PERCHED on a river bend in an unfashionable expanse of central France, Châteaudun is in many ways a typical French town. It boasts a 15th-century chateau, an unemployment rate of 10%, a fine main square shaded by plane trees and a Turkish kebab restaurant. This town of 13,000 inhabitants also happens to have a record of voting in line with the rest of the country. In 2007 locals backed the winner, Nicolas Sarkozy, on the right. In 2012 they voted for the victor, François Hollande, on the left. Today, as the first round of this year’s presidential election approaches on April 23rd, voters once again seem to reflect the national mood.

“I’m perplexed,” says Bertrand, a pensioner shopping on the main square, who voted for Mr Sarkozy in 2012 but has yet to make up his mind this time. He thinks François Fillon, the centre-right candidate who is under investigation for abuse of the parliamentary payroll, may be “competent” but has behaved “disgracefully”. Bertrand’s wife Geneviève, a retired librarian carrying geraniums from the market, voted for Mr Hollande last time. But she dismisses the Socialist candidate, Benoît Hamon, as “Utopian”. She says she is tempted to vote for Jean-Luc Mélenchon, a 65-year-old Communist-backed firebrand, who vows to bring about a “citizens’ revolution”, take France out of NATO and impose a top income-tax rate of 100%.

Candidates who promise to overturn the system have captured the imagination. Didier Renard, a retired construction worker, declares unabashedly that he will vote for Marine Le Pen, the candidate for the anti-immigrant National Front: “She’s the only one who will help people like us.” In a town that lost a big electronics factory a few years back, disillusion is marked. Nobody respects any of the candidates, says a woman running a fruit-and-vegetable stall: “People are totally fed up.” So much so, growls a man with tattoos enjoying a morning beer at a terrace café, that he refuses to vote. Alain Venot, Châteaudun’s centre-right mayor, who was first elected back in 1983, says that he usually has a good sense of how his town will vote, but not this time: “This is the most uncertain presidential election I have ever known.” This urge to back an insurgent matches national trends. In a matter of weeks, Mr Mélenchon, who has a big YouTube following and attracts voters to some rallies to watch his hologram beamed in live, has surged in some polls from fifth place to third, overtaking Mr Fillon. He trails only a few points behind Ms Le Pen and Emmanuel Macron, a pro-European liberal who founded his En Marche! party just a year ago. If these polls are right, candidates from non-traditional parties are set to capture the top three places. Guillaume Kasbarian, the En Marche! representative for the area around Châteaudun, says there is an edge for the candidate who sounds the most dégagiste—that is, the most eager to throw the bums out. Two elements make predicting the results especially precarious. One is turnout, which averages about 80% for presidential votes. Polls suggest that it might drop to as low as two-thirds this year, which could further damage traditional candidates and help Ms Le Pen. Equally unusual, only 60% of voters say they are sure of their choice, a figure that is highest among Ms Le Pen’s voters (76%) and low among those who back Mr Macron (55%). This not only hints at the fragility of Mr Macron’s vote. It also leaves a big chunk of volatile voters close to voting day, possibly ready to vote tactically depending on the final polls. Last-minute deciders, says Edouard Lecerf, of Kantar TNS-Sofres, a pollster, used to reflect the national averages; this time, they may not.

Until recently, the odds were clearly on a run-off between Ms Le Pen and Mr Macron. A former Socialist economy minister under Mr Hollande, Mr Macron has reinvented himself as an insurrectionary leader, promising to break down old divisions between left and right and inject new life into politics. Indeed, at the Châteaudun weekend market, En Marche! was the only party out campaigning. Sophie Zeugin, an entrepreneur and volunteer, toted red, white and blue balloons and a basket of leaflets. Locals called the 39-year-old candidate “sympa” (nice). But one worried that he might be “an opportunist”. Another, testifying to the depth of a nation’s indecision, said she was hesitating between the campaign’s polar opposites: Mr Macron and Ms Le Pen.

It may yet be that Mr Macron keeps his lead and makes it into the run-off. There, in all likelihood, he would meet—and beat—Ms Le Pen. But this election has begun to look like a four-horse race. No scenario can be ruled out, says Jérôme Fourquet of Ifop, a pollster. This includes the possibility that Mr Fillon makes a comeback, thanks to disillusioned voters who claim they will not vote for him but could change their minds on polling day. Or even what Mr Fourquet calls the “craziest” possibility: that Mr Mélenchon squeaks into the run-off. In 2002 Jean-Marie Le Pen, Ms Le Pen’s father, made it to the second round with just a half-point lead over the third-place candidate.

The French have sprung electoral surprises before. They voted against a draft European constitution in 2005. In recent presidential primaries, on the right and the left, they kicked out the favourites, eliminating a former president (Nicolas Sarkozy) and two former prime ministers (Alain Juppé and Manuel Valls). This time, three-quarters of voters could be about to back a candidate who hails from neither of the two political groupings that have run France for the past 60 years. This has already been the most unorthodox French election ever, but even more improbable twists may be yet to come.