The children are all sixth formers at state schools and further education colleges in 13 London boroughs with predicted Bs or Cs at A level. The idea behind the Saturday schools is that the extra two hours will give them a leg-up to achieve the grades they need to get into a good university.
The other requirement is that their parents must not have been educated to degree level in this country.
"We are not catering for students who would get good A-levels and would get into good universities anyway," says Hazel Pennell, of the university's centre for educational research, and who helped set up the scheme. "The main aim of the scheme is not to get students into the LSE , it is just to encourage them to apply to university."
The school, which employs state schoolteachers of maths, economics and sociology on a rota so that they can teach their favourite aspects of the syllabus, is funded by the Sutton Trust and John Lyons Trust.
It is unusual in that it also pays children pound;7 to attend to compensate in part for their loss of earnings from Saturday jobs and also because the money to set it up initially in 1998 came from an anonymous donor.
"It came from the kind of person that the scheme was intended for," says Ms Pennell. "He hadn't done A levels at school and was doing evening classes when the lecturer said he ought to apply to the LSE which he did."
The first year's intake was from Barking and Dagenham but the scheme quickly expanded to cover schools in almost half the London boroughs.
Ms Pennell says the scheme's success is as much about raising aspirations as improving grades. "There are students who are maybe scared of universities like the LSE and don't think it's the kind of place for them.
But when they get here they see this is the most diverse university in the country in terms of background - we take students from all over the world.
"If they can show that they are sufficiently motivated to come along every Saturday for two hours that helps with their university application too."