The Setup

  • A straight line segment connects Alice and Bob.
  • Bob sends 50 ants toward Alice.
  • Alice sends 20 ants toward Bob.
  • All ants move at equal speed along the segment.
  • Whenever two ants collide, they bounce back and reverse direction.

Question

  1. How many ants reach Alice?
  2. How many ants reach Bob?
  3. How many collisions occur?

Key Insight: Collisions Are Indistinguishable

When two ants collide and bounce back, it’s as if they pass through each other.

  • Because ants are indistinguishable, we can pretend they don’t bounce—they just keep moving in their original directions.

So:

  • Every ant continues along its path unless it collides—but we treat them as if they don’t.
  • That means:
    • All 50 ants from Bob continue to Alice.
    • All 20 ants from Alice continue to Bob.

Final Answers

  • Ants reaching Alice: \( \boxed{50} \)
  • Ants reaching Bob: \( \boxed{20} \)

How Many Collisions?

Each pair of ants traveling in opposite directions will collide once.

  • 50 ants from Bob × 20 ants from Alice = \( 50 \times 20 = \boxed{1000} \) total collisions

Summary

Quantity Value
Ants reaching Alice \( \boxed{50} \)
Ants reaching Bob \( \boxed{20} \)
Total collisions \( \boxed{1000} \)

Reference