### Part (a)
1. Define the Concentric Squares:
Let the concentric squares be denoted by $S_1, S_2, \ldots, S_n$. Each $S_k$ is a square of side length $2k$ centered at the origin.

2. Initial Step from Center:
There are 4 ways to move from the center to the first square $S_1$.

3. **Path from $S_k$ to $S_{k+1}$:**
- Once on $S_k$, the path can wander around $S_k$ in one of two senses (clockwise or counterclockwise).
- If the path reaches a corner of $S_k$ (there are 4 corners, each with 2 ways to reach, totaling 8 ways), there are 2 ways to step out to $S_{k+1}$.
- If the path reaches a non-corner point of $S_k$ (there are $8k - 8$ such points), there is only 1 way to step out to $S_{k+1}$.

4. Counting the Paths:
- For each $S_k$, the number of ways to step out to $S_{k+1}$ is $16k - 9$ (since $8 + (8k - 8) = 8k$ and $8k - 1 = 8k - 9$).
- Therefore, the total number of paths is given by:
$$4 \prod_{i=2}^n (16i - 9)$$

### Part (b)
1. Restricting the Paths:
- For pairs of paths that divide the grid into two congruent figures, each path must be symmetric with respect to the center.
- This restricts the wandering on $S_k$ such that no point is touched by more than one path.

2. Counting the Pairs:
- The number of such pairs is given by:
$$2 \prod_{i=2}^n (8i - 5)$$

### Part (c)
1. Restricting the Paths Further:
- For quadruples of paths that divide the grid into four congruent parts, each path must be symmetric with respect to both the center and the axes.
- This further restricts the wandering on $S_k$ such that no point is touched by more than one path.

2. Counting the Quadruples:
- The number of such quadruples is given by:
$$\prod_{i=2}^n (4i - 3)$$