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 Solution(?) by Dej Mar)

looking at your solutions I get the following exact values for the distances

2 seedlings
1+Sqrt[2]

3 seedlings
1+Sqrt[2]+Sqrt[2-Sqrt[2]]

4 seedlings
2+Sqrt[2]+Sqrt[2-Sqrt[3]]

5 seedlings
2+Sqrt[2]+2*Sqrt[2-Sqrt[3]]

having trouble finding an exact value for 6 seedlings without knowing what 0.3632 is exactly,  I was able to reverse engineer the other decimal values used but this one seem to evade me :-).

Now as for seedlings 1-5 the square roots scream to me that there is a pattern to the total distances, but I am unable to discover this pattern as yet.

Posted by Daniel on 2006-08-30 22:00:42