There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube?

$ 
\textbf{(A)}\ 54
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 72
\qquad\textbf{(D)}\ 90
\qquad\textbf{(E)}\ 96
$
1. **Understanding the Problem:**
   We have 27 unit cubes, each with points marked on their faces. We need to construct a \(3 \times 3 \times 3\) cube using these unit cubes and determine the least number of marked points visible on the faces of the new cube.

2. **Classifying the Unit Cubes:**
   - **Corner Cubes:** These cubes are at the corners of the \(3 \times 3 \times 3\) cube. Each corner cube has 3 faces visible.
   - **Edge Cubes:** These cubes are on the edges but not at the corners. Each edge cube has 2 faces visible.
   - **Face Cubes:** These cubes are on the faces but not on the edges or corners. Each face cube has 1 face visible.
   - **Internal Cubes:** These cubes are completely inside and have no faces visible.

3. **Counting the Different Types of Cubes:**
   - **Corner Cubes:** There are 8 corners in a cube.
   - **Edge Cubes:** Each edge of the cube has 3 cubes, but excluding the corners, there are \(3 - 2 = 1\) edge cube per edge. Since there are 12 edges, there are \(12 \times 1 = 12\) edge cubes.
   - **Face Cubes:** Each face of the cube has \(3 \times 3 = 9\) cubes, but excluding the edges and corners, there are \(9 - 4 = 5\) face cubes per face. Since there are 6 faces, there are \(6 \times 1 = 6\) face cubes.
   - **Internal Cubes:** There is only 1 internal cube.

4. **Marking Points on the Faces:**
   - **Corner Cubes:** Each corner cube has 3 faces visible, and each face has 1, 2, and 3 points respectively. Thus, each corner cube contributes \(1 + 2 + 3 = 6\) points.
   - **Edge Cubes:** Each edge cube has 2 faces visible, and each face has 1 and 2 points respectively. Thus, each edge cube contributes \(1 + 2 = 3\) points.
   - **Face Cubes:** Each face cube has 1 face visible, and each face has 1 point. Thus, each face cube contributes \(1\) point.

5. **Calculating the Total Number of Points:**
   - **Corner Cubes:** There are 8 corner cubes, each contributing 6 points. Thus, the total contribution from corner cubes is \(8 \times 6 = 48\) points.
   - **Edge Cubes:** There are 12 edge cubes, each contributing 3 points. Thus, the total contribution from edge cubes is \(12 \times 3 = 36\) points.
   - **Face Cubes:** There are 6 face cubes, each contributing 1 point. Thus, the total contribution from face cubes is \(6 \times 1 = 6\) points.

6. **Summing Up the Points:**
   \[
   \text{Total points} = 48 + 36 + 6 = 90
   \]

Thus, the least number of marked points on the faces of the new cube is \(\boxed{90}\).