HOW do you square 27 very different and competing circles?

That was the conundrum EU officials tried and failed to solve in Brussels.

With 27 countries able to wield a veto, any agreement had to obey a fundamental rule: that everyone would be able to claim a win – or fail.

For the UK the sticking points were many and varied. Despite offers of further cuts, and a brief flurry of hope on that issue during a lunchtime meeting yesterday, ultimately they were still not close enough to achieve David Cameron's stated aim of forcing the EU to freeze its budget.

But there were many other issues affecting the UK's decision. The proposed deal would still have seen money cut from the UK's rebate – an outcome the Prime Minister had threatened to veto on Wednesday.

And the lack of even a tiny cut in the administration budget would have been seen as a humiliation for Mr Cameron after he made such a play of the issue on Thursday, the first day of the talks.

In the final reckoning, it might be considered impolitic for the Commission not to even offer a fraction of what the British leader had called for, especially as his stance was backed by a number of other countries.

For all the summit's 40 hours of negotiations the leaders spent just a handful of those talking together.

The proposals were just too far from what many wanted. A number of countries were disappointed, along with the UK, that the cuts to the headline figure did not go far enough.

But at the same time the Poles and those who get more out of the EU budget than they put in, such as Lithuania, were concerned they had gone too far.

France also protested against the potential scale of the cuts and was pushing for the restoration of scaled-back farm subsidies.

Those who wanted to reform the complicated system of payments, of which the UK's rebate forms a part, included France and Germany.

For the main part, however, Germany was aligned with the UK in arguing for the budget cut.

The UK-German axis between Mr Cameron and Angela Merkel was so close that the pair were quickly dubbed Camerkel, evoking memories of the Merkozy tag applied to the German Chancellor and former French president Nicolas Sarkozy during the eurozone crisis.

The positioning bodes well for Mr Cameron when all 27 EU leaders meet next year to try to thrash out another agreement.

Traditionally, EU summits like these have been won by the French and the Germans who have banded together.

With Chancellor Merkel on side, Mr Cameron might just get his way yet on freezing the EU budget.