1. **Stating the problem:** We are given a rectangular prism with dimensions 4 cm (height), 12 cm (length), and 3 cm (width). We need to find the length of the diagonal inside the prism, labeled as $n$. 2. **Formula used:** The diagonal $n$ of a rectangular prism with sides $a$, $b$, and $c$ is given by the 3D Pythagorean theorem: $$n = \sqrt{a^2 + b^2 + c^2}$$ where $a$, $b$, and $c$ are the lengths of the edges. 3. **Applying the formula:** Here, $a=4$, $b=12$, and $c=3$. Substitute these values: $$n = \sqrt{4^2 + 12^2 + 3^2}$$ 4. **Calculate the squares:** $$n = \sqrt{16 + 144 + 9}$$ 5. **Sum the values inside the square root:** $$n = \sqrt{169}$$ 6. **Find the square root:** $$n = 13$$ 7. **Conclusion:** The length of the diagonal $n$ inside the prism is 13 cm.