Plot

Tintin and Haddock set out to rescue Professor Calculus.

Tintin and Haddock are discharged from the hospital, and they plan with Calculus to return back home. Tintin notices that Calculus is still being followed by Bordurian agents. Abducted at first by Bordurians, Snowy manages to get Calculus's umbrella which he had dropped on the ground while being forced into a car. Tintin and Haddock stake out the Bordurian embassy and find a helicopter landing there at night. Haddock attacks when he sees that the Bordurians are trying to force Calculus into the helicopter. Calculus is then snatched by Bordurian agents in spite of Tintin and Haddock's efforts to rescue him. The agents force Calculus into a motorboat. Pursuing the agents in the Bordurian helicopter across Lake Geneva into France, they chase a boat and then a car carrying Calculus, but the helicopter runs out of fuel, and they are forced to land. The Bordurians force Calculus into a car and escape Tintin who had blocked the road. Tintin and Haddock take help from an Italian driver and pursue them. The Italian manages to find their car, but Tintin finds no sign of Calculus.

After being pursued by Tintin and Haddock through the French countryside, the Bordurians escape in a plane, with Calculus as their prisoner. Tintin and Haddock set off for Szohod, Borduria in hope of finding their friend again.

The Bordurians are alerted to their arrival by the two men in Geneva (who were Bordurian secret agents), and they are intercepted at the airport by the Bordurian Secret Police (ZEP). Colonel Sponsz of the ZEP orders that Tintin and Haddock are not to leave Borduria alive. ZEP assigns two minders who take Tintin and Haddock to a luxury hotel and keep them in bugged rooms, Tintin and Haddock manage to escape (by getting their minders drunk) and hide in the Szohod Opera House, where Bianca Castafiore is performing. Charmed by Captain Haddock, she invites them into her dressing room but is visited by Colonel Sponsz, chief of ZEP, in her dressing room. Tintin and Haddock hide in Bianca's closet, overhearing the conversation between Sponsz and Castafiore.

Sponsz says that his men are looking for 2 spies, who are after a prisoner held by Borduria. Sponsz reveals Calculus's location, a jail in the fortress of Bakhine, and the stress on him to surrender his plans. If he does give them up, then he will be handed over to two officials from the Red Cross, to whom he must swear that he went to the Bordurians of his own accord and gave them his plans voluntarily. Sponsz also reveals that the papers for the officials and Calculus' release are in his overcoat, hanging in the closet in which Tintin and Haddock are hiding.

Overhearing all this, Tintin and Haddock steal the papers and, disguising themselves as the two Red Cross officials, acquire Calculus' release. When Sponsz is told of this, he quickly raises the alarm, but the three friends manage to escape to the border in a car and later, a tank (Their car was attacked by the tank, but Tintin is able to commandeer the tank when its crew goes down the mountain to check on the crashed car). Together, they cross the border in the tank. Back in Geneva, Snowy finds Calculus's umbrella at a lost and found. When they arrive back in Marlinspike, they find that Jolyon Wagg's family is staying there and has nearly wrecked the house.

Realizing the destructive potential of his invention, Calculus burns his plans (which was supposed to be hidden inside his Umbrella, but Calculus had left them on his side table at his lab) by lighting them with Haddock's pipe while it is placed in Haddock's mouth. Haddock is incensed, calling Calculus a "jack-in-a-box". The hard-of-hearing Calculus thinks that Haddock has said "chicken pox", and tells Jolyon Wagg that Haddock is suffering from this disease. While Wagg at first interprets it as a joke, he then remembers that chicken pox is infectious, and Wagg doesn't want to be infected, so he and his family leave Marlinspike.