Sixth-form teachers often tell their pupils Oxbridge entry is a lottery.

They are wrong. It is a game, played to obscure and unwritten rules, osmotically understood by those with a lifetime’s exposure to them, but baffling and intimidating to those who have never met them. That is why “access” schemes, as operated by the two universities, will never work.

They just bring a few new horses to the water. They don’t make these strange waters more palatable. Elfi Pallis is angry about this, and lists steps these ancient institutions must take if they seriously want to attract bright students from across the school system.

But that is an afterthought. Most of this lively, fascinating book is spent teaching bright kids from ordinary schools how to play the Oxbridge entrance game.

Pallis, a sociologist and journalist whose daughter is at Oxford, has conducted extensive interviews with students, teachers and admissions tutors. She guides readers through the process, from deciding if Oxbridge is worth trying for, to what happens if a student does - or doesn’t - get in. And she does it with wit, flair and the kind of down-and-dirty detail students, teachers and their families need.

She looks at what applicants must have in their “kit bag” to show they are well-rounded people (arts, drama, music and volunteering are all good), and which A-levels are best for which subjects. Choosing a degree subject is dealt with in detail. Few people know, for example, that fewer than one in five applicants are accepted to do economics or law, while more than half get in to do chemistry or classics. Picking the right college can also be crucial. It might be a good idea, says Pallis, to hunt down one that has not traditionally welcomed state school applicants, but is now being pressured to do so.

Then there is the interview. Oxbridge loves fast mental footwork, which is a disaster for anyone from a culture where it is seen as impolite to challenge elders, or who has spent the past seven years at school hiding their intellect for fear of ridicule.

Pallis explains it all. What type of questions might be lobbed your way.

How to treat the interview like a conversation, rather than a quiz. How to explain why you hold the views you do. How reading a single “key book” - she lists some that have worked for others - can tip things your way.

She talks about dress, accent (don’t change it) and even table manners, including instructions on how to deal with a whole Dover sole. “Oxbridge dons keener on wider participation,” she observes tartly, “might also wish to learn something new. Fish is eaten by most of the population in the form of easily dissected, breadcrumb-covered fillets without bones. Perhaps the hall menu in interview week could take account of this fact.”

She moves on to look at medicine at Oxbridge, and to questions of race, class and gender. She also examines what unites “non-traditional” students who have navigated their way into Oxbridge. Interestingly, many have teachers for parents. From this Pallis extrapolates that parents who understand how schools work, and who can help their children get the most out of what’s available, are the ones who lay the firmest foundations for confident, high-aspiring children.

There are some slips - not all private schools are single-sex, for example, or even most of them - but overall it is excellent. It makes no false promises, and it makes available to everybody the kind of information private schools have had at their fingertips for generations. It is bound to make for more clued-up teachers and parents. And it strips bare the enduring snobberies and arrogance of Oxbridge, in a way Pallis clearly hopes will make many more “ordinary” students determined to step forward and take them on.