Ant Collision Puzzle

The key trick: when equal-speed ants bounce off each other, the overall motion is equivalent to the ants simply passing through one another.

Ants reaching Alice
50
Ants reaching Bob
20
Total collisions
1,000
50 × 20
The identity-swap trick
A bounce can be viewed as two ants exchanging identities.
Alice Bob 50 ants from Bob → Alice 20 ants from Alice → Bob Bounce = exchange identities Equivalent view: trajectories pass through

1. How many reach each side?

Ignore the ants' identities. Each collision just swaps which ant follows which trajectory.

50 left-moving trajectories eventually reach Alice.
20 right-moving trajectories eventually reach Bob.

Therefore:

Alice gets 50 ants.

Bob gets 20 ants.

2. How many collisions?

Pick one of Bob's 50 ants. Its trajectory crosses each of Alice's 20 opposing trajectories exactly once.

50 × 20 = 1,000

So there is one collision for every pair consisting of one ant initially moving left and one ant initially moving right.

Small example: 5 from Bob × 2 from Alice

Every cell below is one opposing pair, hence one collision.

B₁ B₂ B₃ B₄ B₅
A₁ × × × × ×
A₂ × × × × ×

2 × 5 = 10 collisions → therefore 20 × 50 = 1,000 collisions

Alice: 50 ants · Bob: 20 ants · Collisions: 1,000