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Bounces among two opposing ant colonies

Question

Alice releases her 4040 ants at one end of a thin string and Bob releases his 8080 ants at the other end, all walking toward the opposite end. When two ants meet they both reverse direction. How many collisions take place in total?

Short answer

3200. Treat a collision as two ants passing through each other and swapping identities; the set of positions over time is the same. Then every Alice ant meets every Bob ant exactly once and no other meetings occur, giving 40×80=320040\times80=3200.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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