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MONTHLY NEWS November 1996 Math Olympics Question |
| There is a village with a population of
1,000, each of whom lives in a separate house. Each resident sends just one New
Year greeting
card to his or her closest neighbor. The distances
between houses are varied. No one receives any cards from people living outside
the village. What is the maximum number of cards one villager can receive? Explain
your answer.
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Answer 1 = 5 
Assuming the houses are all the same size, resident A will receive cards from residents B and C if the latter two live further apart from each other than from A. Imagine A, B, and C forming a triangle. For the line BC to be longer than either AB or AC, the angle formed at A must be over 60 degrees--the angle of an equilateral triangle. On a 360-degree plane, the greatest number of such angles is five.
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Answer 2 = 999 
If resident A lives in a huge palace, however, while the 999 others live in small huts, it is possible for the 999 huts to be scattered just outside A's estate so that they are all closest to A.
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