There is an injustice haunting exams in England, Wales and Northern Ireland. No one mentions it because no one has any clue how to solve it. Last week, however, a thought struck me. Let’s see what you think.

On Thursday, thousands of young people will log on to computers and find out their A-level grades. We already know that pupils from state schools will receive proportionately fewer top grades than those in private schools. But what most people don’t know is that a bit of the exams process that happens after pupils have received their grades makes the inequity just that tiny bit worse.

Exam grades are subject to appeal. If a school has enough cash, it can return a paper to be marked again. And tests completed by the exams watchdog, Ofqual, show that when examiners re‑mark they are more generous than they are first time around. The presumption is they know the stakes are higher and are subconsciously more cautious.

Hence, re-marking doesn’t just buy a second spin – it buys an easier ride.

The cost of exam appeals – about £50 each – puts schools off sending back too many. Money is returned if the score is changed, but if it’s not, the cash is gone. Hence, most schools are careful about which re-marks to allow. Parents and pupils have little say in the process: it is for schools to decide how many papers will be marked again.

Private schools, with the ability to recoup the cost from parents via invoicing, are less affected by the cost barrier. They are more able to ask for re-marks – and they do.

Last year, 13% of exams taken at independent schools were re-marked, compared with 8% in the state sector. Some subjects differ widely. For example, one in five French exam papers sat at independent schools last year were re‑marked. In the state sector the figure was just one in 20.

In an attempt to limit the problem, Ofqual has ordered exam boards to re‑mark in a less generous way. It makes score changes less likely so, in theory, the £50 gamble is less worthwhile. But those with limited budgets are the most risk averse. As it stands, private schools – with the ability to gamble even at higher risk – are likely to continue piling in for appeals.

Ultimately, as long as exam appeals have a price attached to them, richer schools will always have an advantage.

So, how to solve the problem? Here’s a possible solution.

The process needs to be entirely free to schools and students, but re-marking should be allowed only if an independent person agrees it is justified. Independent assessors – teachers or examiners working as self-employed consultants – would be asked by schools or pupils to check over papers. Only if the assessor is willing to sign it through to an exam board would it be re-marked. But here’s the twist: the assessor stumps up the initial £50. If their punt on a re-mark is wrong, they would lose out. But, if their hunch is correct, and the exam mark is changed, they would be paid, say, £150 by the exam body. A smart assessor, with good judgment and a quick eye, could make a tidy sum.

Exam boards will probably shriek at these payouts. But the current situation is unfair. At the moment, if schools make a wrong call, they lose £50. If exam boards are wrong, they just have to pay the £50 back. Sure, they re-mark the exam paper. But they should have got the job right in the first place. There needs to be a higher penalty than re-doing the job free of charge.

Practically, the independent assessment would be easy to arrange. Exam papers are already uploaded and marked online. From next summer, one of the largest exam boards, Pearson, will share marked exam papers with students on the morning of their results. These online papers could be opened so schools can forward them to independent exam checkers who scour papers flagged for review and send through those they were willing to bet on.

Examiners could then start re-marking immediately, speeding up the whole process. In 2014, students waited an average of 10 days to get the re-mark results. After that length of time university accommodation is often full and places are lost.

From the student and school’s perspective, this system has the benefit of being free. From a social justice perspective, it means everyone has the same access to a re-mark. From a cost perspective, it penalises the exam boards who mess up most.

It’s not perfect. There are kinks in the plan. But it would at least reduce the injustice we know exists yet dare not stare too hard at, lest it defeat us with its intractability. It’s got to be worth a pilot study at the very least.