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.

(In reply to re: Solution(?) by Daniel)

In my earlier post, I had made an error in transcribing the fraction (now corrected).  The fraction 0.3536 is an approximation of the value of SQRT(2)/4.

I have found a longer path for the hungry slug: which is approximately 4.59898 meters:
(0, 0) --> (0, 0.3876) --> (0.3876, 0) --> (0.5, 0.6124)
         --> (1, 1) --> (0, 1) --> (1, 0) 

There yet maybe longer trails of slime.

Edited on August 31, 2006, 5:39 am

Posted by Dej Mar on 2006-08-31 00:47:42