WHEN the Tories meet in Blackpool next week for their annual conference, one thing will be on their minds: how to pick a leader who can pull off the next election. The odds on Ken Clarke—favourite with the Conservative associations—shortened, after the party rank-and-file voted on Tuesday against handing over the final choice of the next leader to MPs. But whether the victor is Mr Clarke or David Davis, the bookies' favourite, the party has a Herculean task simply to overcome Britain's electoral arithmetic.

In 1997 Labour won a general election with 43.2% of the votes cast, and gained a majority of 179 in the Commons. In 2005 Labour's share of the vote was only 35.3%, but even so it managed a majority of 66. This was the lowest share of the vote to have brought a single party to power since the partition of Ireland in 1921. Despite doing nearly as well as Labour, with 32.3% of the vote, the Tories won far fewer seats.

Labour does better out of the country's first-past-the-post electoral system thanks to a number of demographic oddities. One is that Labour constituencies are on average smaller than Tory ones. Labour is strong in Wales and Scotland, which punch above their electoral weight in Westminster, and the party does well out of long-term demographic trends, because, over time, voters tend to move out of urban (Labour-voting) areas and into suburban and rural (Tory) ones.

As a result, the average Labour-held constituency contains around 6,200 fewer voters than the average Tory-held one. Constituency boundaries are due to be redrawn before the next election to take account of such things, but this is likely to hand at most another 14 seats to the Tories. And Labour's votes go further than Tory ones because turnout is lower in Labour constituencies—so much lower that that there is an 8,500 difference in votes cast.

Some Tory MPs think that the party's salvation lies in a far more radical redrawing of constituencies. David Cameron, one of the leadership candidates, recently suggested legislating to equalise the size of all constituencies, even if that resulted in some odd geographical entities. It wouldn't make much difference. The Tories' main problem is the large number of votes they waste because their voters are inefficiently distributed: Tory MPs tend to have been elected with considerably larger majorities than Labour ones.

John Curtice, professor of politics at Strathclyde university, says a swing from Labour to the Conservatives of 1.5%, spread evenly across the whole country, would give the two parties identical shares of the vote, but, on current boundaries, leave Labour with an overall majority of 22. It would take an implausibly large swing of around 8% for the Tories to form a government. Anything in between would mean a hung Parliament. The most likely way this electoral arithmetic will change is if the two main parties win back votes from the smaller parties, whose share of the vote, at 30.6%, is at a high.

But there is one intriguing possibility. On a swing of just over 1.5%, the Tories would win the popular vote, but find themselves in opposition. Misery for the Tories, but a fillip for electoral reform.