At a time when housing markets across the West are contracting and American architects’ billings are at their lowest point in 12 years, according to the American Institute of Architects, Mr. Cai (pronounced sigh) was offering his guests a rare chance to build big — and paying them, improbably, in wads of cash.

“Basically, Ordos is Texas,” explained Michael S. Tunkey, an American architect based in Shanghai whose firm has designed an opera house that, along with half a dozen museums and a boutique hotel, will anchor Mr. Cai’s new cultural district.

He was referring to the wide open spaces, the frontier attitude and the seemingly endless flow of money (at least in good times) from natural resources. Ordos has rapidly become wealthy, largely because of huge deposits of coal, the primary fuel for China’s economic expansion.

Not long ago, residents of this region 350 miles west of Beijing lived in elaborate tents called yurts. Now, with a population of 1.5 million, many live in homes that would make New Yorkers jealous. According to Bao Chongming, the regional vice-mayor, they have the second highest per-capita income in China (trailing only Shanghai, the country’s financial capital) and an annual economic growth rate of 40 percent.

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A few hours after the first 28 teams presented their designs, 69 more  arrived (a final three were selected later, making an even 100). The  five-day junket that followed included trips to the site, a dinner in a  giant yurt near Genghis Khan’s tomb (Mr. Cai, who is ethnically  Mongolian, claims Genghis Khan as an ancestor) and karaoke in the bar of  the Holiday Inn, a lavish hotel that bears no resemblance to Holiday  Inns in the United States.

At times, as the black-clad architects made their way around the barren  landscape, it was hard for some to escape the feeling that the entire  event was a kind of performance, with architects as hired players. Last  year Mr. Ai sent 1,001 Chinese citizens to Kassel, Germany, where they  lived on cots during the Documenta art fair. “Are we just performers in  another of Weiwei’s pieces?” said Keller Easterling, 49, an architecture  professor at Yale and a practitioner based in Manhattan.

The effect was heightened by the presence of camera crews — one making a  documentary for Mr. Ai, another making a documentary for Mr. Cai, and  many others taping the event for Chinese television. Everywhere the  architects traveled, their parade of gold-colored Mercedes buses had  police escorts, which served no purpose other than to emphasize  something that was obvious from the presence of numerous public  officials at the Holiday Inn: that Mr. Cai has government support for  the Ordos 100 project.

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Many architects were worried about the way the project broke from their usual way of working “with context, with a desire to make buildings that are part of the connective tissue of the city,” as Mr. Rosbottom put it. In Ordos, the buildings will stand alone, “100 sculptures competing for attention,” said Preston Scott Cohen, 46, an architect in Cambridge, Mass., and a professor at Harvard, on a cluster of lots ranging from just a quarter-acre to a half-acre. He proposed that there be restrictions on materials and colors, so that the houses would form a coherent whole, but Mr. Ai and the other organizers politely ignored the suggestion.

Mr. Ai, who is known as a provocateur, encouraged the architects to keep asking questions, though he rarely provided answers.

But he did offer some specific comments on the houses by the first 28 teams. At one point, he told Mr. Meredith and Ms. Sample that a garage building on their property seemed a bit too big and would overpower a neighboring house. “Why don’t you take some time and see if you can adjust it,” he said gently.

But they didn’t need time. Mr. Meredith simply reached over to the cardboard model and ripped the garage off its base, exposing a patch of blue cardboard.

“Good, a swimming pool,” said Mr. Ai, smiling.

He was less impressed with the few houses that had curved walls meant to evoke yurts. “When I see that, I have to take it as a joke,” he said later. After all, Ordos is hardly a tent city. Indeed, Mr. Ai said, pointing to the architects who had traveled thousands of miles looking for work: “These days, it’s the architects, dressed in black, who are the tribe of nomads.”