The House is back for its first business day after a 10-day break, and the first item of business is a big one: finding a leader.

Speaker John Boehner has said he is resigning at month’s end. The Republican conference met to choose a successor, but Boehner ended the session when his No. 2, Majority Leader Kevin McCarthy, withdrew as a candidate.

There’s no shortage of potential candidates. After all, the post is the most powerful position in Congress and second only to the vice president in White House succession.

But none of the declared candidates has anything like the 124 votes needed to be nominated in the Republican conference, much less the 218 needed to win a vote on the House floor. (See an explanation of the vote here.)

So most of the talk has been about one man who says he does not want the job. That would be Paul Ryan of Wisconsin, who by all accounts loves the job he has — chairman of the Ways and Means Committee.

What Ryan does not love is the prospect of dealing with the Freedom Caucus, a core group of confrontational conservatives who made Boehner’s life miserable enough to make him retire as speaker.

The Freedom Caucus has a list of changes in House rules and procedures that it wants prospective speakers to endorse and embrace. Neither Ryan nor any member of the current leadership team has been willing to accept the job under these conditions. The caucus’s favored candidate, Daniel Webster of Florida, would presumably do so, but he has limited support beyond the 40 or so members of the caucus itself.

The conference is to meet Tuesday night and again Wednesday morning to discuss its way forward. No date has been set for another vote on a candidate.

So what will happen? At this point, it is clear that no one knows.

Here are six possible scenarios, in approximate order of plausibility.

1. Ryan runs and the right relents

Many Republicans both inside and outside the House have been urging Ryan to save the situation before it interferes with crucial House business or bruises the party’s image with swing voters. Ryan has been a good party soldier both before and since his nomination as Mitt Romney’s running mate in 2012. (Romney has been among the most vocal on Ryan’s behalf.)

If he gives in to the pressure, he will probably be able to unite roughly 200 votes on Day 1. But he will need to add at least half the Freedom Caucus to ensure 218 on the floor. And Ryan has said this number is not enough. He has said he would need something closer to a unanimous GOP vote to feel confident the job was truly his and he could execute its duties.

For that kind of margin to happen, the Freedom Caucus would probably have to back off its key demands. Speaker Ryan could offer an olive branch by agreeing to some changes and setting up a study committee to look at the rest.

2. Ryan runs and the right resists

If Ryan runs without settling with the Freedom Caucus, he will face a firestorm of criticism not only from that group but from the dozens of conservative and ultraconservative bloggers, talk show hosts, fundraisers and activists who began sharpening their knives for him last week. Ted Cruz, the senator from Texas running for president, took a shot at Ryan for past support of budget deals with Democrats and for supporting a comprehensive immigration reform bill.

It is conceivable Ryan could brave these slings and arrows, face down his critics and run for speaker without the support of the Freedom Caucus. After a series of votes behind closed doors, the Republican conference might find itself without options other than to accept Ryan on his own terms. There would surely be support for this from most Republicans.

3. Ryan refuses, Boehner remains

If Ryan says no, there is at present no fallback establishment candidate. The Freedom Caucus can stand behind Webster, despite his lack of seniority, low national profile and the real danger his district in Florida will be redrawn before the next election. Another champion of the right might emerge in his stead, but none is apparent. The Freedom Caucus chairman, Jim Jordan of Ohio, has staunchly refused to seek a leadership office. So have some of the other outspoken members of the group such as Tim Huelskamp of Kansas. Knowing they would not be elected, they prefer the power they now wield by withholding their votes en bloc.

Under this scenario, it will simply not be possible to hold a floor vote on Oct. 29 as originally planned. And it will be similarly difficult for Boehner to leave. He has said he will not leave the House without a speaker. If the full GOP conference has not chosen a nominee by month’s end, it appears Boehner will simply have to delay his retirement.

At that point, will the Freedom Caucus back down and allow him to run the show? Or will it continue to threaten him with a revolt on the floor? It was just such a threat in September — prompted by his plans to pass a spending bill with Democrat votes — that led Boehner to announce his resignation.

4. The House finds a new star

If Ryan says no, perhaps a dozen candidates will spring forward. Perhaps more. All will lack the necessary votes to be nominated or elected, but in the melee that will follow, someone might catch fire. It could conceivably be a member of the current leadership team such as the current majority whip, Steve Scalise of Louisiana, or the conference chair, Cathy McMorris Rodgers of Washington.

Or what about this? When the Select Committee on Benghazi holds its climactic hearing with Hillary Clinton on Thursday, someone emerges as her most effective critic. Suddenly everywhere on TV and social media, that person would be an instant star in a House badly in need of a new hero (or heroine). The panel’s chairman, Trey Gowdy of South Carolina, is an obvious prospect for this role. Several well-known members have already urged him to step forward, including the current speaker himself.

5. The House turns to an outsider

The Constitution provides for the House to be run by a speaker but it does not require that person to be a member of the House. No outsider has ever won the job, but there could be a first time. The names of several icons of American conservatism have been mentioned — Dick Cheney and Newt Gingrich among them. But the notion of such a person reaching 218 votes, including at least half the Freedom Caucus, is probably fanciful. And no such person has shown a willingness to play ball, at least so far.

A somewhat more likely scenario would be a kind of caretaker speaker chosen from outside with the understanding that he or she would serve for a limited time only. Such a person might even be elected with some support from Democrats. But again, there is no precedent for such a procedure.

6. Boehner leaves House in chaos

No one is predicting this. But if Boehner fulfills his plan to depart at month’s end, no one automatically takes the big gavel in his place. There is no succession other than by election by vote of the full House. Boehner could designate a temporary substitute, as is done routinely to conduct the House’s business, but that person’s authority would come from Boehner and expire with his departure.

This would mean the House could not proceed with its regular order of business, pass bills, respond formally to the Senate or negotiate with the president. It would mean the Congress as a whole could not make laws.

UUnpopular as Congress has become, with the lowest approval ratings in the history of polling, this option would have to be considered even less attractive than the status quo.