After a 36-ballot debacle to resolve the 1800 election, Congress proposed the 12th Amendment to set up new procedures for electing a president and vice president. When no candidate receives a majority of votes in the Electoral College (currently, 270 of 538), the House of Representatives elects the president, choosing from among the top three candidates on a one-state, one-vote basis.
In the current House, majority GOP delegations outnumber majority Democratic delegations 34 to 15, with the Minnesota delegation evenly divided. But it's the newly elected House that will elect a president, if the Electoral College splits. Even if Democrats see a net gain of 25 seats and take control of the House – highly unlikely if current polls hold – Romney is nonetheless favored to carry such a House vote, because mainly red states outnumber blue states by more than 2 to 1.
Split-state delegations typically submit blank ballots. It takes a majority of state delegations, or at least 26, to elect a president under these rules.