How do I structure hexagon edge data? In my program, it draws them by offsetting every other row by half of the width, as pictured above. Each tile can be referenced by coordinates, also shown above.



I want to know how many blue tiles are accessible from a certain starting tile and a series of "roads." In this example, three blue tiles can be reached. 

How would I represent roads? This is what I would need to know about it:


how it is rotated (for drawing)
which hexagons it borders
which other roads it touches, if any
where it should be drawn


I assume I could use recursion to to count the accessible blue squares if the above conditions are met.
Let us first calculate the coordinates $C(m,n)$ of the center of hexagon $(m,n)$. We assume that the origin is at the center of hexagon $(0,0)$ and that the hexagon radius (distance from center to vertex) is $r$.  A short calculation shows that hexagon $(1,0)$ has its center at $$C(1,0) = a = (\sqrt{3} r, 0)$$ and hexagon $(0,1)$ has its center at $$C(0,1) = b = (\sqrt{3} r/2, 3 r/2).$$
Therefore, hexagon $(m,n)$ has its center at
$$C(m,n) = m \cdot a + n \cdot b = (\sqrt{3} r (m + n/2), 3 r n / 2).$$
Notice that we should allow negative $m$ and $n$ if we want to cover the whole plane.

To get from the center $C(m,n)$ to one of the adjacent centers we need to add to $C(m,n)$ one of the vectors $a$, $b$, $-a$, $-b$, $a - b$, or $b - a$. (If we add $a + b$ or $-a-b$ we go too far). If we travel half the distance we will end up exactly in the midpoint of a side. Therefore, midpoints of sides have coordinates of the form
$$C(m+1/2, n), C(m,n+1/2), C(m-1/2,n), C(m,n-1/2), C(m+1/2,n-1/2), C(m-1/2,n+1/2).$$
We have discovered how to encode sides: just use half-integer coordinates. But this is silly, it is better to multiply everything by two, which leads to the following system.

We represent both hexagons and sides as pairs of integers $(m,n)$. When $m$ and $n$ are both even this encodes a hexagon whose center is at $$C(m/2,n/2) = (m \cdot a + n \cdot b)/2.$$
If either $m$ or $n$ is odd then $(m,n)$ represents a side of a hexagon whose midpoint is at $C(m/,n/2)$.

It is also easy to tell which two hexagons a given side belongs to: the side $(2 i + 1, 2 j)$ belongs to hexagons $(2 i, 2 j)$ and the side $(2 i + 2, 2 j)$, while the side $(2 i, 2 j + 1)$ belongs to hexagons $(2 i, 2 j)$ and $(2 i, 2 j + 2)$.

Similarly, the sides of hexagon $(2 i, 2 j)$ are: $(2 i + 1, 2 j)$, $(2 i - 1, 2 j)$, $(2 i, 2  j + 1)$, $(2 i, 2 j - 1)$, $(2 i + 1, 2 j - 1)$ and $(2 i - 1, 2 j + 1)$.

We now have a good system for representing hexagons and sides as pairs of integers. We still need to solve the original problem, namely which blue hexagons are reachable from a starting hexagon by traveling along certain "black" paths. This is a graph-theoretic reachability problem if we think of sides as edges in a graph and their endpoints as vertices of a graph (as we reach each black side we verify whether it belongs to a blue hexagon). Many algorithms are known on how to solve this. Let me know if you need more details.