Can you cover the center of a unit circle, such a Indi's circular pit?

You have a number (N) of identical planks of length L (0 < L <= 2). The planks have small width and negligible thickness, but very high strength and rigidity. No plank extends outside the circle. Both endpoints of each plank, with the exception of the N-th plank, must rest on either the circle or another plank. Neither weaving of planks nor cantilever designs are allowed. Covering the center means that the N-th plank crosses over the center of the circle.

For a given N, L is the minimum length plank necessary.
Obviously if N=1, then L=2.

What is the smallest L when N=2? N=3? N=4?

For larger N, is there an optimum algorithm to minimize L?

Can you determine a general relationship between N and L?

N=2 you just make a T.  The horizontal is laid first as a chord and the vertical crosses the center.

Connecting the center of the circle with a upper end of the T forms a right triangle.  Solving the equation (1-L)^2 + (L/2)^2 = 1 gives the solution
L = 8/5

N=3 is the letter H.  With two opposite chords and a third plank spanning.
The equation is (L/2)^2 + (L/2)^2 = 1 which gives the solution
L = sqrt(2)

For N>3 things get more complicated.  (This problem has been listed as the highest difficulty for a reason.)

Posted by Jer on 2010-06-04 16:11:15