At the Cabaret of Magic, the star performer, the magician calling himself the Great Santini, goes to the basement and loads a revolver and attaches a silencer to it. He then covers it with a cloth and puts it on a tray, and then removes a radio transmitter from his bag. Santini then places the transmitter in a hidden compartment. He then goes upstairs and says hello to the stage manager, George, who welcomes him back. George tells him that the manager, Jesse Jerome, was looking for him. Neither one of them like Jerome, and Santini tells George that they'll go out some night.
Santini then goes through the kitchen up to Jerome's office, where partner and maître'd Harry Blandford is meeting with Jerome. Harry leaves and Jerome invites Santini to have a drink with him. Santini refuses and Jerome asks if he broke into the office a month ago before he left for New York. The magician jokingly claims that he did and Jerome apologizes for the accusation. The manager then points out that Santini only paid him $5,000 of his salary for the tour, and Santini says that he only owes him 5%. Jerome wants 50% but Santini refuses to pay him blackmail anymore. The manager reminds Santini that his real name is Sgt. Mueller and he knows where he came from. Santini insists that he was only 21, but Jerome says that no one, even a SS officer, was just a boy. He warns that he'll tell the newspapers and the Israelis if Santini pushes him, but Santini figures that he's too valuable to the manager. Jerome figures that Santini broke in to find the letter from an old man that confirmed that Santini was Mueller. The manager paid the old man off, took the letter, and is now using it to blackmail Santini. Jerome tells Santini to have the other $45,000 for him by the performance that night or it will be his last performance. Santini walks out without a word, smiling.
The magician goes back downstairs and hears his daughter Della flirting with the club singer, Danny Green. Santini interrupts them and Danny apologizes, but promises to meet Della after the show for supper. Once Danny leaves, Della insists that she'll continue seeing him and Santini tells her to check the props with him.
That night, Santini watches the audience from behind the curtain. Jerome, out front, confirms that Santini didn't ask about something and then goes up to his office. Santini watches as Jerome goes upstairs with the take and then enters his office, locking the door behind him. He puts the money in the safe and takes out the letter.
On stage, Danny finishes his song and goes backstage, telling Della that he has to contact his agent about a job. Santini begins his performance, working his way through the audience and then going up on stage to perform with Della, his assistant. He calls two volunteers up on stage to help him.
In his office, Jerome puts a piece of paper into his typewriter and begins a letter.
Santini continues his act and brings out his water tank escape illusion. After demonstrating that the trunk is solid, he steps in and Della locks it close. There is only enough air to keep Santini alive for nine minutes. The trunk is lowered into the water tank, but the audience is unaware that Santini has already escaped through a trick panel in the bottom and descended into the basement. He checks his watch and then removes the radio transmitter from the hidden compartment. The magician then dons a waiter's uniform, and alters his hair and mustache as a disguise. Santini then puts a microphone in his lapel flower and puts a rotating light behind a screen, and places the radio transmitter nearby.
Once everything is in place, Santini locks the door to the basement behind him and goes up through the kitchen. The staff are dealing with the busy house and don't notice him passing through. Once Thackery, the waiter with Santini's brandy, goes to the basement, Santini goes upstairs, checks his watch, and waits until the waiter goes to the office. Speaking over the radio, Santini bets the waiter $5 that he can guess a number he's thinking of from 1 to 4. The waiter gives a number and Santini directs him to a skull with a piece of paper beneath it saying that he'd pick the number 4.
Jerome is typing a letter to the Department of Immigration telling them who Santini really is. With time running out, Santini picks the lock on his office door, comes inside, and murders Jerome with his silenced gun as the manager walks over to see who it is. Santini puts the gun on the floor, takes the incriminating letter, and sees Jerome's letter to Immigration on the typewriter. He removes it and calls Harry. Impersonating Jerome's voice, Santini tells him to send up some coffee.
Time runs out and Della has the hooded assistants remove the trunk from the water tank. They unlock the trunk and... Della emerges. One of the assistants removes his hood, revealing that he's Santini. Thackery comes in and tells Harry that he's found Jerome while Santini watches with satisfaction.
The police are called to the Cabaret and Columbo arrives, wearing a new raincoat. He meets with Sgt. John J. Wilson, the officer in charge of securing the scene. Wilson is holding all of the customers so that Columbo can talk to them, but doesn't recognize Columbo at first in his new raincoat. Columbo explains that the raincoat is a present from his wife but thinks that it's too stiff. Wilson brings him up to speed and Columbo notices that Jerome was sweating heavily before he died. As they work, Wilson says that it's a great honor to work with Columbo, and he learned a lot since their last case.
