Let's break down the given problem:

We have a board with squares numbered from 1 to 40. The letters in the squares light up according to a certain pattern:

1. At minute 1, all letters light up.
2. At minute 2, only letters in squares with numbers that are multiples of 2 (i.e., even numbers) light up.
3. At minute 3, only letters in squares with numbers that are multiples of 3 light up.
4. At minute 4, only letters in squares with numbers that are multiples of 4 light up.
5. This pattern continues for 40 minutes, and in the 40th minute, only the letter in square 40 lights up.

The problem asks for the square number whose letter lights up the most times.

Solution:

Each square number will light up whenever the current minute is a divisor of the square's number. Thus, the square number with the most divisors will light up the most frequently.

The number with the most divisors between 1 and 40 is 36, which has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Thus, the letter in square 36 will light up the most times.

Answer:

The correct answer is C) 36.

Would you like a more detailed explanation on finding divisors or any other questions?

Here are some related questions:

1. How do you find the divisors of a number?
2. What is the significance of perfect squares in this type of problem?
3. How can you generalize this approach for a larger board?
4. What is the pattern of numbers that light up at the same minute?
5. How can you optimize finding the number with the most divisors?

Tip: Always remember that perfect squares tend to have the most divisors, which can help in such problems!