Question

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?

Answer 1

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