryptography and computer programming” came from David Flannery, who also put her on to the huge potential of the computer programme Mathematica. Her initial project on cryptography won an Intel award for science in 1998 and taught Sarah that: “If you really want to learn something, learn it with a view to explaining it to someone else.”

It was also her first solo plunge into the world of maths. “I could never have believed, a few years ago, how much maths I have done, how much time I have spent on maths,” she says, adding: “I was enthralled by the work.” From that project, Sarah went on to work experience with the Dublin office of premier Internet company Baltimore Technologies, where she got the idea of how to speed up the internationally used encryption algorithm, the RSA.

Devised in 1976 by mathematicians Ron Rivest, Adi Shamir and Leonard Adleman, the RSA uses very large numbers and feeds them through several kinds of patterning to make coded messages which would take thousands of years to decipher without a key. But the key to encode is public knowledge. Sarah’s idea was to use matrices, a way of multiplying groups of numbers, as part of the “trap-door”, one-way encoding. The idea, worked on over one summer, was to bring her the Irish Young Scientist of the Year award, headlines screaming of fame and fortune and a range of opportunities, such as a visit from an eccentric American billionaire.

Yet in the end, the algorithm was flawed. It is clever, but it will not make millions, for Sarah or anyone else. “It was horrible,” says Sarah. “It would have been fine - after all, the algorithm was still speculative; exciting, visionary, but not a finished product - had there not been the ‘genius’ tag and everyone saying ‘Sarah can make millions’. I worried that I might seem a fraud.” On the contrary, the scientific community took Sarah to its heart. She spent months examining her work in the light of an intellectual attack, discussing it with her mentors, and concluded that it had a flaw which made it suitable for private but not public key cryptography.

She then talked about the flaw at the European Young Scientist award presentation last summer. “It was the right thing to do,” she says sombrely. “Dad says doing the right thing is hard. And it was.” But, to great applause, Sarah won one of the first prizes - and that invitation to the Nobel Prize dinner which she describes as being “a fairytale, almost”.

Virtue and scientific thinking were rewarded. As for the money, Sarah says with characteristic common sense: “The newspapers said ‘Sarah will not register copyright, and will give her invention for free’. I never said I wouldn’t take the money. Of course I like having money. It’s just that I agree with Dad that you shouldn’t make profit out of intellectual advances. And anyhow, how could we patent it? Everyone had seen it at the Young Scientist display.”

Meantime, as “the more maths I’ve done, the more I’ve become interested,” Sarah says, she is doing her leaving certificate in English, maths, physics, chemistry, German, accountancy, applied maths and Irish. She is also doing English A-levels in maths and further maths. (“There is so much more maths than you learn in school.”) She might like to teach, but she likes to learn more. The failure of her algorithm was “disheartening at first” but the issue is “fairly closed” now. There are her friends, many of whom she’s known all her life, her interests, Cork to support at hurling and football and fun with the family.

She plays ladies’ Irish football for her school, rides her neighbour’s horses every Saturday, plays the piano and listens to her brother play long Jimi Hendrix solos on his guitar “all the time, I love it”. She really enjoyed Jane Austen’s Emma (interestingly, for a cryptographer who transforms information into secret code, she says: “I love the way you know a bit more than Emma all the time”) and Orwell’s Animal Farm. She loves to experiment with clothes and hair, and talks with her friends on the phone.

Then there is the book. Sarah is exhausted by writing it. So are her family. “Every word, every phrase” has been chewed over thoroughly by Mr and Mrs Flannery and Sarah: does the joy of maths come through, is it too personal, is it too difficult? (I would answer: yes, no, repays hard work). And, says Sarah, it’s a relief the writing is over because writing a book is a million times harder than devising a new encryption system and now the boys can have their mammy back, even if they do pull a face at the appearance of their baby pictures in the book.

The Flannerys are profoundly sensible. You don’t need to be a genius to have a happy and fulfilled intellectual life, they say, and anyhow, it’s more important to be yourself, have moral courage to admit when you are wrong, learn from it and go on. Maths is exhilarating, fun, addictive and never better than when you can discuss it with someone else who understands. “It’s thrilling to explain it when someone understands your thinking,” Sarah says. But life is fuller even than this - there to be embraced. As one of her classmates said to Sarah: “Fair play to ya, girl.”

Last Word Lectures and The TES have five pairs of tickets to give away for Sarah Flannery’s LastWord evening lecture, ‘Maths as You’ve Never Seen it Before: learning through puzzles’, at the Royal Geographic Society, 1 Kensington Gore, London SW7, on Wednesday April 5 at 7pm. Be one of the first five to call Colette O’Neill on 020 7782 3269 to claim your tickets.Those who fail to win tickets on the TES phoneline can book for pound;12.50 (pupils pound;5) from Last Word Lectures on 020 7792 9512