To find the third side length of the corresponding triangle given that the triangles are similar and two sides correspond, we can use the principle of proportionality resulting from similar triangles.
Given the first triangle with side lengths 1, 4, and 6, we have the corresponding sides of the second triangle as 3 (which corresponds to side length 1 of the first triangle) and 12 (which corresponds to side length 6 of the first triangle).
Let the third side of the first triangle be \( s \) (the side length corresponding to the second triangle's unknown side), and we denote the length of this third side of the second triangle as \( x \).
The ratios of the corresponding sides must be equal due to the triangles being similar:
\[ \frac{1}{3} = \frac{6}{12} = \frac{4}{x} \]
From the information provided, we can simplify \( \frac{6}{12} = \frac{1}{2} \) to show that the scale factor from the first triangle to the second triangle is \(\frac{1}{3}\).
We need to solve for \( x \):
\[ \frac{4}{x} = \frac{1}{3} \]
Cross-multiplying gives:
\[ 4 \cdot 3 = 1 \cdot x \] \[ 12 = x \]
Thus, the third side length of the second triangle corresponding to the third side length of the first triangle is \( \boxed{12} \).