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 The gardener's woe is continuous (Posted on 2006-08-30)
In Gardener's Woe http://perplexus.info/show.php?pid=4770 the seedlings could only come up at integral locations. Now let's allow for the seedlings to be anywhere within a unit square:

A gardener has broadcast seeded a 1 meter plot and only some of the seeds sprouted. A hungry slug is eating the first seedling, which is at position (0,0). When it is finished it will go directly to the next closest* seedling and so on until all have been consumed. Where are the seedlings located if the total distance travelled is maximized?

Solve for 2,3,4,5,6 etc. seedlings inside the square garden?

*An important change needs to be made from the original version:
The slug may now have to choose between equidistant seedlings. In any situation where it would have to choose, you may choose the desired path for it.

 No Solution Yet Submitted by Jer No Rating

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 Solution(?) | Comment 1 of 5

2 seedlings:
approximate distance: 1.4142 meters
path: (0, 0) --> (1, 1)

3 seedlings:
approximate distance: 2.4142 meters
path: (0, 0) --> (0, 1) --> (1, 0)

4 seedlings:
approximate distance: 3.1796 meters
path: (0, 0) --> (0.7071, 0.7071) --> (0, 1) --> (1, 0)

5 seedlings:
approximate distance: 3.9319 meters
path: (0, 0) --> (0.5, 0.8660) --> (1, 1) --> (0, 0)
--> (0, 1)

6 seedlings:
approximate distance: 4.4320 meters
path: (0, 0) --> (0.5, 0.8660) --> (0.8660, 0.5)
--> (1, 1) --> (0, 1) --> (1, 0)

7 seedlings:
approximate distance: 4.4959 meters
path: (0, 0) --> (0, 0.3536) --> (0.3536, 0) --> (0.5, 0.5)
--> (1, 1) --> (0, 1) --> (1, 0)

*(correction made...slug begins at the the first seedling!)

Edited on September 19, 2006, 5:55 pm
 Posted by Dej Mar on 2006-08-30 18:56:11

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