His children use the "modern art installation" as an impromptu goal, but it also serves as a reminder of the constant need for the Prime Minister to keep thinking of radical solutions to the unprecedented economic and political challenges he faces.

Many in his own party are increasingly of the opinion that Downing Street and the Government's key figures may be failing to rise to the challenge.

Britain is back in recession and the crisis in the euro looks likely to condemn this country to a lost economic decade. Mr Cameron does not "see a time" when the austerity programme will end.

The budget was regarded as a presentational disaster and George Osborne, the Prime Minister's key lieutenant and ally, is facing questions over his position that seemed unthinkable last year.

Mr Cameron is struggling to persuade even his own MPs to support government policy and there is a growing clamour for a more Right-wing agenda.

The relationship with the Liberal Democrats appears to be in a state of permanent limbo with relaunches for the Coalition an almost monthly occurrence. After a frenetic first 18 months bringing radical new policies for welfare, schools and health, Downing Street aides talk of "mid-term blues" as Labour remain consistently ten points ahead in the polls.

TORY DISQUIET WITH THE LIB DEMS

Mr Cameron says he understands the grumbling and that even he becomes "frustrated" that he cannot always get his own way. But he urges his supporters and the public to give him time to see the fruits of the Coalition's work.

"If you're in government, you don't get everything you want," he says. "I'm the Prime Minister; I don't get everything I want. Sometimes you have to, we have to, hammer out compromises around the Cabinet table, or in your department. And, if you lose the argument, you have to support the position.

"Obviously it's been a very tough time to be in government but I think the most important thing is we've given Britain a secure plan for getting out of the debt and the deficit and the difficulties that we inherited.

"The sense I get of the mood of the British public is of very fair-minded people: they say it's very difficult, they're quite pessimistic about the state of the world and the state of the world economy and their own finances. They know it's tough, they know it's difficult, they know that none of these things are going to be easily done."

Mr Cameron appears to have some sympathy for the dozens of Conservative MPs who defied the whips to vote against policy on Lords reform and Europe.

Although the rebels are highly unlikely to be offered government jobs in the reshuffle expected this autumn, the Prime Minister indicates that their careers have not been forever blighted.

"If no one who ever voted against the Government [got a] ministerial office ... [Iain] Duncan-Smith wouldn't be in the Cabinet. I think politics has changed, there are people who come with more strong and independent views. It's difficult in coalition because sometimes you're coming forward with proposals that your coalition partners are more enthusiastic about than you are." The comments are likely to intensify Lib Dem suspicions that the Prime Minister is not really intent on a hard-line approach to ensure that the Coalition's policy deals become reality.

THE FUTURE OF THE COALITION

Mr Cameron has obviously been dedicating a lot of time to thinking about the changing nature, and possibly form, of the Coalition over the next three years. Key decisions on airports, wind farm subsidies and social care have been delayed in recent weeks, and future assumptions about how the relationship will function are being reconsidered.

For example, the Coalition was due to agree a programme for public spending from 2014 to 2017. Discussions are due to begin later this year or early next year, but the prospect of a wide-ranging, three-year deal now seems remote. Instead, the Prime Minister suggests a shorter agreement may be under consideration.

"We talked about the need to have one. But how many years it runs for, how exactly it works, we're going to spend more time talking about that. I think what matters is taking the big decisions about the future. I much prefer to approach it from the decisions, and then work out how we are going to [do it], how many years. I want us to make some very big decisions about housing policy, welfare policy, how we fund infrastructure. Let's work out how these decisions trade off against each other."

He also indicates that the Coalition may function differently in the year before an election. His preference is for the current arrangement to run to May 2015. But asked whether it may switch to "confidence and supply", where Lib Dems would not vote to bring the Government down but would not support a policy agenda, he replies: "I think the Coalition will run to 2015. How exactly you separate before an election and fight an election