Reference: J.R.R. Tolkien, The Hobbit, George Allen and Unwin, 1937.

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After reading through The Hobbit many times, I once tried to determine whether or not the celestial events in The Hobbit could ever be reconciled. After prompting from Daniel Smith of Beyond Bree, I tried once again. I have now come up with a solution to the problem of the events described in The Hobbit. Unfortunately, the events in The Hobbit are irreconcilable, as my article describes, and thus I present a potential solution that satisfies given constraints and criteria as much as is reasonably possible: in short, the proposal is that Durin's Day is the last full moon before winter. This article gives a much more thorough presentation of this material: A timeline for The Hobbit. I wish Christopher had lived long enough to read it.

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As a point of interest, one of Tolkien's earliest drawings entitled the Death of Smaug contains what was originally a full moon, but at some later point, Tolkien drew in a crescent, and added the side note "The moon should be a crescent: it was only a few [nights?] after the New Moon on Durin's Day." In my document, however, I show that it is exceptionally unlikely there would be sufficient light from the slight sliver of a waxing moon only a few days old, and such a moon would have quickly set after it became in any way significantly visible following twilight. 

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Oh yes, I got it wrong, Mr. Ken Grosskurth, the reason the hole was there in Smaug's armor is that, although he was armored "above and below with iron scales and hard gems," no doubt the constant beating of his heart over the century-and-a-half after he established himself on the Lonely Mountain caused sufficient vibrations so as to first loosen and ultimately dislodge those scales and gems in the immediate vicinity of his heart. Without a constant threat, Smaug likely relaxed and failed to maintain the integrity of his armor, and it is through this chink that the black arrow flew, penetrating his softer underbelly, striking his heart, and killing him almost instantly: "With a shriek that deafened men, felled trees and split stone, Smaug shot spouting into the air, turned over and crashed down from on high in ruin."

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Below is the image drawn by Tolkien, and as this is a criticism of that book and and at least an attempt at scholar research, this should be considered fair usage; however, another source of that image is available here.