Maine does not like to do things by the book. The state’s officeholders include two fairly conventional liberal Democrats, Representatives Chellie Pingree and Mike Michaud; two extremely moderate Republican Senators, Olympia Snowe and Susan Collins; and a conservative, Tea Party-backed governor in Paul R. LePage.

The state has also been fertile territory for independent candidates in the past. An independent, Angus King, was its governor from 1995 to 2003. And in 1992, Ross Perot received a higher share of the vote there than in any other state. Perhaps only Alaska shares Maine’s history of ignoring the left-right spectrum of politics when a candidate suits its interest.

That could represent an opportunity for Representative Ron Paul, the only remaining Republican candidate who has yet to win a state, when Maine announces its Republican caucus results on Saturday night.

Mr. Paul has spent considerable time in Maine — in fact, he departed the campaign trail early in Florida in order to visit it. He had also performed relatively well in the state’s caucuses in 2008, when he received 18 percent of the vote.

Although 18 percent would not be enough to win the caucuses, Mr. Paul has so far considerably overperformed his 2008 figures in every caucus state. In Colorado and Nevada, he improved his share of the vote by 40 percent, in Minnesota by 74 percent, and in Iowa by 116 percent.

This allows us to extrapolate a reasonable range of figures for how Mr. Paul might perform in Maine this year. If he improved on his 2008 vote share by 40 percent, as he did in Colorado, Mr. Paul would finish with about 25 percent of the vote in Maine. But if he improved on it by 116 percent, as he did in Iowa, he would get 40 percent of the vote there, very possibly enough to win.

The other factor helping Mr. Paul is that turnout is likely to be extremely low — so voter enthusiasm could go a long way. Only 5,431 Republicans cast votes in the presidential preference poll there in 2008, an exceptionally low total even considering the state’s small population. A personal appearance that might motivate another 50 or 100 Republicans to caucus makes quite a lot of difference in that context.

(Like most other Republican caucus states, Maine conducts two separate votes at its caucuses: a nonbinding straw poll for presidential preference, and another vote where delegates are selected to regional conventions. The eventual delegate allocations are likely to bear some resemblance to the straw poll results, but it will be an imperfect one.)

The other favorite in Maine is Mitt Romney, who has made a late set of appearances in the state. Mr. Romney won the Maine caucus in 2008, getting 52 percent of the vote in the straw poll.

However, Mr. Romney might not have a natural advantage in Maine just because he comes from New England. Maine shares some affinity with New England in the sense of rooting for the Boston Red Sox and liking its clam chowder. But its demographics are somewhat different. It is more working class than the other New England states, particularly in the winter when the tourists leave. About a quarter of its residents claim French-Canadian or Acadian ancestry. Like Vermont and New Hampshire — but not Massachusetts or Connecticut — it is rural and can have an anti-establishment streak.

Although there have been no recent polls of Maine, Mr. Romney did relatively poorly in surveys of registered voters there in 2011. And he has raised just $77,000 there, much less than the $290,000 that he raised in New Hampshire.

Mr. Paul’s name has gotten about five times more Google search traffic in Maine than Mr. Romney’s in the last 30 days, an indicator that has had some predictive power in other states. (Although Mr. Paul gets a surprisingly large amount of Google search traffic everywhere, the candidates are much closer together elsewhere in the country.)

Rick Santorum, meanwhile, is unlikely to win Maine, even though he is surging nationally. Maine is not as socially liberal as the rest of New England — in 2009, its voters narrowly repealed a same-sex marriage initiative that its legislature had passed. Still, its preferences run more toward the libertarian side than to Mr. Santorum’s version of conservatism.

In addition, the state’s caucuses have been conducted over a rolling period — some areas vote on Saturday, but others voted last weekend, before Mr. Santorum’s surge in the national polls. Given the unpredictability of the results so far, I suppose that we should not rule anything out — but if Mr. Santorum were to win there, or come close to it, it would be a sign that few states were off-limits to him.

In all probability, however, it is Mr. Paul and Mr. Romney who have the most on the line. If Mr. Paul fails to win in Maine, perhaps only Alaska offers an equally good opportunity to get on the map.

Meanwhile, Mr. Romney could regain some momentum with a win. For the reasons I have outlined, Maine presents a mix of favorable and unfavorable characteristics for Mr. Romney; winning there would be a real accomplishment. But a loss there, also, would be a signal that he is starting to fall off the track that would lead to his getting a majority of delegates — and leave him with just one victory in five February contests, a month that was thought to favor him.

The state will announce the straw poll results at 7:30 p.m. on Saturday.