Write the locker numbers that remain open. [Condition: 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.]
Lockers that remain open: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
These are all the perfect square numbers from 1 to 100. A locker remains open if it is toggled an odd number of times.
This happens only for perfect squares because they have an odd number of factors.
Explanation
The chapter explains that each locker is toggled by people whose numbers are factors of the locker number. A locker toggled an odd number of times remains open. Only perfect squares have an odd number of factors because one factor (the square root) pairs with itself.
Solution Steps
Step 1: A locker is toggled once for each of its factors
Step 2: Lockers toggled an odd number of times remain open
Step 3: Only perfect squares have an odd number of factors
Step 4: Perfect squares from 1 to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100