CandidGamera: I can’t express this very mathematically nor tie it into some ultimate proof, but it seems to relate to what you said…
If you imagine a square, n by n, and you want the next square. Visualise that square - you need to add a line on top, a line on bottom and one in the corner. Hence, n^2 2n + 1. 2n + 1 is a function that gives all odd numbers, which is why they appear as the differences between terms.
Visualise a cube - to get the next cube along, you need to add three squares (on top, and two sides), three lines to go in between and another corner piece. Hence n^3 + 3n^2 + 3n + 1
Now visualise a 4 dimensional cube…
…j/k. But I think you see what I’m stabbing at. We can extrapolate that turning a 4D cube of order n into one of order n + 1 would require 4 Cubes, 6 Squares, 4 Lines, and the customary corner piece.
…
As for my own deductions, I can’t really think of any truly sherlockesque moments, but…
I came up with the formula for the sum of n terms of an arithmetic sequence back in 7th grade.
In one of my primary school classes, we had a tank with snails in. Our teacher explained how “the snails are both male and female”, yet it took me to deduce that when we found one set of eggs, there had to be another. There was… damn, I was proud of that.
This is far from a deduction, and I’ve actually reversed part of this thinking since… but in one of my first years at school, I decided that God didn’t really make sense. This went down well in a song where it was still customary to sing Hymns. To be honest, I can’t have been basing it on anything serious, and it’s funny how it took longer for me to question Santa… 
