This episode was based on the novel Three Act Tragedy, first published in 1934. It was adapted for television by Nick Dear and directed by Ashley Pearce, who also teamed up for the adaptation of Mrs McGinty's Dead.
Script versus novel
Nick Dear's script stays remarkably true to the novel, with only some minor changes. First, Dear introduces Poirot much earlier than in the book (as has become the norm in the adaptations). He becomes a long-time friend of Charles Cartwright (a change that actually adds to the depth of the story), and he consequently replaces Mr. Satterthwaite (who is deleted). Second, Egg Lytton-Gore gets a somewhat more active part in the investigation, joining, for instance, the search of Ellis's room. Third, Egg's interview with the model from Cynthia Dacres' store is removed (it added little to the plot). Fourth, the character of Angela Sutcliffe (Charles' former lover) is removed - and she doesn't feel missed. Other than that, only minor changes are made to the plot, with some interview sections shortened and some lines changed. As a result, I think we can safely say that Dear's adaptation is a faithful retelling of its source material.
Directing, production design, locations, soundtrack
Ashley Pearce's direction here is probably not to everybody's taste, but I think the slight artificiality works well. I enjoy the many hints towards the link between the case and the theatre (the theatrical opening scene, the presentation of the dramatis personae, the murders, and the denouement, fittingly set on a stage. Also, I think the scene in which Poirot builds the house of cards (a homage not only to the novel but also to the adaptation of 'The Disappearance of Mr Davenheim') is absolutely exquisite. The production design for this episode is impressive, with several different sets (both real and artificial) beautifully dressed. The locations used are perfect for the adaptation. These include St. Anne's Court in Chertsey (previously seen in 'The Plymouth Express' and Mrs McGinty's Dead) and the entrance hall of Eltham Palace (previously seen in Death on the Nile) doubling as Crow's Nest. Also, the Novello Theatre in London, Knebworth House in Hertfordshire (doubling as Dr Strange's house), Villa Maria Serena in Menton, France (previously seen as Villa Marguerithe in The Mystery of the Blue Train, doubling as the Majestic in Monte Carlo), Wandsworth Town Hall (doubling as the interior of the Majestic), Claridge's Hotel in London (doubling as Ambrose store), Paddington train station (footage last scene in the Marple episode 4.50 to Paddington, I believe), the Bluebell Railway Pullman carriages (used for the trian journeys), Wansford Station in Peterborough (doubling as Loomouth station), and Little Marlow St John the Baptist Church (used for the exhumation scene). Christian Henson, who took over from Stephen McKeon as composer for Series Twelve and Thirteen, does an excellent job with the soundtrack, perfectly mixing Gunning's theme with his own musical trademarks.
Characters and actors
By now, Poirot (or rather, Suchet) is starting to move into real retirement, with lunches at the Ritz and yearly visits to Monte Carlo. His friendship with Cartwright somehow seems perfectly natural in this stage of his life. I like that David Yelland (George) is given a tiny cameo in which he is clearly familiar with Cartwright. A small continuity thing I've only noticed re-watching the episode is the fact that Egg is seen reading a book called Travels in Arabia by Dame Celia Westholme (a reference to the adaptation of Appointment with Death from Series Eleven). Of the guest actors, Martin Shaw is the main standout, with a performance perfectly balancing the theatricality of the character (love his reference to Stanislavsky, and the roles he have played in the theatre) with the human being, particularly in the denouement scene, which he manages to make both touching and slightly creepy. However, Kimberley Nixon (Egg Lytton-Gore), Art Malik (Sir Bartholomew) and the rest of the cast do an excellent job, too.