In his basement office, Santini burns the old man's letter and pockets the radio transmitter. Harry comes down to bring him upstairs to meet with the police and Santini goes up. Meanwhile, Columbo finds it interesting that Jerome was shot from the front, and Wilson explains that Harry provided the time of death by confirming when he received the request for coffee, and when Zachery found the body. None of the night's take is there, meaning robbery isn't the motive. Columbo points out that the body isn't near the door, meaning Jerome didn't open it to let in the murderer even though he was shot from the front. When Wilson explains that he held all of the customers, Columbo tells him that they don't need them and that they can send them home. Wilson points out that the door lock is new and the only key was on Jerome's person. According to Harry, Jerome locked the door every night when he counted the receipts. Columbo has Wilson take the lock to the lab and leaves. He almost manages to "forget" the new raincoat until Wilson helpfully brings it to him.
Della and Santini are preparing to leave when the magician notices Columbo preparing to go backstage. Santini refuses to let him go back and Columbo realizes who he is. The lieutenant introduces himself and Santini makes sure that Columbo knows that he was on-stage doing his water tank illusion when Jerome was murdered. The magician offers any assistance that he can, while Danny arrives and meets with Della.
The next day at the lab, the technician informs Columbo that the lock was picked. The gun was stolen from a wholesaler in San Francisco and there are no prints or record of ownership. Columbo takes the lock with him and then goes back to the Cabaret, leaving his coat in the car. Santini is on stage practicing when the lieutenant comes in and watches him. The magician finally takes a break and asks if Columbo has discovered anything. Columbo asks for his opinion and Santini suggests that a pro left the gun and was hired by the mob, who were connected with Jerome. The lieutenant then asks if he's in the trunk during the illusion and Santini doesn't want to give away his trade secrets. Columbo persists and Santini says that he has the best alibi of all and the worst, since he can't prove where he was without ruining the illusion. The magician finally says that he didn't murder Jerome but won't produce an alibi unless he absolute has to. As Santini leaves, Columbo remembers one more thing and asks if he can come to see the show. Santini says that he'd be glad to have him there.
Wilson tracks down Columbo at a locksmith and brings his coat. He explains that the lieutenant left it in the cafeteria. Columbo then asks the locksmith, Mr. Lassiter, to have his request for handcuffs ready by 8 that night. Wilson asks to talk to Columbo privately and explains that Danny was having an argument with Jerome to get out of his contract, and has no alibi. However, Columbo explains that Danny couldn't have picked the lock so he couldn't be a murderer. As Wilson leaves, Columbo tells him to leave the new raincoat in his car.
That night, Danny is performing while Santini flirts with a customer and then claims that he'll try to contact Jerome's spirit. Columbo comes over and asks the magician to get him a table on the main floor. Santini arranges for Columbo to use the guest of honor table, and Columbo explains that he brought a trick that he wanted to challenge Santini with. The magician brushes him off to talk to an old friend, Lally, and then goes backstage to prepare for his performance while Columbo takes his seat.
Danny finishes his song and Santini begins his act, stopping at Columbo's table briefly before going on stage. The magician then "produces" Della and proceeds with his on-stage routine. He then calls Columbo up as a volunteer and tells the audience that they have a police lieutenant as a guest of honor. Santini produces cards from out of Columbo's clothing. Columbo finally challenges Santini with a pair of handcuffs and the audience applauds. Santini egotistically accepts the challenge and Columbo explains that the handcuffs are supposed to be inescapable. The magician turns his back and after a few moments of concentration, frees himself and returns the handcuffs to Columbo. Columbo assures the magician that he knew he could do it.
Santini next begins his water tank illusion. As Della introduces the act, Columbo slips away and talks to the bartender, who confirms that Santini does the same act at the same time every night. Once the trunk is lowered into the case, Columbo tries to go downstairs but George refuses to let him go. Wilson is there and wants to talk about Harry, who owned 25% of the club and now owns it all with Jerome's death. Columbo says that he'll talk about it later and goes to the kitchen. He talks to Harry and gets directions to the basement. Once there, Columbo calls out to Santini, who calls him into his office. Santini isn't surprised to see him, and Columbo realizes how the illusion works. Columbo then asks if Santini can verify if he was in the basement office. Santini points out that no matter which of the two staircases he went up, someone would have seen him. Columbo, apparently satisfied, congratulates him on the handcuff escape and notes that the timing must be critical. As Santini leaves to "appear" on stage, he offers Columbo his nightly brandy and explains that Thackery brings it to him at the same time each evening.
