Last Wednesday, the U.S. command thought it had Saddam Hussein in its crosshairs and tried to kill him with a massive air strike in the opening salvo of the war.
But by Saturday, Commanding General Tommy Franks had to acknowledge that -dead or alive - the United States had lost track of him again.
In response to a reporter's question, General Franks replied bluntly, "Actually, I have no idea where he is right now."
Saddam has managed to elude the United States for the better part of a dozen years, and has survived attempted coups and assassination plots, and several wars. He has a network of tunnels and underground bunkers, a massive security apparatus, a cadre of look-alikes and a capital of five million people to hide in.
Former CIA Director James Woolsey appreciates how difficult it's been to get rid of Saddam Hussein: "Trying to do something like that from outside the country with a leader who changes places every night and who very rarely sleeps in any of the 50-some palaces that he has built, and has doubles, is just a titanic task."
It may, in fact, require an element of luck. And so far, Saddam has had all the luck.
During the first Gulf War, U.S. warplanes launched more than 260 attacks against what the Pentagon called leadership targets -- bunkers, command centers and presidential palaces -- where Saddam could conceivably have been. He survived them all, plus another round of bombings in 1998, when William Cohen was secretary of defense.
"We really couldn't have a good fix on where he was at any given time," Cohen recalls, "and the circle around him was so tightly controlled that it was virtually impossible to get information, access to people who would know where he is."
Why is it so difficult?
"You have to have agents who are supporting you, working for you, in order to gain access to information in terms of where individuals are going to be," Cohen says. "It's very hard to take out an individual with a cruise missile."
With U.S. intelligence on Iraq coming from electronic eavesdropping and satellite photography, Saddam decided to go underground, building elaborate bunker and tunnel complexes. A German firm designed and later built a 20,000-square-foot bunker under one of Saddam's palaces, at a cost of $90 million. It has luxurious bedrooms for Saddam, his family, and dozens of bodyguards and staff. It's stocked with enough food and water to last a year. Giant shock absorbers and redundant air filtration systems are designed to withstand multiple bomb blasts and missile strikes.
Michael Vickers, a consultant to the Pentagon who spent 10 years in the Army's special forces and later became a C.I.A. operative, says there are "some very, very hardened, deep underground facilities that have 20 or more feet, maybe 100 feet in some cases, of dirt, and then 6 to 20 feet of reinforced concrete, and then prefabricated steel."
In some cases, Vickers says, the bunkers are 300 feet or so deep, and "almost impervious to anything but nuclear attack."
Worse, some of these bunkers may be connected by tunnels, allowing Saddam to move from underground facility to underground facility.
"One of the things that makes these underground structures difficult-is the labyrinth network of them," Vickers says. They have blast doors in between, too, so "even if you penetrate down into one compartment, say, and destroy that, then the steel doors may contain the effects of a blast. If you're on the other side you're a quarter mile away, you're perfectly safe."
The U.S. identified more than two dozen of these bunkers 12 years ago. There may be more now, and they may be located under schools, hospitals, even mosques - targets that the U.S. is not likely to bomb.
And, if and when U.S. forces finally take Baghdad, they won't necessarily find maps and blueprints of the bunkers and tunnels, or even the engineers that designed them.
"When Saddam realizes you're an engineer and you know too much about where he could hide, he's going to have you killed," says Ibrahim al-Marashi.
Ibrahim al-Marashi is an Iraqi-American scholar at the Monterey Institute of International Studies. He's pored over 300,000 pages of Iraqi intelligence documents captured in the last Gulf War, and he is considered one of the foremost authorities on the Iraqi command structure.
He says if Saddam was present when the U.S. hit a suspected hideout last Wednesday, it would be a major intelligence breakthrough.
"There's only a select few people around Saddam Hussein that know his whereabouts," he says, "maybe 55 people who could have known his whereabouts."
That suggests to him that the Americans may have penetrated one of the security units in Iraq that is the most difficult to penetrate.
With the skies over Baghdad filled with smart bombs, Saddam may forgo presidential bunkers and other past hiding places, he says. He believes it's possible, perhaps even likely, that Saddam is holed up in a private home in one Baghdad's many residential neighborhoods.
"I heard stories from family and friends in Baghdad that during the 1991 Gulf War that he could just show up at your home, just show up at your doorstep and knock and say, 'Do you mind if I stay with you?' I remember a joke from my family friends saying, 'You know we have to keep the house clean because most likely Saddam will be coming over.'"
In other words, if the U.S. is targeting places where it believes he might be, the safest refuge would be somewhere he has never been among people he has never met.
He has been preparing for this day, perhaps for a long time.
"Oh, I'm sure he has tricks up his sleeve," Cohen says. "There have been concerns as to whether the person who would surrender would in fact be Saddam Hussein, whether it would be a double. The doubles - at least, I'm under the impression - they're very, very good doubles."
It could even be possible for Saddam to escape, to get out of Baghdad and maybe out of Iraq. The question is, notes Cohen, "where would he go? How would he live? I think there are very few places he could go where he would be welcome and safe."
There are no friendly nations on Iraq's borders, and Ibrahim Marashi says Saddam has so many enemies in his own country that the only place he could safely hide is a small triangular area in the middle of the country.
"His options are pretty much limited," Marashi says. "He wouldn't want to venture too far to the north here. So, either he has to go through his hometown, Takrit, or stay in the environs of Baghdad."
It's too dangerous now for Saddam to travel with full security details in large convoys. If he is moving around Baghdad at all, it may be with a few bodyguards in a civilian vehicle. Panamanian dictator Manuel Noriega drove around Panama City for days eluding U.S. forces before finally requesting asylum at the Vatican Embassy.
But is there an embassy in Baghdad that Saddam Hussein could go to and expect shelter?
"Perhaps for hours to days, but not for weeks," says former CIA director Woolsey. "Not to really, really protect him. There might be some Arab state. But generally speaking, he's not well liked in the Arab world."
He laughs at the notion that the French embassy might shelter Saddam. "I don't think even the Chirac government, even in the aftermath of the last couple of months, would be willing to do that," he says.
Eventually, we'll see the endgame. Woolsey suggests that Saddam probably would like "to disappear, or go out in some way in which people didn't know what happened to him, and he could try to believe that a myth would be kept alive. He has a very heroic self-image."
But Marashi notes that Saddam "is obsessed with his legacy. He wants to go down in the history books as, you know, this great Babylonian leader." That points to a preference, he says, for "going down fighting" rather than accepting the humiliation of surrender.