Context: Queen Ratnamanjuri left a puzzle in her will for her son Khoisnam and 99 relatives. They are in a room with 100 lockers, numbered 1 to 100. - Person 1 opens every locker. - Person 2 toggles every 2nd locker (closes if open, opens if closed). - Person 3 toggles every 3rd locker (3rd, 6th, 9th, ...). - Person 4 toggles every 4th locker (4th, 8th, 12th, ...). This continues until all 100 get their turn. Q. Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer? Hint: Find out how many times each locker is toggled.

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Question 1

Context: Queen Ratnamanjuri left a puzzle in her will for her son Khoisnam and 99 relatives. They are in a room with 100 lockers, numbered 1 to 100.

Person 1 opens every locker.
Person 2 toggles every 2nd locker (closes if open, opens if closed).
Person 3 toggles every 3rd locker (3rd, 6th, 9th, ...).
Person 4 toggles every 4th locker (4th, 8th, 12th, ...). This continues until all 100 get their turn.

Q. Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer?

Hint: Find out how many times each locker is toggled.

Solution

Khoisnam figured out that the lockers that will be open at the end are those whose numbers are perfect squares.

Step 1 — Understanding how locker states change

Each locker starts in a closed state. When a locker is toggled, its state changes. If it was closed, it becomes open. If it was open, it becomes closed.

Let us consider the final state of a locker. 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.

Now, let us determine how many times each locker is toggled. Person 1 toggles every 1st locker. Person 2 toggles every 2nd locker. Person 3 toggles every 3rd locker. This pattern means that a locker, say locker number $L$ , is toggled by person $P$ if $P$ is a factor (or divisor) of $L$ . For example, locker 12 is toggled by Person 1, Person 2, Person 3, Person 4, Person 6, and Person 12. These are all the factors of 12. So, the number of times a locker $L$ is toggled is equal to the total number of its factors.

Diagram 1

Step 2 — Identifying lockers that remain open

We need to find which lockers are toggled an odd number of times. This means we need to find numbers that have an odd number of factors.

Let's examine the number of factors for different types of numbers: Consider a number that is not a perfect square, like 6. Its factors are 1, 2, 3, 6. These factors can be paired up: (1, 6) and (2, 3). There are 4 factors, which is an even number. Consider a number that is a perfect square, like 9. Its factors are 1, 3, 9. These factors can be paired up: (1, 9) and (3, 3). The factor 3 is paired with itself. This means it is counted only once in the list of distinct factors. So, there are 3 factors, which is an odd number.

In general, factors of a number always come in pairs $(a, b)$ such that $a \times b = \text{the number}$ . If the number is not a perfect square, all its factors will have a distinct partner. This results in an even number of factors. If the number is a perfect square, one factor (its square root) will be paired with itself. This means that factor is counted only once, leading to an odd total number of factors. Therefore, only perfect square numbers have an odd number of factors.

The lockers that will be open at the end are those whose numbers are perfect squares between 1 and 100. Let's list these perfect squares: $1^2 = 1$ $2^2 = 4$ $3^2 = 9$ $4^2 = 16$ $5^2 = 25$ $6^2 = 36$ $7^2 = 49$ $8^2 = 64$ $9^2 = 81$ $10^2 = 100$

Khoisnam figured out that the lockers corresponding to these numbers would be open.

Answer

Khoisnam figured out that a locker would be open at the end if it was toggled an odd number of times. A locker is toggled by every person whose number is a factor of the locker's number. Therefore, a locker is open if its number has an odd number of factors. Numbers with an odd number of factors are perfect squares. So, Khoisnam knew that the lockers numbered with perfect squares would be open.

The lockers that will be open at the end are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Question 1

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