"Just when I thought I was out ... they pull me back in!"
- Michael Corleone, "The Godfather: Part III" 

House Speaker John Boehner deployed a reliable, pragmatic strategy to govern during his four-and-a-half-year tenure wielding the gavel. When it came to non-essential measures, Boehner turned to Republicans to provide the votes.

Yet with the big stuff, Boehner repeatedly assembled small teams of Republicans and leaned on Democrats to usher those bills to passage.

The House of Representatives now faces one of those "must-do" agenda items: the election of a speaker. Republicans are at an impasse in settling on a Boehner successor -- stymied at finding the right vote mixture to elect a speaker on the floor. But this time, Republicans can't bank on Democrats to help. They have to settle this one on their own.

Therein lies the problem. What happens if Republicans can't agree on a speaker?

"I wouldn't be surprised if we don't get a speaker until the next Congress," predicted one Republican lawmaker who asked to not be identified.

Republicans huddle Thursday to select their nominee to put on the floor for a final vote Oct. 29. But a deadlock then would create a sticky situation for Congress -- and indeed for Boehner, who may have to weigh whether he can leave just yet, despite plans to resign at the end of the month.

Understanding the mathematical equation to the speakership is complex yet essential. The political algebra grew more complicated late Wednesday afternoon when the conservative House Freedom Caucus declared it would support -- mostly as a bloc -- Rep. Dan Webster, R-Fla., for speaker, in a blow to front-runner House Majority Leader Kevin McCarthy, R-Calif.

Here are the mechanics:

Though Republicans will nominate their candidate Thursday, someone can only become speaker if elected by an absolute majority of all House members who vote for a candidate by name. 218 is generally considered to be the magic number.

McCarthy certainly enjoys the lion's share of the vote -- somewhere around 200 supporters. But at least 30-40 votes could go to Webster since the Freedom Caucus threw its support behind him. That could mean House Oversight Committee Chairman Jason Chaffetz, R-Utah, brings up the rear with just a handful of votes or more.

It's doubtful McCarthy can command 218 supporters in the conference. Indeed, it's daunting for any candidate to hit 218 votes on the floor. Consider that Republicans consistently leaned on Democrats to support major initiatives, but Democrats can't help Republicans get to 218 now.

McCarthy downplayed the potential vote gap he's expected to face. "You don't need to get to 218 until you get to the floor," McCarthy said.

But it's unclear if McCarthy can move some conservatives.

Further complicating this is yet another noxious vote to hike the debt limit by Nov. 5. Boehner desperately wants to craft a package to lift the debt ceiling and address a cavalcade of other fiscal issues. But Boehner is mindful not to leave the next speaker scrambling to raise the debt limit during their first hours in office.

The House is prohibited from conducting additional legislative business until it taps a speaker. That sets up the possibility of a brutal congressional collision course: a vacancy in the speaker's chair draped against a debt ceiling emergency.

There is increasing chatter on Capitol Hill -- and it's just that, chatter -- that Boehner may have to stay as speaker for a time if the House can't agree on a successor. Moreover, some argue that only Boehner can forge a coalition of some Republicans and Democrats to stave off a possible fiscal shock if Congress fails to raise the debt limit.

Those close to Boehner assert he is out the door as of Oct. 30, no matter what. Other Boehner confidantes indicate the Ohio Republican knows better than to leave the House in a lurch if members can't agree on a replacement.

There is precedent for this.

In 1989, the late-House Speaker Jim Wright, D-Texas, tendered his resignation contingent upon the House selecting a new speaker. Wright's resignation became effective "on the election day of my successor."

If the House remains gridlocked in selecting a new speaker, it's also possible a unique, post-9/11 provision -- drafted to assure continuity of government in case of national emergency -- could kick in.

House rules state that "In the case of a vacancy in the Office of Speaker, the next Member on the list described in subdivision (B) shall act as Speaker pro tempore until the election of a Speaker."

This is new ground. It is unclear if a "vacancy" is conditional, created by an act of terrorism. It's also unclear if "vacancy" applies to all situations.

That said, the current House speaker is supposed to have a top-secret list, presumably composed of current members and non-members who are qualified to serve as a speaker. There's speculation as to whether McCarthy is on that list -- or perhaps others, which could add an utterly fantastic twist to the entire state of the affairs ... like former House Majority Leader Eric Cantor, R-Va., or former House Speaker Denny Hastert, R-Ill.

Boehner even joked about staying longer during a GOP leaders meeting Tuesday. When asked Wednesday, Boehner wouldn't commit to lingering if the House failed to elect a speaker.

"There's going to be an election for Speaker on October 29th," said Boehner. "I am confident that we'll have a new speaker on that day."

In the meantime, McCarthy has to make inroads with conservatives.

"Whoever gets nominated by this conference has to have leadership skills to pull 218 members to support him. If he can't get 218 himself, it's going to be a problem," said Rep. Blake Farenthold, R-Texas.

This is ironic. Democrats usually provide the necessary votes on big issues. And while the entire House votes for speaker, avoiding a leadership vacuum is one crisis Republicans must solve by themselves.

Capitol Attitude is a weekly column written by members of the Fox News Capitol Hill team. Their articles take you inside the halls of Congress, and cover the spectrum of policy issues being introduced, debated and voted on there.