35. $[\leq \mathbf{2 5}]$ The Algorithm. There are thirteen broken computers situated at the following set $S$ of thirteen points in the plane:
$$
\begin{aligned}
A & =(1,10) \\
D & =(377,422) \\
G & =(941,500) \\
J & =(3,713)
\end{aligned}
$$
$$
\begin{aligned}
B & =(976,9) \\
E & =(535,488) \\
H & =(225,583) \\
K & =(504,872) \\
M & =(22,997)
\end{aligned}
$$
$$
\begin{aligned}
C & =(666,87) \\
F & =(775,488) \\
I & =(388,696) \\
L & =(560,934)
\end{aligned}
$$

At time $t=0$, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and $A$ and fixes computers instantly, what path does he take to minimize the total downtime of the computers? List the points he visits in order. Your score will be $\left\lfloor\frac{N}{40}\right\rfloor$, where
$$
N=1000+\lfloor\text { the optimal downtime }\rfloor-\lfloor\text { your downtime }\rfloor,
$$
or 0 , whichever is greater. By total downtime we mean the sum
$$
\sum_{P \in S} t_{P},
$$
where $t_{P}$ is the time at which the repairman reaches $P$.
Answer: ADHIKLEFGBCJM. This is an instance of the minimum-latency problem, which is at least NP-hard. There is an easy $O(n!)$ algorithm, but this is unavailable to teams on computational grounds ( $100 \mathrm{MHz}$ calculators used to seem fast...) The best strategy may be drawing an accurate picture and exercising geometric intuition. The distribution of the points somewhat resembles a short, four-pronged fork with its outermost prongs bent apart; it is plausible to assume that the optimal order respects this shape. The optimal downtime is 24113.147907, realized by ADHIKLEFGBCJM, though a number of others also receive positive marks.