Article : La route des fourmis - Colegiul National Emil Racovita (Cluj, Roumanie) Lycée d’Altitude (Briancon)

The problem we were presented with is:
The first ant makes 10 random moves, each having a length of one unit and, at each moves, she leaves a mark. After the tenth move, she reaches food and sends a radio signal back to the anthill. A second ant goes along the first half of the first segment, then, from the middle of it to the middle of the second and so on, until she reaches the middle of the last segment, from where it goes to the food. A third ant follows the same strategy, following the second ant’s path.
a) Is the second path shorter than the first?
b) Will the ants tend to find the shortest itinerary between the two points?
What question B asks for is to actually describe howa given polygonal chain behaves, asymptotically, after the above described algorithm, is repeated infinitely.