Problem: Rectangle $A B C D$ has sides $\overline{A B}$ of length 4 and $\overline{C B}$ of length 3 . Divide $\overline{A B}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline{C B}$ into 168 congruent segments with points $C=\underline{Q_0}, Q_1, \ldots, Q_{168}=B$. For $1 \leq k \leq 167$, draw the segments $\overline{P_k Q_k}$. Repeat this construction on the sides $\overline{A D}$ and $\overline{C D}$, and then draw the diagonal $\overline{A C}$. Find the sum of the lengths of the 335 parallel segments drawn. Answer: By unpacking the problem, we get $2 \sum_{k=1}^{167} \sqrt{\left(\frac{3 k}{167}\right)^2+\left(\frac{4 k}{167}\right)^2}+5$ which, after simplification, gets us to $2\sum_{k=1}^{167}\frac{5k}{167}+5 = 845.$ From this, we subtract $5$ since the extra diagonal $\overline{AC}$ is already covered in this sum. So, we get $840$