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At the Los Alamos, New Mexico, atomic energy plant and housing complex, where security is extremely tight, a new television set is delivered to the family of scientist Dr. Frank Addison. While installing the set, the delivery man, Pattiz, chats with Frank's young son Tommy, who reveals that his father is a physicist. Excited about his class's field trip to the Fiesta Day carnival in Santa Fe, Tommy then dashes off to school. During the carnival, a drawing is held for a free bicycle, and Tommy's ticket is selected. When Tommy fails to respond, his teacher, Ellen Haskell, becomes concerned and begins searching for him. At the Addisons', meanwhile, Frank returns home early, disturbed over a work-place accident that sent a co-worker to the hospital. Frank's wife Martha is also distressed about life in Los Alamos, noting that Tommy had earlier used the phrase "if I grow up" instead of "when I grow up." Frank tries to reassure her, but is interrupted by the arrival of a telegram, announcing that Tommy has been kidnapped and directing the Addisons to attend the Fiesta Day dance that night to obtain "details." Just then, Ellen telephones to report Tommy missing, and Frank tersely informs her that he picked his son up before the drawing and forgot to tell her. Later, at the dance, the Addisons wait nervously at a table, while Ellen complains to her boyfriend, undercover FBI agent Russ Farley, about Frank's thoughtlessness. Finally, Martha discovers a second telegram under her plate, and she and Frank race home to read it. The kidnappers direct Martha to go to a cathedral the following morning, alone. Although Frank wants to notify the police, Martha convinces him not to, hoping they can meet the kidnappers' demands on their own. The next morning, in a phone booth outside the cathedral, Martha is called by Emil Jablons, one of the kidnappers, who gives her instructions for Frank and tricks her into thinking that Tommy is on the phone. Back at home, Martha relays the kidnappers' demands to Frank, who reluctantly agrees to comply. The next day, Frank requests a classified file from his office, but before he can leave with it, he is stopped by Russ and Inspector Harold Mann of the FBI. Russ explains that Frank's anxious demeanor at the dance made him suspicious, and when he checked on Frank's story about picking up Tommy, he discovered that Frank had not left the plant. Thus trapped, Frank reveals that the file contains information about an old, failed experiment and would require at least two days to decipher. After Frank convinces Mann to give the kidnappers the data, reasoning that it will buy them some time, agents are sent to the Los Angeles hotel to which Frank has been instructed to mail the file. The file is picked up by petty thief David Rogers, who is then tailed by a number of agents to a baseball game. There, Mann orders a television crew to make a kinescope of Rogers. In the parking lot after the game, Rogers' car explodes when he turns on the ignition, and Mann deduces that the thief must have passed the file to someone at the ballpark. Later, Mann shows the kinescope footage to Frank and a group of undercover agents, and two people identify a hot dog vendor as Communist Party member Donald Clark. Clark is brought in for questioning, but refuses to talk. Frank, however, beats Clark into revealing where he took the file, and the location, 19 Elevado Street, is raided. The agents deduce that the kidnappers recently abandoned the apartment, having worked out Frank's formula on a blackboard, and Martha cries hysterically when she hears a tape recording of Tommy and realizes that her earlier phone conversation with her son was faked. At the Puye Indian ruins in the New Mexico desert, meanwhile, Jablons and fellow kidnappers Robert Kalnick and Arnie Molter, who have disguised themselves as park rangers, are keeping Tommy captive inside one of the cliffside dwellings. After Jablons gets rid of the Fentons, a family of tourists, kidnapper Dr. Peter Rassett, a Communist-sympathizing physicist, races up to report that Frank's formula is a phony. Rassett orders that Tommy be killed and the operation dismantled. Unknown to them, Tommy has climbed out the ruin's chimney and is running away. The kidnappers soon spot the boy and give chase, and Tommy dashes into a cave and slips into a crevice, away from the adults' grasp. In a Santa Fe bicycle shop, meanwhile, the Fentons' son discovers that he has the winning ticket from the Fiesta Day drawing and asks for his prize. An agent who has been watching the shop asks the boy where he got the ticket, and after the child reveals that he found it near the Puye ruins, the FBI sends a helicopter to scour the area. The spies are spotted among the ruins, and with Frank and Martha in tow, the FBI quickly surrounds them. Faced with capture, Rassett shoots his cohorts, then is arrested. Tommy, meanwhile, escapes out the other side of the cave but finds himself dangling on a steep cliffside. As his strength is about to give out, Tommy is rescued by an agent and reunites with his relieved parents.