2006/07 AHSMC Part 1 Problem 7

Problem 7 of 16

A country has three provinces, each province has three cities, each city has three wards and each ward has three electors. In a two-way election, a candidate wins a ward by getting more votes in the ward, wins a city by winning more wards in the city, wins a province by winning more cities in the province, and wins the election by winning more provinces. Only electors may vote, and they must vote. The minimum number of votes needed to guarantee winning the election is

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We can steal the election by winning as little as $16$ votes. We can win $2$ provinces, $2$ cities within each of these $2$ provinces, $2$ wards within each of these $4$ cities, and $2$ electors within each of these $8$ wards. Hence $(81 - 16) + 1 = 66$ votes are required to guarantee winning the election. The answer is (c).

University of Alberta

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