As the partial government shutdown enters its third day, many House Republicans are determined to keep fighting, even though they see no plausible way out of the current impasse, because they've come so far they cannot imagine backing down now. "I think there's a sense that for us to do a clean CR now -- then what the hell was this about?" one Republican House member told me. "So I don't think it's going to end anytime soon."

Wednesday was the best day in a while for Republicans, but not because of any progress they made toward their goal of defunding, delaying, or limiting Obamacare as part of a resolution to fund the government. Rather, Wednesday was a good day because Democrats handled their end of the crisis so badly. First, the Obama administration inexplicably went out of its way to barricade popular open-air monuments and memorials on the Washington DC mall, in particular closing off the World War II memorial to groups of elderly, Greatest Generation veterans who had come to the capital to pay their respects to a heroic moment in American history. Republicans couldn't believe what they were seeing. "That looked atrocious," said one well-connected GOP strategist, who suggested it proved once and for all that the ham-handed Barack Obama is no Bill Clinton when it comes to handling a government shutdown.

Then, Senate Majority Leader Harry Reid badly bungled a question about a House Republican bill that would fund, and re-open, the National Institutes of Health. "If you can help one child who has cancer, why wouldn't you do it?" asked a reporter. "I have 1,100 people at Nellis Air Force base that are sitting home," Reid said. "They have a few problems of their own. This is -- to have someone of your intelligence to suggest such a thing maybe means you're irresponsible and reckless." Republicans jumped on Reid's gaffe -- and why shouldn't they, since they often face such why-are-you-so-heartless questions themselves? By late Tuesday, Reid was reduced to pledging that he does, indeed, care about children with cancer.

"I think they're stepping in it a bit," said the GOP congressman, clearly relieved that Democrats were taking a turn making mistakes. But as much as the Democratic missteps buoyed Republicans, they didn't do anything to solve the problem. "They were positive tactical moments that gave some breathing room," said the strategist. "But there has yet to be a strategic moment that is a clear, outright win."

"It's sort of like when you have a temperature and you take an aspirin and your temperature goes down," the strategist continued. "You still have a problem."

For Republicans, the problem is that they don't know how to get out of the impasse they created when they embarked on a strategy to attach Obamacare measures to continuing resolutions needed to fund the government after Sept. 30. Senate Democrats and the president rejected all of them, Sept. 30 came and went, and now the government is partially shut down.

The GOP's latest tactic, to pass small, targeted bills to fund the National Institutes of Health, which would take care of the children-with-cancer problem, and the National Park Service, which would take care of the veterans-who-can't-visit-their-memorial problem. The House passed those bills Wednesday night and now lawmakers are waiting to see whether Senate Democrats will vote down measures to fund public health and the nation's historic and natural heritage. Reid's statements Wednesday suggested Democrats will do just that, but Republicans are waiting to see if they really do.

In any event, the small-bore bills are just a small-bore solution to part of the problem. There is still no government funding, and beyond the targeted measures, Republicans have no new ideas about what to do. One possibility is to pass yet another continuing resolution, this one with just one Obamacare measure attached to it. Perhaps it would be the repeal of the medical device tax, or perhaps it would be some variant of the Vitter Amendment, to forbid members of Congress and staff from receiving special subsidies when they purchase health insurance on the Obamacare exchanges. Maybe Democrats will finally accept one of those, some Republicans think, since there is clear and substantial Senate Democratic support for repealing the device tax, and clear and substantial public opposition to Congress making special deals for itself.

So perhaps Republicans will try one more continuing resolution, with just one attachment. It would fit the pattern. In the last week, House Republicans have made progressively weaker demands with the continuing resolutions they have passed, first to defund Obamacare entirely, then to delay it for a year, then to delay just the individual mandate, and now this. Perhaps the latest idea will work, although there's no real reason to think it will. And beyond talk about perhaps passing a continuing resolution with a non-Obamacare attachment -- maybe a measure to finally build the Keystone Pipeline -- Republicans don't have any other plans to fund the government.

Of course, the House GOP leadership could always decide to propose a "clean" continuing resolution that doesn't touch Obamacare, a move that insiders estimate would win the support of perhaps 175 of the House's 232 Republicans. With Democratic support, a clean measure would win overwhelming passage. But the leadership is apparently dead set against that, at least as the shutdown begins Day Three.

So what now? House Speaker John Boehner and leading Republicans like Paul Ryan and Dave Camp are apparently reviving the old goal of a "grand bargain" -- a budget deal that would include entitlement reforms, tax reform, and a new budget agreement, while also restoring government spending and raising the debt ceiling. The idea is that with the debt ceiling deadline coming up on Oct. 17, Republicans and Democrats could fix all, or at least many, of their problems in one fell swoop.

Such "grand bargain" attempts have failed in the past, and there is little reason to believe one will succeed now. And not just because Obama and Democrats are intransigent, which they are. The fact is, this is Oct. 3, meaning there are just two weeks before the nation hits the debt ceiling. Could Republicans get even their end of a "grand bargain" together in time? "Look at tax reform," said the GOP strategist. "If you took Democrats out of the mix entirely, I don't think Republicans could come up with a tax reform package by Oct. 17. There is a huge range of opinion on what direction to take." The first "grand bargain" Republicans would have to reach would be among themselves.

Nevertheless, some Republicans are grasping at the idea that a "grand bargain" will somehow solve their problems. Meanwhile the shutdown goes on. Before October began, Republicans privately expressed the opinion that a shutdown of two or three days would probably be politically acceptable; after that, they said, the repercussions could be quite serious and entirely unpredictable. Now, some Republicans are talking about a shutdown that drags on while both sides work on a "grand bargain." It has become an attractive course mostly because GOP leadership, having rejected the option of passing a clean continuing resolution, simply doesn't have any other ideas on how to get out of the current mess.