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20 October, 09:58

There are 120 seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?

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  1. 20 October, 10:18
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    Let's name the seats 1 through 120.

    Occupy seat 2.

    Leave 2 empty seats. You can't leave 3 empty seats because then the middle seat of the three empty seats is not adjacent to an occupied seat. You can leave only 2 seats empty. Seats 3 and 4 are empty.

    Occupy seat 5.

    Leave 2 empty seats. Seats 6 and 7 are empty.

    Keep on going like this to the end, occupying 1 seat and leaving 2 seats empty.

    Now we need to find the number of occupied seats.

    Think of the entire row being divided into groups of 3 seats.

    The middle seat of each group is occupied.

    Since there are 120 seats in the row, there are 40 groups of 3 seats whose middle seat is occupied. There are 40 middle seats, so there are 40 occupied seats.

    Answer: 40 seats
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