M. Bouc and Dr. Constantine attempt to think, though their minds wander. Poirot, on the other hand, stays still and appears to be asleep.
Poirot thinks of something. He opens his eyes. He tells his friends that he has a hypothesis now, but that he'll have to make "certain experiments" (3.3.21) to test them.
The first suggestive point he mentions is that he is surrounded by "people of all classes, of all ages, of all nationalities" – not common this time of year (3.3.22). Also, one passenger didn't show up.
Other suggestive points are discussed. First? The dirty mark on the Countess's passport. What if her first name was actually "Helena" and not "Elena" – she could be tied to the handkerchief. Confirmation of this hypothesis: her damp luggage label on top of her initial on the case.
What else? The murder was planned to look like an outside job, but the train stopping changed all that.
The threatening letters? Not real.
The burned letter? It must mean someone on the train was intimately connected to the Armstrong family, and the letter would have suspicion cast on them.
The handkerchief – was it dropped to cast suspicion?