Columbo goes back up to the chaotic kitchen and finally yells, getting Thackery to identify himself. The waiter explains that he brings Santini the brandy each night, and that Santini was there when Jerome was murdered. However, Thackery admits that he only heard Santini, he didn't see him, and that they did their nightly bet on a mental trick. Harry comes over and Columbo asks if he would have noticed a new man on duty. The maître'd says that they were running short and the temp agency sent over new waiters, and he wouldn't have noticed a stranger.
Next, Columbo goes to a magic shop and asks the owner how someone could have thrown his voice. The owner explains that a radio transmitter could be used, like one in a mind-reading act.
Columbo then goes to see Michael Lally, the performer from the cabaret. Lally says that Santini has always been great and that they go back to Europe where Santini ran a mind-reading act. He can't remember the name that Santini used then but tells Columbo that Santini had a German accent but an English name. Five years later, he met Santini again and the magician used an English accent and an Italian name. Lally admits that Santini is an expert mimic but doesn't know his real name.
Santini is at a restaurant chatting with a potential new female assistant and explains that his daughter has gone soft over a singer. Columbo comes in and apologizes for intruding, but insists that he has to check out every lead. He notes that Santini uses his assumed name on his passport but Santini ducks the question and refuses to provide any further information on his background.
Back at the station, Columbo calls in Wilson, who suggests that Thackery is the killer. Columbo tells him that Santini is the killer and demonstrates that Santini could have pulled off his alibi using a radio transmitter. The lieutenant demonstrates to Wilson that the "pick a number from 1 to 4" trick is simple, and that he bought the secret from the magic store owner. Wilson concedes that Santini could have done it but they need a motive, and Columbo admits that he doesn't know it yet. As they go, Columbo goes back to get his raincoat and notices a man drinking water from a fountain. He comes back with his old coat and tells Wilson that the new one didn't fit. Columbo then approaches the man at the fountain and points out that his back is covered in perspiration. The officer confirms that he was sitting in a leather chair, and Columbo realizes that they have to figure out where Jerome was sitting.
Columbo and Wilson go back to Jerome's office and look for a leather chair. There are two of them but one is too high to match the perspiration on Jerome's shirt. The remaining chair is in front of Jerome's typewriter and he had reading glasses on the desk. Jerome never counted the money in the money box, and they figure he had something more important to do. Santini storms in and demands to know how long they're going to continue their investigation. Columbo has Wilson explains the "1 to 4" trick, and then tells the magician that they know where Jerome was sitting. The lieutenant admits that they consider Santini the suspect but the magician bluffs it out and says that he isn't going anywhere. Once he leaves, Wilson points out the typewriter but Columbo notes there was no typewritten documents present. However, the lieutenant looks at the typewriter and notices the type ball inside. Wilson demonstrates how it works and how there is a disposable carbon ribbon inside. Intrigued, Columbo congratulates Wilson and smiles in satisfaction.
Later, Santini arrives at the cabaret and asks Harry what's going on. Harry tells him that Danny is in the office with Columbo, and Columbo summoned Santini there. The magician goes up and hears Columbo and Danny discussing how Santini murdered Jerome because he was going to expose the magician as a SS guard. Santini goes back downstairs but a policeman stops him. Wilson comes down and Santini goes to the stage, but Columbo emerges from backstage and tells him there's nowhere to run to. The lieutenant explains that they used a radio transmitter to make it sound like he was upstairs, just like Santini used it in the basement.
Santini congratulates Columbo but tells him that he's wrong. Columbo produces the letter that Jerome was typing. Santini takes it, reads it, and then makes it disappear in a burst of flame. However, Columbo trumps him by producing the same letter from his coat pocket. He tells Santini he can burn it but Columbo produces several others, and Wilson has a copy as well. Columbo tells the magician that he made one mistake: he didn't check the disposable ribbon. The type ball left an impression on the ribbon, and Columbo shows Santini the ribbon with the entire letter's content on it. Santini admits that he thought he'd performed the perfect murder, but Columbo points out that a perfect murder is just an illusion and has him taken away.