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Some treasure is hidden underground in a small square-shaped island of area $64 km^2$. You have no idea where the treasure is exactly, and no time to dig the whole island anyway.

But, luckily, you do have a very advanced device which can find any underground treasure in the area for a radius of 1 km.

You need to start somewhere and do the search. You can start up the device whenever you want on the island but you don't want anyone to interfere with your search, so you need to find a route which is as short as possible to cover the whole island.

So in the worst case scenario,

at least how many kilometers will it take to find the treasure on the island?

Note that; when you start to use the device, it is continously searching the area while you are moving.

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2 Answers 2

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I do that every week-end to water my garden with a remote-controlled robot sprinkler! In my case it is in meters, not kilometers though.

Here is the best path I came up with over the years. It has length

35.626097 km

And here is how it is done:

![![enter image description here

This is not completely optimal. You should be able to win a few meters by fine-tuning the positions of the critical points. But I think it is close.

PS: Just joking, I don't have a square garden, let alone a robot sprinkler.

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  • $\begingroup$ Did you solve this with the help of a computer using brute force? If not, then I would like to know of your thought process and why you believe that this is close to optimal. $\endgroup$ Commented Aug 20, 2022 at 20:55
  • $\begingroup$ No computer (except Excel for the drawing). Near optimality is just a gut feeling. I have tried a few different paths, this ended up the best. The process was to scan most of the area without overlap and then fix the holes at the angles and optimize locally. The centers of the circles are critial points that are difficult to reach and constrain the path. $\endgroup$
    – Florian F
    Commented Aug 20, 2022 at 21:21
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Here's an approximate solution, obtained by solving a set covering problem on a randomly generated subset of $10000$ points in the $8 \times 8$ island and then solving a traveling salesman problem through the resulting $29$ points. There are some tiny uncovered areas, but I think this solution can be tweaked to cover them.

The total distance traveled is

40.79 km

\begin{matrix} \hline \text{order} & x & y \\ \hline 1 & 0.03269 & 3.89347 \\ 2 & 1.63970 & 3.79365 \\ 3 & 3.12915 & 4.10851 \\ 4 & 2.32068 & 5.49250 \\ 5 & 0.88738 & 5.48734 \\ 6 & 0.38842 & 7.08688 \\ 7 & 1.70432 & 7.21094 \\ 8 & 3.17240 & 7.15717 \\ 9 & 4.53167 & 7.30110 \\ 10 & 5.95387 & 7.28834 \\ 11 & 7.33536 & 7.39629 \\ 12 & 7.06943 & 6.06926 \\ 13 & 7.19784 & 4.82554 \\ 14 & 7.26827 & 3.38731 \\ 15 & 7.18894 & 2.05085 \\ 16 & 7.34469 & 0.57561 \\ 17 & 5.96647 & 0.78093 \\ 18 & 4.46693 & 0.89973 \\ 19 & 3.07901 & 0.75323 \\ 20 & 1.67030 & 0.79909 \\ 21 & 0.22551 & 0.76141 \\ 22 & 0.89378 & 2.29282 \\ 23 & 2.57410 & 2.29616 \\ 24 & 4.02039 & 2.55957 \\ 25 & 5.50652 & 2.49403 \\ 26 & 6.18168 & 3.84952 \\ 27 & 4.86950 & 4.27101 \\ 28 & 5.41861 & 5.58752 \\ 29 & 4.08007 & 5.61891 \\ \hline \end{matrix}

enter image description here

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  • $\begingroup$ @2012rcampion maybe you can apply your approach for covering a square with equal circles. $\endgroup$
    – RobPratt
    Commented Mar 14, 2022 at 20:04
  • $\begingroup$ while you are moving, device is continuously searching all the time. it is not on and off actually. $\endgroup$
    – Oray
    Commented Mar 14, 2022 at 20:21
  • $\begingroup$ OK, then that probably takes care of the parts I thought were uncovered. $\endgroup$
    – RobPratt
    Commented Mar 14, 2022 at 20:26
  • $\begingroup$ @user39583 yes, that's what I get for misinterpreting how the device works. $\endgroup$
    – RobPratt
    Commented Mar 14, 2022 at 20:56

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