The play is set in the Great Hall of Monkswell Manor, in what Christie described as "the present".
Act I opens with the murder of a woman named Maureen Lyon, played out in sound only. The action then moves to Monkswell Manor, recently converted to a guesthouse and run by a young couple, Mollie and Giles Ralston. While waiting for the guests to arrive, Mollie listens to a radio report about the Lyon murder, which notes that police are looking for a man in a dark overcoat, observed near the scene.
Their four guests arrive. Christopher Wren is an unkempt, flighty young man. Giles reacts strongly to Wren with instant dislike and Mollie with instinctual trust. Mrs Boyle and Major Metcalf then arrive together in a taxi from the station. Mrs Boyle complains about everything; Metcalf is an amiable ex-military man. Miss Casewell, a mannish young woman, is the last of the booked guests to arrive, before an unexpected fifth party appears. Identifying himself in a foreign accent as Mr Paravicini, he tells the Ralstons his car has overturned in a snowdrift. He remarks that the snow has blocked the roads and that the denizens of the house are trapped. Uneasy about Paravicini's manner, Mollie nevertheless places him in the last remaining room.
The next afternoon the guest house proves to be snowed in, and the residents are restless. Mollie answers the telephone to Superintendent Hogben of the Berkshire Police. Hogben tells her that he is dispatching Sergeant Trotter to the guest house, and that the Ralstons must listen carefully to what Trotter has to tell them. The Ralstons wonder what they could have done to garner police attention.
Trotter appears at the door on a pair of skis and Major Metcalf discovers that the phone has stopped working.
Trotter explains he has been sent in regard to the murder of Maureen Lyon. In a story loosely based on the real Dennis O'Neill case the dead woman and her husband had mistreated their three foster children resulting in the death of the youngest. Both adults were imprisoned for their actions; the husband died in gaol, while the wife served her sentence and had been released, only to be found strangled. Police suspect the elder boy of the abused children, who would now be twenty-two, of being the killer.
Trotter reveals that a notebook found at the murder scene contained the address of Monkswell Manor and the words "Three Blind Mice". A note reading "This is the First" was pinned to the woman's body. The police have sent Trotter to find out how the Ralstons' guesthouse is connected to the murder, and whether the residents are in danger. Both Giles and Mollie deny a connection to the case, though Mollie is ill at ease answering Trotter's questions and quickly excuses herself. Trotter asks each of the guests to explain why they are at Monkswell Manor and any connection they have to the foster children. All five guests deny any personal knowledge of the case.
While Trotter and Giles tour the house, Major Metcalf confronts Mrs Boyle, revealing that she was one of the magistrates who had assigned the children to the foster parents. Mrs Boyle acknowledges this but denies that she has any responsibility for what eventually happened to the children there.
As the evening wears on, Giles and Mollie become suspicious of each other while the guests snipe at one another. Sergeant Trotter traces the phone wire to find out if it has been cut. Mrs Boyle wanders back into the now-empty room and listens to the radio. The opening notes of "Three Blind Mice" are heard whistled by an unknown party, and Mrs Boyle responds without alarm, speaking to the person only she can see. Suddenly, the lights go out and a scuffle is heard. Moments later, Mollie walks into the room and turns on the lights, only to find Mrs Boyle dead on the floor.
Ten minutes after Mollie found Mrs Boyle dead of strangulation, Sergeant Trotter has taken charge of the household. All the remaining residents are gathered in one room as he attempts to sort out the events of the evening. A shaken Mollie Ralston cannot provide him with any useful clues; the only thing she is sure she observed was the radio blaring. Frustrated, Trotter points out that their lives continue to be in danger; a third murder could very well happen, given the notes left with Maureen Lyon. He insists that everyone tell him where they were when Mrs Boyle was murdered. As each person recounts his or her whereabouts, Trotter takes them to account for inconsistencies or weaknesses in their stories. Finally, he declares that everyone in the house had the opportunity to commit the murder, since each of them was alone at the time. Giles counters that while seven people in the house lack alibis, only one fits the description of the man the police suspect to be the murderer: Christopher Wren. Wren insists that it is all a frame-up, and Trotter acknowledges that he lacks any evidence pointing to Wren in particular.
Mollie later pulls Trotter aside; Trotter says that while the police suspect the elder boy to be the killer, the dead boy also had relatives and loved ones who might be interested in revenge: the children's father, an Army sergeant, for example; or the dead boy's sister, who would now be a young woman. Trotter notes that Metcalf or Paravicini could be the father, Miss Casewell or Mollie could be the sister, and Giles could be the elder boy. Mollie, aghast, objects to the notion that either she or Giles could be a murderer, but Trotter forces her to admit that they know little about each other's pasts.
Mollie soon finds herself in conversation with Christopher Wren, who confesses that he is actually an Army deserter hiding from his past under a false name. Mollie acknowledges that she, too, is running away from her past. Despite the trust Christopher and Mollie are forming, he and Giles each suspect the other and nearly come to blows over Mollie. The situation is only defused by the arrival of Paravicini, who tells the company that Trotter's skis are missing.
Trotter again calls an assembly of the household, declaring that he now intends to check the alibis everyone provided to him after Mrs Boyle's murder. They will re-enact the murder, with each member of the household acting out another's alibi. Trotter's hope is that while the most of the alibis will be verified, one will be proved impossible. Each person is to go to his or her assigned position and stay there until summoned back by Trotter. The household obediently disperses, leaving Trotter alone onstage.
After the role-players scatter, Trotter sits for a moment before calling for Mollie. He tells her that she has risked extreme danger by not identifying herself to him; he now knows that she was once the schoolteacher of the doomed Corrigan children. She failed to answer a letter the younger boy sent her at the time, begging to be rescued from the farm. Mollie protests that she had been seriously ill when the letter arrived, and was unable to even read it until well after the boy was dead. To this day, she says, she is haunted by her failure to help the children out of their circumstances.
Trotter takes a gun out of his pocket and points it at Mollie, telling her that though she assumed he was a policeman, she only believed that because he had rung up beforehand, playing the role of his own superintendent. Trotter is, in fact, Georgie, the elder Corrigan brother, and he intends to take his revenge on Mollie. Falling back into the demeanour of a wounded child who never grew up, he drops his gun and begins to strangle her, but is stopped by the sudden appearance of Miss Casewell. Casewell calls him by name and reveals that she is Kathy, his long-lost sister, come to take him somewhere safe. Major Metcalf, who accompanied Miss Casewell into the room, summons Giles and tells the frightened innkeepers that he had known all along that Trotter wasn't a policeman– because Metcalf himself is a policeman, having arranged to switch places with the real Major Metcalf after discovering the "Three Blind Mice" notebook on Maureen Lyon.