100 lockers are in a row, all initially closed. A student toggles the state of every locker. Then, a second student toggles the state of every second locker. This continues until the 100th student tog

Initially, all lockers are closed. A locker's state is toggled (opened if closed, closed if open) each time a student visits it.

• If a locker is toggled an odd number of times, it will end up open.
• If a locker is toggled an even number of times, it will end up closed.

Student $k$ visits every $k$-th locker. This means locker $n$ is visited by student $k$ if $k$ is a divisor of $n$. Therefore, the total number of times a locker $n$ is toggled is equal to the number of its divisors.

A number has an odd number of divisors if and only if it is a perfect square. This is because divisors usually come in pairs $(d, n/d)$. If $n$ is not a perfect square, all its divisors can be paired up, resulting in an even number of divisors. If $n$ is a perfect square, say $n = m^2$, then $m$ is a divisor that is paired with itself, making the total count of divisors odd.

The lockers that will remain open are those whose numbers are perfect squares between 1 and 100.

• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• $4^2 = 16$
• $5^2 = 25$
• $6^2 = 36$
• $7^2 = 49$
• $8^2 = 64$
• $9^2 = 81$
• $10^2 = 100$

The lockers that will be open at the end are:
$\text{1, 4, 9, 16, 25, 36, 49, 64, 81, 100}$