House Republicans rode to power last fall with a mandate to change how Washington works.

Now a potential government shutdown looms and no one knows who the public will blame if President Obama and Congress can't forge an agreement on spending levels.

Perhaps this explains why the GOP is making sure everyone knows they're not completely in charge here.

"We control one-half of one-third of government," declared House Speaker John Boehner (R-OH) Thursday.

In other words, the GOP is responsible for precisely 16.5 percent of government. The rest is apparently run by the Democrats, although one may need an algebraic equation to deduce the "value of X" when determining who "controls" the Supreme Court.

In short, Boehner is warning people that if the government closes down, the Republicans are not really responsible. The GOP is merely a fraction.

What's next from Professor Boehner? A story problem?

If 87 House Republican freshmen leave Washington for their districts on April 8 with less than \$61 billion in spending cuts, how many will receive primary challenges from the tea party next year?

Isn't math fun?

You see, the biggest calculation in Washington these days is determining the coefficient that keeps the government running past the polynomial of April 8. That's the day the federal government runs out of money unless both houses of Congress and the president can strike a deal on spending. The question is whether the House and Senate can find collinear points along the spending axis (and enough votes) to keep the federal lights burning.

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For starters, this is an binary computation, dealing with both the House and Senate. And to most Republicans, the House portion of the equation is balanced. The Senate portion is not.

In today's budget battle, one constant will always be the inevitable tension between the House and Senate. Which is why a number of House GOP freshmen marched across Capitol Hill to the Senate steps on Friday to denounce Senate Majority Leader Harry Reid (D-NV), even though senators weren't meeting.

"The fact that the Senate is not in session today shows that they're not serious about this," charged freshman Rep. Bill Flores (R-TX).

Others used a Thursday tea party rally on Capitol Hill to stoke anti-Senate hostilities.

"Let's go pick a fight," proclaimed Rep. Mike Pence (R-IN). "If liberals in the Senate play political games, then shut down government. I say shut it down."

But not all Republicans are aligned with Pence.

"Just shut ‘er down. Is that the right thing?" asked Rep. Mike Simpson (R-ID). Simpson's a senior member of the House Appropriations Committee, which determines how much money the government spends on individual programs and writes earmarks. Simpson is incredulous at some of the fissures among the GOP ranks.

"I go home and get calls on the radio calling me a RINO (Republican In Name Only) and a liberal," vented Simpson. "This is the first time anyone voted to cut \$10 billion and is called a RINO."

Simpson doubts the public realizes how a government shutdown could arrest everyday events in a myriad of ways. He notes that ranchers in Idaho and other parts of the west will soon look to release cattle to graze on Bureau of Land Management (BLM) property. But if the government is shuttered, BLM can't issue ranchers the necessary permits.

John Boehner certainly doesn't want a shutdown and told reporters Friday he wasn't planning on one. In fact, Boehner said a shutdown would wind up "costing more than you save" because of how it would interrupt federal contracts.

3.14 is nothing like \$61 billion. But to House Republicans, \$61 billion is a constant as sacred as pi. After all, \$61 billion accounts for the cuts the House agreed to in February. And the Senate simply cannot go that deep.

"We want to see \$61 billion," said freshman Rep. Jeff Landry (R-LA), who doubted that many of his GOP classmates could tolerate a figure below that. But Vice President Biden and Senate Democrats have compiled a package which axes \$33 billion in spending for the rest of this fiscal year.

Boehner insisted this week that "there is no agreement on a number." But the speaker added this caveat: "We are going to fight for the biggest cuts we can get."

Boehner emerged as the consummate dealmaker during his years in the House before assuming the speaker's chair. And that's why some Capitol Hill analysts believe that Boehner's statement about battling for "the biggest cuts we can get" sounds as though he could accept a lower figure in exchange for some policy "riders." Riders are just that: extras attached to a bill. And many House Republicans are demanding that the GOP leadership Velcro a litany of riders to any legislation that keeps the government open for business.

The problem is that many of those riders are toxic to Democrats. Potential riders include defunding Planned Parenthood, barring the Environmental Protection Agency from regulating CO2 emissions and cutting off money to implement the new health care law.

But Boehner insists, there's no deal yet.

Part of that is because the "deal" is still developing. House and Senate appropriators are toiling throughout the weekend to prep a potential package. Then it will be up to Boehner and others to conjure up the votes to keep the government open. The House GOP leadership lost 54 of its own members on the last stopgap bill. In fact, the House would not have adopted that measure were it not for Democratic support.

So could a number of Republicans bolt again this time? Yes. Which is why a coalition may be essential to keep the government at speed.

When Democrats controlled the House, the most pivotal bloc of votes on the "cap and trade" climate bill and health care reform came from moderate "Blue Dog" Democrats. As swing votes, the fate of those two bills hinged on Blue Dog support. There are significantly fewer Blue Dogs in the House after the 2010 electoral purge. But conservative and moderate Democrats could be just as crucial to Republicans now as they were to Democrats in the last Congress.

"They may very well need us," predicted one moderate House Democrat. "If they don't, there's a government shutdown."

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Counting votes may be one of the most critical mathematical components to averting a government shutdown. However, there's another variable at work now: time.

The government is scheduled to run out of money at 11:59:59 pm on September 8. On its face, it seems like Congress has nearly a week to hammer out an agreement. But that timeframe is an illusion. In their Pledge to America, House Republicans promised to release the text of legislation "online for at least three days before coming up for a vote in the House of Representatives." On the first day of this Congress, Boehner said that everyone "will have three days to read a bill before it comes to a vote." A literal interpretation of those statements would mean 72 hours, start to finish. But top GOP House aides insist that hypothetically, if a bill is released on Monday, it could conceivably be debated as early as Wednesday.

The bottom line is this: even if the House has a bill ready as early as Monday, lawmakers can't consider it until midweek. And then there's a problem with the Senate. Cutting off a possible filibuster in the Senate could easily chew up three days there, too. This is why accomplishing any of this by April 8 seems to be a stretch.

So what to do? Under most circumstances, lawmakers might prep a "miniature" interim measure, perhaps to keep the government open for as little as 24 hours or just a few days. But scores of Republicans and Democrats are already on record saying they won't vote for another short-term measure.

One freshman Republican said leadership "would have to do some serious persuading." But freshman Jeff Landry said he'd be willing to entertain another emergency measure to stave off a shutdown.

"I certainly am not opposed to doing that," said Landry of a limited patchwork bill. "Whatever I can do to give the leadership the tools necessary to get the American people the cuts they want."

But Landry signaled that even a "bridge" bill to avoid a shutdown would have to carry pro-rated spending cuts.

Senior aides on the House Appropriations Committee tell FOX that crafting a miniature stopgap bill with commiserate cuts would be nearly impossible.

It is that scenario which could dramatically bolster the likelihood of a government shutdown next weekend, even if the sides manage to break the impasse and have an agreement in the works.

Still, some lawmakers argue there's something nefarious going on.

"I believe that Harry Reid, (House Minority Leader) Nancy Pelosi (D-CA) and Barack Obama orchestrated a government shutdown," assailed Rep. Paul Broun (R-GA). "It's a diabolical plan they hatched out."

But Rep. Pete Sessions (R-TX) fretted that the public may declare a pox on lawmakers, regardless of who deserves the blame.

"I think we look silly if the government does shut down," opined Sessions. "The American people will think that we have lost it."

The next few days could be decisive for this Congress. This is where governing and mathematics share similarities: both have "problems" that demand "solutions."

The voting public is keeping an eye on this. And just like a tough teacher back in math class, they're watching to see if everyone shows their work.