Why do Lombard, Armstrong, and Blore decide to search the house and the Island?
Agatha Christie's mystery stories were characteristically limited to a set group of characters, one of whom had to be the murderer because there was nobody else available. "And Then There Were None" was one of her best stories. She set it on an island where there were only ten characters present and they were cut off from the mainland. One of them was murdering the others. She has several of those characters make a thorough search of the house and the island for their own reason, because they suspect that someone else is hiding somewhere; but the real purpose of having them conduct the thorough search is to establish for the reader that that there was no eleventh person on the island and that the murderer had to be one of the ten who are all known to the reader by name and description. Otherwise the conclusion to the story could be jeopardized. At the last moment the real murderer, someone the reader knew nothing about, could come out of hiding and spoil the effect. This would be somewhat like a deus ex machina ending, which is usually a letdown.
When they all decide to search the island, they have decided that the deaths that have already occurred, Anthony Marston and Mrs. Rogers, "fits too damn well to be a coincidence" (pg 118) They figure out the there is a Unknown soldier amongst them. U.N. Owen (unknown), their host. However, they know that the person is not among them, and they figure that he is out on the island, lying in wait to kill each of them. The boat is not coming back for them, and they know that they need to find him before he takes their lives. Blore meets General Macarthur and says to him "We're just making a tour of the island so to speak. Just wondered, you know, if someone might be hiding on it" (pg 124)