It was Sunday morning, so they started with a prayer, along with a re-worked version of the hymn Amazing Grace by a country music band. In a way, that set the stage perfectly for the arrival of Mitt Romney, a candidate whose whose final hours on the campaign trail revolve around a big leap of faith.

The Republican nominee was bright-eyed, bushy-tailed, and had his sleeves rolled firmly up as he greeted a small but energetic crowd in the swing state of Iowa. Then he rattled optimistically through what is now established as his "closing argument" to the American people: that he, and not Barack Obama, is the true candidate of change.

"The question of this election comes down to this," Romney said. "Do you want four more years like the last four years? Or do you want real change? President Obama promised change, but he couldn't deliver it. I have promised change and I have a record of achieving it."

Romney will deliver a similar sales-pitch throughout this final stretch of the election race. And he will stand at lecterns emblazoned with the logo: "Real Change From Day One." But behind the scenes it is perhaps not "Change" that is now fuelling his campaign, but another term successfully co-opted by Barack Obama four years ago: hope.

The Republicans must hope that their rival's narrow, but solid lead in almost every swing state poll is misleading. They must hope that data showing a slight decrease in early voting by Democrats heralds a wider shift in public sentiment. And most of all, they must hope that a small, but crucial slice of the electorate can yet be tempted to make a late switch.

It's these presumably-moderate voters that Romney was targeting today when he talked about "reaching across the aisle," and promised that when he takes the White House: "I won't just govern for one party, I'll govern for one nation." It was their ears that he hoped to prick when he asked voters not to "settle" for more of the same.

"Obama has tried to convince you that the last four years have been a success," Romney told the Iowa crowd. "So his plan for the next four years is to take all the ideas from the first term. The stimulus, borrowing, Obamacare and the rest, and do them over again. He calls that plan 'forward.' I call it 'fore-warned.' The same course that we've been on will not lead to another destination."

In the room, they lapped it up, banging inflatable "thunder-sticks," waving stars-and-stripes flags, and cheering and booing at all the right moments, like the audience at a Christmas pantomime. At Obama's rallies, they've lately been chanting "four more years." Mitt's crowd in Iowa today offered a riposte: "two more days."

Team Romney retains similar enthusiasm. His strategists remain publicly convinced they can win this thing, arguing that the President's lead is soft, and vulnerable to a late collapse. Indeed, they are so bullish about their prospects that Romney this afternoon swept into Pennsylvania, a theoretically safe Obama territory which has barely been contested so far.

The blue-collar State carries 20 electoral college votes, and a Romney victory there would upend projections about the outcome of this race. But it also has large urban and ethnic minority populations, who in theory are solidly Democratic. Many analysts see his late play there as a sign of desperation.

For Pennsylvania to turn red on Tuesday, every poll would have to be out, by a historic margin. But for Romney to make it to the White House, the same would have to be true. His chances of re-election are now 16/5 according to the betting exchanges. Those odds equate to roughly 31 percent.

Tomorrow, the pursuit of that small, but still real possibility will bring a last, frantic push across the country. Romney begins the day in Florida, before passing through Virginia, Ohio, and Wisconsin. After clocking up several hundred more air miles, he'll wind up in New Hampshire, at an election eve "victory celebration" headlined by the musician Kid Rock.

Win, or lose, it will have been a long, hard road. "We've journeyed far and wide in this great campaign for America's future, and now we're almost home," began his closing statement. "One final push will get us there. The door to a brighter future is there, open, waiting for us. I need your vote. I need your work. I need your help. Come! Walk with me! We'll build the future!" Romney hopes, in other words, that against the apparent odds his greatest adventure is yet to come.

The election in numbers

270 Electoral-college votes needed for victory

95 The number of electoral college votes at stake in the key swing states of Iowa, Wisconsin, Ohio, New Hampshire, Virginia, Florida, Colorado and Nevada

1956 The last time Ohio backed the loser in a presidential election. The state has seen more visits from the candidates than any other during this campaign

$52m Barack Obama’s advertising bill in Ohio

$347m Obama’s total advertising bill

$852.9m Obama’s total campaign spending, from a total fundraising pot of $1.076bn

$32m The amount the Republicans have spent on a publicity blitz in Ohio

$386m Mitt Romney’s total advertising bill

$752.3m Romney’s total campaign spending, from a total fundraising pot of $1.125bn

100m: Expected number of votes that will be cast in the 2012 election

7.8%: US unemployment rate, the same as when Obama took office, below the symbolic 8% threshold. The rate touched 10% in 2009 at the height of the financial crisis

$1.1trn: US budget deficit, up from $438bn in January 2009

67,000: Troops currently in Afghanistan

34,400: Troops in Afghanistan when Obama took office in January 2009

173,900: Troops in Iraq when Obama took office in January 2009

200: Troops in Iraq today

26m: Number of people who have cast their votes early, including 3.5m in Florida and 1.6m in Ohio. Early voters traditionally favour the Democrats

52% of men polled by The Wall Street Journal said they would vote Romney