After 39-year-old John Skelton tried to hang himself, he told police he gave his three sons - Andrew, Alexander and Tanner - to a woman with whom he had an online relationship.
Now, police are saying that this woman never existed, and the whereabouts of the boys remains a mystery.
Profiler: Dad's Story of Missing Kids "Bogus"
Search to Resume for 3 Young Mich. Brothers
CBS News National Correspondent Jeff Glor reported that on day three of the desperate search that as hundreds of volunteers joined police, firefighters, and the FBI in looking for the missing brothers, their father's story fell apart.
Skelton's sons were last seen on Thanksgiving during a court-ordered visitation with their father. He claimed he turned his children over to a woman named Joann Taylor because he didn't want them around when he committed suicide.
Now, police are saying Skelton's story is a lie, that he made Taylor up out of thin air.
Morenci, Mich., Police Chief Larry Weeks said, "What I'm saying is a reported relationship between Mr. Skelton and Joann Taylor does not exist."
Search parties are concentrating their efforts in an area along the Ohio turnpike, an area where Skelton's minivan was spotted around the time the boys disappeared. One search leader was overheard telling searchers Skelton "may have just dumped them on the road," Glor reported.
Weeks said, "Since we can last pinpoint the boys in the backyard on Thursday afternoon and we know that the vehicle wasn't at the home at a period of time there, you certainly want any leads that we could gather for that period of time on where that vehicle is at."
Police have searched Skelton's home and vehicle, and though he remains a suspect, no arrests have been made. Skelton, currently under observation at a local hospital, has yet to explain what actually happened to his three young sons.
Wendy Murphy, a former prosecutor, told CBS News, "We don't know why he lied, but we're certain that he lied, and when a father is not telling the truth to the cops about the disappearance of all of his children, that's a good reason to be suspicious."
This is not the first time Skelton has taken his children without permission, Glor reported. Skelton took the boys to Florida in September on the same day his wife filed for divorce.
Glor said Skelton kept his children there until a judge ordered him back to Michigan.
Ernie Allen, president and chief executive officer for the National Center for Missing and Exploited Children, said on "The Early Show" Tuesday his team has been helping in the search for the boys.
He said, "We have a program called Team Adam that uses retired law enforcement experts to advise and assist on the scene. We've been helping with grid searches of rural areas, looking for clues, looking for indications that the children have been there. We're circulating their photographs, widely to try to generate those key leads and information. We need to do more to bring these three little boys home."
"Early Show" co-anchor Maggie Rodriguez said, "You know better than anyone that the more time that passes, the less likely it is that these boys will turn up alive. In these cases time really is the enemy, isn't it?"
Allen responded, "It really is, Maggie. We know that in 90 percent of these cases the children are recovered, the cases are resolved within the first 24 hours. And we know that when children are abducted by strangers, and murdered, that in three-fourths of the cases the children are dead within the first three hours. So law enforcement has moved quickly. There's been enormous attention to it. We remain hopeful, but the passage of time is not encouraging."
Allen explained in cases when a parent abducts a child, the motive for taking them is anger or revenge directed at the other spouse.
He said, "What is perhaps most encouraging in this case is we also know that it is very rare for an abducting parent to do physical harm to the children. This is usually directed more at the other spouse. So it's not unprecedented, but there certainly is hope that these three little boys are still alive and still out there somewhere."
Rodriguez said, "But in this case, do you fear that it is different because the father is not physically with the children, and we know that he made up a story about what happened to them?"
He replied, "Absolutely. I mean, this is a really troubling scenario. We're very concerned for the safety of these children. Our hope is that somebody knows, and that somebody will come forward and provide information. It's really important that people respond now and not feel intimidated because they know someone or have a relationship. Somebody knows where these children are."
Allen said he thinks the boys will be recovered.
"I think we've put such emphasis on the importance of time and rapid response that, to lots of people, that means that if you don't recover them in 24 hours or 48 hours there is no hope," he said.
Allen continued, "There are lots of examples of children being found and coming home after long periods of time. Most recently Jaycee Dugard after 18 years. So there is hope that we can find them. Law enforcement, state, federal, local law enforcement in two states, Ohio and Michigan, are working aggressively. There are volunteers helping. We think there is still hope that these children can be found and can be returned to their mom."