This is partly because the number of applicants from all backgrounds is at an unprecedented high. The Government’s various access schemes, funded to the tune of nearly pound;8 million over the last three years, deserve much of the credit.

What the schemes have not done, though, is perceptibly raise the number of state school students getting in, or that of working-class students within that group. In short, many of those who won’t be able to pay are not being admitted anyway.

Not all of this is the fault of Oxbridge dons. Most try to spot talent in even the poorest pupils, although many insist that “working class” is an outdated term. But they do not have any way of knowing what background a student comes from.

Although the Universities and Colleges Admissions Service records the social class of every applicant’s parents, it does not pass on that information. Unless told by the school, admissions tutors can only guess.

It is like trying to hit a target with a sack over your head while doubting that the target is actually there.

Other obstacles to acceptance are inherent in the Oxbridge entrance process, however. The crucial one is its assumption that “the brightest and best”, which the two universities explicitly seek, share a set of characteristics that are easily discerned during the main acceptance test: the informal, unstructured verbal interview. But the reality is that these characteristics, which include intellectual curiosity, a gift for analysis, lateral thinking and determination, can be far from obvious in this setting.

That is why - despite Oxbridge assurances that acceptance is a matter of luck, not preparation - applicants from private schools are coached for months in how to display these characteristics convincingly in interview.

The result can be disappointment for even a brilliant state school student.

And when applications from their school then dry up, the blame is often put on teachers. Some dons perceive them as unreliable suppliers of goods: they fail to answer access circulars, don’t send enough candidates or send the wrong ones.

There are also dark mutterings about anti-elitism. “These people think that Oxbridge is a posh place, so one mustn’t encourage pupils to go there,” I was told by a magisterial academic figure. In fact, one of my strongest impressions was of teachers’ extraordinary efforts to keep up with Oxbridge requirements and persuade hesitant pupils to apply.

Equally obvious was their frustration with recurrent failure. Many of them had formed a connection with a specific Oxbridge college, but found that it did not bear fruit, at least on an annual basis. A college which used to accept two or three students out of a sixth form of almost 400 would suddenly stop doing so. Access funding was tied to fresh initiatives, which encouraged Oxbridge colleges to forge new links rather than rewarding those with whom they already had a relationship.

Government intervention now needs to turn away from increasing applications. Perhaps funding for access projects could initially come from Oxbridge itself, which would then be repaid by the Government if it resulted in an acceptance rate of, say, 30 per cent.

The Department for Education and Skills may also wish to stipulate that all candidates undergo some formal, scientific ability test. Alternatively, of course, the Government could simply impose a new entrance quota, higher than the present, unacknowledged UK one of 55 per cent state school students and 10 per cent of students from working-class homes.

Those Oxbridge academics who insist that such steps would amount to a disastrous intervention in the autonomy of the two universities need only to look to the past for reassurance. It was legislation, or at least the threat of it, which compelled the two universities to accept Jews and Catholics, dissenters and women. A-levels, too, only became an entrance requirement because the Government wished it.

There is no doubt that the academic reputation of both places rose immeasurably as a result.

Elfi Pallis is the author of Oxbridge Entrance: the Real Rules, published by Tell Books