3 + 9, x (7+12) Divided by Eleventy-one... oh dear

[the guy who be short]

Here in the Books, we enjoy endlessly dissecting little phrases and sentences to the point of sadism. But what of those numbers...?

I'm aware that a few threads already exist on the significance of numbers in Tolkien, and will provide a few links. However, I've created a new thread for new input and also because I want to discuss a few things specifically rather than numbers as a whole.
The Power of Nine
Unoriginality
Deeper symbolism and hidden meanings in the books

Well, enjoy dragging yourself through those.

Now, the first question likely to cause dissent is: Did Tolkien consciously choose numbers in his works as some sort of symbolism?

The first part of the question can be answered with a resounding "Yes." The numbers 3 and 7, in particular, are repeated so often that there is almost no chance of it being down to chance! Observe:

3 Silmarils, 3 rings, 3 houses of the Eldar, 3 houses of the Edain, 3 sons of Finwe, 3 Unions of Elves and Men.

7 Dwarf Houses, 7 Durins, 7 Rings, 7(?) Balrogs, 7 Valar, 7 Valier, 7 Sons of Feanor, 7 Days in the Numenorean week, 7 Stars, 7 Palatíri.

Two other numbers can also be given a mention. 9, obviously, as the number of Nazgul, rings and Walkers.

12 and 144 (the former's square) as the Elves preferred form of counting, the Number of Original Elves and the combined age of Bilbo and Frodo.

Obviously there is far too much here to be disputed. For some reason, Tolkien specifically chose 3, 7 and 12 (and perhaps, though less clearly, 9) to play large parts in his mythology.

Why?

And that's where the fun begins! I have one or two ideas of my own, but I need to do a little research. I'm more than likely to dismiss the usual ideas of numerology - I just don't see Tolkien being interested in that sort of thing. I think his numbers would be likely to have more mythological or religious connotations (I think 7 in particular has some sort of Christian connotation?) than be related to numerology. Until I do get round to doing a little research on the matter, I leave it to you, my dear BarrowDowners, to discuss the issue, as I'm sure that several amongst you will have the necessary knowledge regarding mythology and religion.