In the end of the book it is said that the culprit believes, even though the crimes committed were perfect and the police will be unable to solve it, there are three clues left that can help identify the killer.
What are these three clues?
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Spoilers for the book follow
First, the killer is found with a bullet through the forehead. This is the "mark of Cain". Cain was the first murderer:
And the Lord said to him, Therefore, whoever kills Cain, vengeance shall be taken on him sevenfold. And the Lord set a mark on Cain, lest anyone finding him should kill him (Genesis 4:15).
The precise nature of the mark is not revealed, and there is uncertainty as to whether it was in fact a physical mark due to the difficulty of translating from the Hebrew. However, if it was a physical mark it must have been very obvious to serve its purpose.
How, then, do we deduce that the wound is to be interpreted as a mark? Because the killer in the book is playing God. As Lombard remarks in the book:
That is to say, he’s played God Almighty for a good many months every year. That must go to a man’s head eventually. He gets to see himself as all powerful, as holding the power of life and earth—and it’s possible that his brain might snap and he might want to go one step farther and be Executioner and Judge Extraordinary.
The second clue is that the actual killer is the only one who is not actually guilty of murder. His "crime" was sending Seton to the gallows in spite of a lack of conclusive evidence. However, our killer justifies this by pointing out that Seton was certainly guilty of a great many other things and nevertheless deserved to hang.
The final clue harks back to the poem that gave the book its original (now seen as offensive) title:
Four little Indian boys going out to sea;
A red herring swallowed one and then there were three.
This is the fate of Dr. Armstrong, who is discovered washed up on the beach. The killer had identified Armstrong as the weakest and most gullible member of the group and had tricked him into helping further his plan by deception. A "red herring" is a trick or a distraction and the inference is that the clue should help identify Armstrong as somehow being implicated in the plot.
When I was reading I discovered a constant repetition of certain phrases that allowed me to deduce who it was through the random thought sequences we were shown. I also remarked that he was the only one sent to the island by someone other than U.N. Owen and he was the one who noticed that it sounded like unknown, which reminded me of the way I hint at my siblings where the Easter Bunny hid their chocolates because I can't stand the waiting, and I want my brilliant hiding spots to be shown. It is evident that Lawrence also enjoyed being thought of as brilliant, and that is the reason for his confession. I read a story once about a dentist who murdered his associate and he planned it out very carefully and intelligently. The detective got him to confess by saying he stumbled across things by 'accident' and he got 'lucky'. The accused flung himself into a rage and detailed his entire plan, giving a full confession. That's why Lawrence revealed the name and sent the confession in a bottle.