The spring constant $k$ is inversely proportional to the length of the spring ($k \propto \frac{1}{L}$).

1. Cutting the Spring: Let the original length be $L$ and the original spring constant be $k$. The spring is cut in the ratio $1:2:3$. The lengths of the segments are: $L_1 = \frac{1}{6}L, \quad L_2 = \frac{2}{6}L = \frac{L}{3}, \quad L_3 = \frac{3}{6}L = \frac{L}{2}$ Since $kL = \text{constant}$, the spring constants of the segments are: $k_1 = 6k, \quad k_2 = 3k, \quad k_3 = 2k$

2. Series Connection ($k'$): When connected in series, the reciprocal of the equivalent spring constant is the sum of the reciprocals of individual constants: $\frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3}$ $\frac{1}{k'} = \frac{1}{6k} + \frac{1}{3k} + \frac{1}{2k} = \frac{1 + 2 + 3}{6k} = \frac{6}{6k} = \frac{1}{k}$ $k' = k$ (Note: Connecting parts back in series restores the original spring).

3. Parallel Connection ($k''$): When connected in parallel, the equivalent spring constant is the sum of individual constants: $k'' = k_1 + k_2 + k_3$ $k'' = 6k + 3k + 2k = 11k$

4. Ratio: $\frac{k'}{k''} = \frac{k}{11k} = \frac{1}{11}$

Thus, the ratio is $1:11$.