The answer is 178 and a half hours.

The question is: What’s the maximum amount of time it could take to secure the confirmations of all six prominent nominees President Barack Obama wants to get on the job in the new year?

Only one of their timetables has been set, and it’s likely to be the exception that proves the rule: On Monday afternoon, senators will spend just 30 minutes “debating” the virtues of Patricia Ann Millett, a prominent 50-year-old Washington appellate litigator, before confirming her for a seat on the U.S. Court of Appeals for the District of Columbia Circuit.

However she distinguishes herself during that lifetime appointment, Millett will be remembered by congressional historians for this: She’s the first person to benefit from the limitations on Senate filibuster rules muscled through by the majority Democrats three weeks ago.

Since Millett was the nominal subject of five dramatic roll calls during the parliamentary maneuvering that put the “nuclear option” into effect — lowering from 60 to a simple majority the number of senators required to cut off debate on almost all nominations — Republicans agreed to not delay her final vote for the 30 hours they still have available for such protests. The duration of the Thanksgiving recess, they conceded, would suffice.

But the GOP minority has not decided how much of a fuss it will make about the other five: Federal Reserve Vice Chairwoman Janet L. Yellen to take the helm of the central bank, former top Pentagon lawyer Jeh Johnson to be the fourth-ever secretary of the Department of Homeland Security, Democratic Rep. Melvin Watt of North Carolina to run the Federal Housing Finance Agency and, for the two other vacancies on the D.C. Circuit, Georgetown law professor Nina Pillard and federal trial Judge Robert Wilkins.

Each may be slowed, but no longer can any of them be stopped. Minority Leader Mitch McConnell will use Tuesday’s weekly caucus lunch to press for a consensus game plan.

The challenge is how to honor the desire of many Republicans who want to protest their profound anger at seeing one of their most potent powers stripped away, while avoiding a pair of pitfalls. First, they are fearful of reviving the storyline about “GOP obstructionism,” which could divert the attention they want focused on the troubled rollout of the health care law. Second, they are wary of looking as though they’re imperiling passage of  a potential sequester-relief budget agreement, a stopgap farm bill extension and a stripped-of-controversy defense authorization measure.

In addition, not all that much time on nominations could guarantee a grouchy finish to a desultory year, with late-night roll calls, this coming weekend spent at the Capitol and adjournment delayed beyond the current target of Dec. 20.

The more dilatory the Republicans decide to be, the more logistically punitive Majority Leader Harry Reid is likely to be in return.

Such mutual brinkmanship is pretty standard fare for the final days before the end of a session. Usually, the threats give way to some mutual accommodation that permits the losers to save some face, and allows everyone to make their flights home. But that routine hasn’t been tried since the most important changes to the way the Senate works in four decades, so it’s not easy to predict what it will take for the minority party’s righteous bile to get vented — and then mopped up.

Of the big five nominees remaining on Reid’s agenda after Monday evening, the GOP is likely to be most retaliatory toward the three they stopped earlier in the fall under the old filibuster rules.

As justification for the almost-unheard-of snub of denying confirmation to a sitting member of Congress, Republicans say Watt lacks the technical expertise necessary for overseeing Fannie Mae and Freddie Mac. But they also know the 21-year congressman would push an Obama regulatory agenda they disdain. Republicans could force a cloture do-over, which they could drag out over a series of roll calls. But after that, thanks to a little-remembered Reid-McConnell deal in January that limits debate for some executive branch nominees after cloture is invoked, opponents could delay Watt’s confirmation for only eight additional hours.

The GOP argument against the judicial nominees is that the D.C. Circuit doesn’t have enough work for them to do, but it’s also undeniably true that filling all 11 seats would alter the ideological balance of what’s properly called the second most important bench in the nation. Even if McConnell is willing to accept the inevitable, conservative rhetorical warriors such as Ted Cruz of Texas and Mike Lee of Utah will be tempted to insist on watching the clock run for a combined 60 hours before allowing final votes on Pillard and Wilkins.

Johnson has a solid reservoir of GOP support, although Arizona’s John McCain wants to block confirmation until he gets records regarding Border Patrol apprehensions. And South Carolina’s Lindsey Graham wants to hold the nominee hostage until the administration cooperates with the senator’s investigation into the 2012 attack on the U.S. mission in Benghazi, Libya.

They don’t have the votes to stop Johnson, even under the old filibuster regime, but those two could compel the Senate to spin its wheels for 25 hours or so before taking a 51-vote-threshold cloture vote they’d be sure to lose, then wait 30 hours more before acquiescing in his confirmation.

The same 55-hour routine could also confront Yellen, the likeliest to see her confirmation postponed for the new year — because Ben S. Bernanke is primed to remain Fed chairman until the end of January. Among the reprisal-minded Republicans are a decent number who oppose her because of a disagreement over monetary policy. This group disapproves of the Fed’s economic stimulus effort, which Yellen has signaled a readiness to continue even while tapering it a bit.

Combining the maximum time on all the countdown clocks works out to an average of 16 hours a day, assuming a Tuesday start and finish 10 days later. And that’s not counting 15 minutes for every required roll call.

Welcome back, world’s greatest deliberative body. It’s your call: Happy holidays — or not?