2020/21 AHSMC Part 1 Problem 7

Problem 7 of 16

A doctor's office has a row of chairs, two of which are already occupied. Ruby and Ollie come in and want to sit somewhere in the row, in adjacent chairs, that is, beside each other without a gap. Because of social distancing, there must be at least two empty chairs between them and the persons already sitting. How many chairs at least must the row contain so that Ruby and Ollie can always find a place to sit together, regardless of the location of the occupied chairs?

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Solution

Let $c_1$ , $c_2$ , $c_3$ , $c_4$ , $\ldots$ denote the chairs from left to right. The most unfavourable scenario is when $c_4$ and $c_{10}$ are occupied. In this case Ruby and Ollie can not sit on the left of $c_4$ nor between $c_4$ and $c_{10}$ . They should occupy $c_{13}$ and $c_{14}$ . Therefore to assure the social distancing the row must contain at least $14$ chairs. The answer is (E).

University of Alberta

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