# Problem 532: Nanobots on Geodesics
Bob is a manufacturer of nanobots and wants to impress his customers by
giving them a ball colored by his new nanobots as a present. His
nanobots can be programmed to select and locate exactly one other bot
precisely and, after activation, move towards this bot along the
shortest possible path and draw a colored line onto the surface while
moving. Placed on a plane, the bots will start to move towards their
selected bots in a straight line. In contrast, being placed on a ball,
they will start to move along a geodesic as the shortest possible path.
However, in both cases, whenever their target moves they will adjust
their direction instantaneously to the new shortest possible path. All
bots will move at the same speed after their simultaneous activation
until each bot reaches its goal. Now Bob places n bots on the ball (with
radius 1) equidistantly on a small circle with radius 0.999 and programs
each of them to move toward the next nanobot sitting counterclockwise on
that small circle. After activation, the bots move in a sort of spiral
until they finally meet at one point on the ball. Using three bots, Bob
finds that every bot will draw a line of length 2.84, resulting in a
total length of 8.52 for all three bots, each time rounded to two
decimal places. The colored ball looks like this: In order to show off a
little with his presents, Bob decides to use just enough bots to make
sure that the line each bot draws is longer than 1000. What is the total
length of all lines drawn with this number of bots, rounded to two
decimal places?