Determinants & Inverses · Superchallenge problem

Compute $\det\begin{pmatrix}\frac12 & \frac13 & \frac14\\[2pt] \frac13 & \frac14 & \frac15\\[2pt] \frac14 & \frac15 & \frac16\end{pmatrix}$, the matrix with entries $\dfrac{1}{i+j}$ for $1 \le i, j \le 3$.