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

To solve this problem, we need to ensure that no person can sit in a seat without being next to someone who is already seated.

### Step 1: Consider the best possible way to arrange the occupied seats
If we want the fewest number of occupied seats such that the next person has to sit next to someone, we should arrange the seats so that there are as many gaps as possible between the occupied seats.

### Step 2: Consider the arrangement
Suppose you occupy every second seat. This means there will be an occupied seat, then an empty seat, then another occupied seat, and so on. 

For example, if the row has 7 seats and you occupy every second seat:
\[
X \, O \, X \, O \, X \, O \, X
\]
Here, \( X \) denotes an occupied seat and \( O \) denotes an empty seat.

In this arrangement, the next person would still have a free seat that does not require them to sit next to someone, so this arrangement does not yet force someone to sit next to another person.

### Step 3: Calculate the maximum number of empty seats
If you occupy every second seat, there would be \( \lceil \frac{120}{2} \rceil = 60 \) seats occupied, and 59 empty seats in between.

In this case, the 61st person who tries to find an empty seat would be forced to sit next to someone, because the 60 available seats are all adjacent to an occupied one.

### Conclusion:
Thus, the fewest number of seats that must be occupied to ensure the next person must sit next to someone is \( \boxed{60} \).

Would you like further clarification or have any additional questions?

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Here are some related questions:
1. What if the row had 200 seats instead of 120?
2. If 30 people were seated randomly, what’s the probability the next person would have to sit next to someone?
3. How many ways can you arrange 60 people in a row of 120 seats?
4. What is the maximum number of seats that can remain empty while ensuring the next person has to sit next to someone?
5. How many different arrangements can force the 61st person to sit next to someone?
6. How would the problem change if the row were a circle instead of a straight line?
7. What if there were 2 rows of 60 seats each?
8. If every third seat were occupied, how many people must be seated to force the next person to sit next to someone?

**Tip:** For problems involving seating arrangements, consider boundary cases—like fully alternating occupied and unoccupied seats—to find the minimum or maximum values.

Discover the fewest number of seats that must be occupied in a row of 120 seats so that the next person must sit next to someone. Learn the strategy behind arranging seats to maximize gaps and ensure adjacency. Suitable for high school students studying combinatorics and seating arrangements.

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