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.

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