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Interview prep / Brainteaser/Logic

A small glass covering three of 51 ants

Question

Fifty-one ants are placed anywhere inside a square of side 11. Prove that a circular glass of radius 17\tfrac17 can always be positioned to cover at least three ants.

Short answer

Split the square into a 5×55\times5 grid. With 5151 ants in 2525 cells, some cell holds at least 33 ants. Each cell has side 15\tfrac15, so its circumscribed circle has radius 210≈0.1414<17≈0.1429\tfrac{\sqrt2}{10}\approx0.1414<\tfrac17\approx0.1429, and a glass centred on that cell covers it entirely.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 2.6 The Pigeon Hole Principle, 'Ants on a square', p.21; question and answer rewritten by SelfMadeQuant.com.

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