Consider a $1$ by $2$ by $3$ rectangular prism. Find the length of the shortest path between opposite corners $A$ and $B$ that does not leave the surface of the prism.
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To find the shortest path between opposite corners \( A \) and \( B \) on the surface of a \( 1 \times 2 \times 3 \) rectangular prism, we can unfold the prism into a 2D plane and then find the shortest distance between the corresponding points.

1. **Unfold the Prism:**
   We can unfold the prism in such a way that the path from \( A \) to \( B \) remains on the surface. There are several ways to unfold the prism, but we need to consider the configurations that will give us the shortest path.

2. **Possible Unfoldings:**
   Consider the following unfoldings:
   - Unfolding along the \( 1 \times 2 \) face and the \( 2 \times 3 \) face.
   - Unfolding along the \( 1 \times 3 \) face and the \( 2 \times 3 \) face.
   - Unfolding along the \( 1 \times 2 \) face and the \( 1 \times 3 \) face.

3. **Calculate Distances:**
   For each unfolding, we calculate the Euclidean distance between the points \( A \) and \( B \).

   - **Unfolding 1:**
     - Unfold the \( 1 \times 2 \) face and the \( 2 \times 3 \) face.
     - This results in a \( 3 \times 3 \) rectangle.
     - The distance between \( A \) and \( B \) is the diagonal of this rectangle:
       \[
       \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}
       \]

   - **Unfolding 2:**
     - Unfold the \( 1 \times 3 \) face and the \( 2 \times 3 \) face.
     - This results in a \( 1 \times 5 \) rectangle.
     - The distance between \( A \) and \( B \) is the diagonal of this rectangle:
       \[
       \sqrt{1^2 + 5^2} = \sqrt{26}
       \]

   - **Unfolding 3:**
     - Unfold the \( 1 \times 2 \) face and the \( 1 \times 3 \) face.
     - This results in a \( 2 \times 4 \) rectangle.
     - The distance between \( A \) and \( B \) is the diagonal of this rectangle:
       \[
       \sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5}
       \]

4. **Compare Distances:**
   - From the calculations above, the shortest distance is \( 3\sqrt{2} \) from the first unfolding.

Conclusion:
The shortest path between opposite corners \( A \) and \( B \) on the surface of the prism is \( 3\sqrt{2} \).

The final answer is \( \boxed{3\sqrt{2}} \)