A cylindrical log has diameter $ 12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $ 45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $ n\pi,$ where $ n$ is a positive integer. Find $ n.$
1. **Determine the dimensions of the cylinder:**
   - The diameter of the cylindrical log is given as \(12\) inches.
   - Therefore, the radius \(r\) of the cylinder is:
     \[
     r = \frac{12}{2} = 6 \text{ inches}
     \]

2. **Understand the cuts:**
   - The first cut is perpendicular to the axis of the cylinder, which means it is a horizontal cut.
   - The second cut forms a \(45^\circ\) angle with the plane of the first cut and intersects the first plane at exactly one point on the log.

3. **Visualize the wedge:**
   - The wedge formed by these cuts is a right triangular prism with the base being a right triangle.
   - The height of the cylinder (and thus the height of the wedge) is the diameter of the log, which is \(12\) inches.

4. **Calculate the area of the triangular base:**
   - The base of the triangle is the diameter of the log, which is \(12\) inches.
   - The height of the triangle is the radius of the log, which is \(6\) inches.
   - The area \(A\) of the triangular base is:
     \[
     A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 6 = 36 \text{ square inches}
     \]

5. **Calculate the volume of the wedge:**
   - The volume \(V\) of the wedge is the area of the triangular base times the height of the cylinder:
     \[
     V = A \times \text{height} = 36 \times 12 = 432 \text{ cubic inches}
     \]

6. **Express the volume in terms of \(\pi\):**
   - The problem states that the volume can be expressed as \(n\pi\). Since the volume we calculated is \(432\) cubic inches, we need to find \(n\) such that:
     \[
     n\pi = 432
     \]
   - Solving for \(n\):
     \[
     n = \frac{432}{\pi}
     \]

However, this does not match the form \(n\pi\) directly. Let's re-evaluate the problem:

7. **Re-evaluate the volume calculation:**
   - The volume of the cylinder is:
     \[
     V_{\text{cylinder}} = \pi r^2 h = \pi (6^2) (12) = 432\pi \text{ cubic inches}
     \]
   - Since the wedge is half of the cylinder:
     \[
     V_{\text{wedge}} = \frac{1}{2} \times 432\pi = 216\pi \text{ cubic inches}
     \]

Thus, \(n = 216\).

The final answer is \( \boxed{216} \).