LAS VEGAS—Throughout history, the desert has been a place for personal reflection and self-discovery, where holy men retreat to seek insights into life's deepest questions.

For example, if you're a Republican in 2012, why your party couldn't get its act together and beat President Barack Obama.

America's Republican governors made a pilgrimage here this week in search of answers. Why did their party lose a contest they were told would be a cakewalk? For two days at the Encore resort and casino, the members of the Republican Governors Association met to lick their postelection wounds and mull over what went wrong. By week's end, there seemed to be more questions than answers: Was it Mitt Romney's fault? Are the Republican Party's ideas outdated? Why did Todd Akin and Richard Mourdock talk about rape so much? What can we do to make minorities like us more? Where's the craps table?

To help guide the beleaguered Republicans, there was no shortage of party brainpower on hand. In attendance was former Mississippi Gov. Haley Barbour, the wise grandfather figure who offered comfort to the afflicted, but not without forcing everyone to swallow a bitter pill first. "We've got to give our political organizational activity a very serious proctology exam," Barbour told the governors Wednesday. (Perhaps the pill was a bit too much to take orally.)

Virginia Gov. Bob McDonnell, the outgoing leader of the association, beamed like a proud father passing on his legacy to a golden son, Louisiana Gov. Bobby Jindal, who became the group's chairman on Thursday. Under McDonnell's leadership, the number of Republican governors swelled to 30, making the office one of the few bright lights of an election in which Democrats dominated almost everywhere else.

Jindal, who will lead the organization until New Jersey Gov. Chris Christie takes over in two years, took on an active campaign to assert his place as a national party leader. One of the most visible governors during the confab, he led discussion sessions about the election, blasted Romney for the way he ran his presidential campaign, and outlined his vision for the future to anyone who would listen.

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Not playing a major public role this year was Christie, a governor burdened by the devastating hurricane that hit his state just two weeks ago. Flanked on all sides by an entourage of bodyguards, Christie walked the halls of the hotel unmolested by reporters. Looking exhausted from sleepless nights handling the relief effort, Christie was in no mood to talk. When one reporter tried to ask him a question after he said he wouldn't answer any, Christie glared at him with tired eyes and boomed, "What part of 'I'm not taking any questions' don't you understand?" Word spread, and few dared ask him anything after that.

During the conference, the governors engaged in an ongoing discussion about the party's need to reach "new constituencies." (Also known as "minorities.") Republicans have long struggled to compete for votes from Hispanic and black voters, but the election was particularly brutal this year on the party of Lincoln. Obama was able to extend his lead among all minority groups. To win in the next cycle, Republicans must recoup the losses. Problem is, they don't seem quite sure how to make this happen.

"We've got to do a much better job of reaching out to the minority community," Iowa Gov. Terry Branstad said in an interview near the end of the conference, echoing a line heard dozens of times in the preceding two days. "Democrats ran a very effective campaign and they did a very effective job motivating interest groups and different people in their base they needed to get out."

Republicans who examined exit poll data after the election were shocked at how much voting demographics had changed. They could hardly believe that Romney won a majority of independent voters but somehow managed to lose the election. The makeup of the nation is changing fast, they found, and the old rule that a candidate just needs to win independents no longer applies.

"There's too many damn Democrats out there," Republican pollster Glen Bolger complained.

The most astute proposal came from New Mexico Gov. Susana Martinez, a Latina Republican with a record of bringing traditionally Democratic voters into the fold. In 2010, she ran as a conservative in a state where a majority of voters have supported a Republican presidential candidate only once in the past 20 years. She won by more than 6 percentage points.

"Republicans need to stop making assumptions, and they need to start talking to younger people, people of color, and ask them—not talk to them—ask them, 'What is it that we can do better? How do we earn your vote?'" Martinez said. "We have to start electing people who look like their communities all the way from city council to county commissioners to county clerks all the way through the state and up into national politics."

Of course, reaching minority voters wasn't the only problem the Republican governors felt the party needed to deal with. Many believed that the party has propped up some embarrassingly bad candidates for Congress in recent years, a series of mistakes that helped bolster the Democratic minority in the House and may have cost the Republicans a majority in the Senate. Two Senate candidates this election, Richard Mourdock in Indiana and Todd Akin in Missouri, lost easily winnable elections in part because they couldn't explain their opposition to abortion in a way that didn't turn off swaths of voters.

"We ought to have some rule where if you start talking about 'legitimate rape,' you're out," Bolger said.

If pressed on social issues, they were told to find a way to pivot back to the economy whenever possible. During a meeting Wednesday, Utah Gov. Gary Herbert warned that he saw the country turning "center-left" on issues like gay marriage, and asked what Republicans could do to survive the shift without violating their principles. The answer: If you must discuss it, stand by your beliefs, but please, for the good of the party, try not to sound like a jerk.

"You can articulate your opposition for example to same-sex marriage," advised Bill Bennett, a former adviser to Ronald Reagan. "But you can do it in a dignified way, in the right language, in a forceful way that shows you're not a bigot or intolerant."

While much of the conference focused on how to bounce back, this was, after all, the gathering for Republican winners, not losers. The governors were optimistic about what they can accomplish at the state level, even if the feds in Washington can't balance a budget or name a post office without some bickering.

"There is a brand, there is a set of ideas that Republican governors are offering that stands in fairly stark contrast to what Washington is offering," McDonnell said Thursday. "Washington is broken."