A couple of weeks ago, a prominent dot-com warrior gave me a hot tip about Google: the next big move of the search phenom would be an assault on eBay. Think about it. Millions visit the Google site daily; why not let them search for items offered by sellers? The company already knows auctions, since it uses a bidding process for ads that accompany its search results. And eBay might be vulnerable, since it recently angered its sellers by raising commissions. The move would be straight out of the "Art of War" playbook apparently distributed by venture capitalists to all ambitious entrepreneurs.

But when I floated this theory to Google's CEO, Eric Schmidt, the Valley veteran who joined the company in 2001, he just laughed. "It's a perfectly reasonable question, but it doesn't compute here," he says. "If I said at a meeting, 'Are we going to enter eBay's space?' everyone would look at me and say, 'Why? They do a fine job.' The genius of Google is that we find new ways to solve problems that were never solved before."

Welcome to Google-think, the mind-set by which Sergey Brin and Larry Page, the brainy Stanford dudes who founded the company in 1998, hope to change our world while incidentally redefining the way corporations should be run. We've learned the mission of the company: to make all the world's information readily available to everybody. But in the past year we've begun to see the quirky management style by which this might be accomplished. The founders first outlined their views in a colloquial "owner's manual" included in the prospectus for the initial public offering that would make Brin and Page multibillionaires. It warned shareholders that Google would not be overly concerned with quarterly results, but focused on the long term. It stated that while Google actively sought profits, it also believed in higher principles, notably the Google version of the Boy Scout Pledge, "Don't be evil."

The IPO, run by an unconventional auction process and plagued by embarrassments like an untimely Playboy interview that piqued the SEC, was somewhat of an ordeal. The company took a battering in the press, and the final price to bidders for the opening stock was $85 a share, much less than the $108-to-$135 range originally sought. On the day Google went public, while Page and Schmidt went to NASDAQ to ring the opening bell, Brin came to work as usual. "I was tired; I didn't want to take a red-eye. It was a very productive day." In fact, that day kicked off a period of intense productivity, with Google introducing a slew of new features and improvements including desktop search, a mapping product and a scheme to put the entire contents of five major libraries online.

"The IPO was work and, yeah, it was a distraction," says Brin. "But now that it's over with I actually see us executing really well."

"I would disagree a little with that," Page says. "I actually think we executed well during that whole time. We always crossed our fingers that we'd be able to continue to operate in the way we wanted to as a public company, and I think that's been largely true."

Certainly by the numbers, Google's results have been awesome. In the last quarter, the company took in more than a billion dollars, twice as much as the previous year. Profits more than tripled. "They haven't missed a quarter," says John Battelle, author of the upcoming book "The Search." And the share price? About $180. All this while sticking to the promise that the company won't kowtow to Wall Street's demands for financial guidance on a quarterly basis. Indeed, when Google hosted a daylong analysts' briefing in February, the chief financial officer, usually the star of such confabs, spoke for less than four minutes, with literally no comment on the numbers.

But Google-think is under big-time pressure, as the company tries to handle dramatic growth--from under 10 employees to more than 3,000 in just a few years--and much tougher competition. Yahoo has undertaken a massive attempt to rebuild its search efforts, and is in a position to lead in the next step in search--personalization, where the search engine gets better results because it knows what you like. "We can leverage our base of 65 million users," says VP Jeff Weiner. Microsoft, on a mandate from Bill Gates to overtake the Google guys, has also launched a huge effort, backed by a $150 million ad campaign. "In 10 years from now, what you see in search today will be like looking at an eight-track tape," says MSN's Bob Visse. And last month the lagging but technology-rich contender Ask Jeeves became revitalized when Barry Diller added it to his Internet port-folio. "The 44 million visitors [to his sites like Expedia and LendingTree] will help Ask Jeeves," says Diller.

The Google-chasers take heart in recent surveys, which indicate that competitors are catching up both in technol-ogy and in market share. "A couple of years ago we were talking of Google and only Google, but that's changed," says Danny Sullivan of Search Engine Watch.

All this has led some critics to wonder whether Google's let-'em-loose theory of planning can withstand these challenges. Google's execs say the critics just don't get it. "Let me tell you why we look so scattered," says head of Web products Marissa Mayer. She draws a big circle on a whiteboard, with a bunch of smaller circles around it. Outside those orbits, she makes a few X's. "We operate on a model of 70-20-10." (This is a formula concocted by mathematician Brin.) The big circle represents the 70 percent of effort Google puts into its core business: Web search and the targeted ads that accompany the results. Included in that category are the things deeply integrated into search, like Google News. The loops outside are search-related products like Gmail (which displays the targeted ads Google sells). (Some of those are the results of Google's policy of letting its engineers use one fifth of their time to work on projects of their own choosing.) The X's are less-integral products like the Weblog tool Blogger, the Picasa photo organizer and Keyhole, a 2004 Google acquisition that uses satellite imagery to provide photographic maps.

While the discrete new products outside the core get attention, Mayer says that the bulk of Google's work is quietly improving the bread-and-butter of search. "We don't put out a press release saying that we got rid of 5 percent of search spam, but that stuff happens all the time. We definitely have a grand plan," she says. Indeed, just this week Google is integrating the Keyhole service into its local search product.

Nonetheless, the evolutionary process by which such products rise and sometimes fall underlines the fact that Google is sort of scattered--by design. "Larry and Sergey's vision, which I think is absolutely brilliant, is, 'Let's get these systems to prove themselves.' It's very Darwinian," says Schmidt. In order to let the products develop organically, sometimes Google forgoes revenues in the short term. "It takes years to become profitable in terms of total dollars invested, but we don't even think about it," he adds. "When we started Google News, we forgot to put ads in it. It's not deliberate. We actually forgot."

If your takeaway from that story is that Google is sloppily leaving money on the table, don't bother applying to work there. Page and Brin both believe that moving forward, fast, will work better in the long run. "With Larry, it's all about big impact," says Google exec Sukhinder Singh.

This year it will be Brin's turn to write the shareholder letter. He's focusing on things like new compensation plans to keep employees motivated now that the IPO is over. (He's already announced a "Founders' Award," with grants that can total more than $1 million to star performers.) "This one's less interesting," he says. "There's not going to be anything terribly shocking in it."

"Well, one thing," Page prompts him. "There's a certain amount of... humor." Brin smiles. "Maybe I made it a little more entertaining." But no matter how outrageous his quips may be, they won't be half as entertaining as Google's larger quest to remake search--and business--by its own rules.