Ants on a 1-Meter Stick (Elastic Collisions Pass-Through Equivalence)

Scenario

There are 100 identical ants placed at arbitrary initial positions along a 1-meter ($100\text{ cm}$) stick. Each ant is independently oriented facing either Left or Right and moves with a constant speed of $1\text{ cm/s}$. When two ants collide, they instantly and elastically bounce off each other, reversing their directions while maintaining speed. When an ant reaches either endpoint of the stick ($0\text{ cm}$ or $100\text{ cm}$), it falls off the stick immediately.

Your Task

1. Explain the physical and combinatorial equivalence between identical 1D elastic collisions and ghost "pass-through" particles.
2. Determine the maximum possible time (in seconds) until the stick is guaranteed to be completely free of ants, regardless of initial placements and initial orientations.
3. State the necessary and sufficient initial configuration that achieves this theoretical supremum.

Constraints

• Stick length $L = 100\text{ cm}$, speed $v = 1\text{ cm/s}$.
• Infinitesimal point-mass ants (zero body length).

Examples If an ant starts at $x = 0.1\text{ cm}$ facing Right and never collides, it falls off at $t = 99.9\text{ s}$. Collisions do not prolong this maximum transit time.