NCERT Book Class 8 Mathematics (Part I) Chapter 1 A Square and A Cube

1 A SQUARE AND A CUBE

Queen Ratnamanjuri had a will written that described her fortune of
ratnas (precious stones) and also included a puzzle. Her son Khoisnam
and their 99 relatives were invited to the reading of her will. She wanted
to leave all of her ratnas to her son, but she knew that if she did so, all
their relatives would pester Khoisnam forever. She hoped that she had
taught him everything he needed to know about solving puzzles. She left
the following note in her will—
“I have created a puzzle. If all 100 of you answer it at the same time, you
will share the ratnas equally. However, if you are the first one to solve the
problem, you will get to keep the entire inheritance to yourself. Good luck.”
The minister took Khoisnam and his 99 relatives to a secret room in
the mansion containing 100 lockers.
The minister explained— “Each person is assigned a number from 1 to
100.
• Person 1 opens every locker.
• Person 2 toggles every 2nd locker (i.e., closes it if it is open, opens
it if it is closed).
• Person 3 toggles every 3rd locker (3rd, 6th, 9th, … and so on).
• Person 4 toggles every 4th locker (4th, 8th, 12th, … and so on).
This continues until all 100 get their turn.
In the end, only some lockers remain open. The open lockers reveal
the code to the fortune in the safe.”

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.