One of the very first number theory concepts introduced to students -- even before primeness, divisibility, etc. -- is the idea that a natural number can either be "even" (that is, evenly divisible by 2) or "odd" (all other numbers). For all intents and purposes, at that time, even and odd numbers were evenly distributed and of the same density in the natural numbers.
There always seemed to be something inherently "special" about a number's evenness or oddness, besides the trivial one that members of one class could be divided equally into two groups and the other cannot. One would memorize addition and multiplication tables of evenness and oddness (an odd number will remain odd when added to ANY even number). You could not iterate through all of the even numbers through multiplication alone, while you could for the odds.
I've heard the chess board being described as possessing evenness and oddness. For example, dark squares and light squares can represent either even or odd. A diagonal move can be an "even" move and a side-to-side move is an "odd" one, and moves are represented as additions to a square.
In this way, a knight's move is an "odd" move (odd+odd+even), and when added to an odd square will yield an even square; when added to an even square will yield an odd square (odd + odd = even, odd + even = odd) A bishop's move can be considered always even, so once a bishop is on an odd square, it can only ever move onto other odd squares. Likewise for bishops on even squares.
Are there any more generalizations of this concept to math? Is it meaningful to talk of even or odd matrices, or even or odd vectors or vector spaces?
I've heard the concept applied to functions (even or odd functions), but I don't know if they are related to this by anything other than their name.