The Locker Problem

Imagine you are at a school that has student lockers Imagine you are at a school that has student lockers. There are 1000 lockers, all shut and unlocked, and 1000 students.

Suppose the first student goes along the row and opens every locker.

The second student then goes along and shuts every other locker beginning with number 2.

The third student changes the state of every third locker beginning with number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.)

The fourth student changes the state of every fourth locker beginning with number 4.

Imagine that this continues until the thousand students have followed the pattern with the thousand lockers. At the end, which lockers will be open and which will be closed? Why?

Click here to simulate this problem. http://hydra. educ. queensu

SOLUTION

These numbers are the square numbers: 12, 22, 32, and 42 These numbers are the square numbers: 12, 22, 32, and 42. So, there is an open locker door at every square number. How many square numbers are there between 1 and 1000? Through a little trial and error, you'll find that 312 is the last square number less than 1000. So, there are 31 open doors (the last one occurring on the door numbered 312 or 961).

Looking at Factors A special property of square numbers is that they always have an odd number of factors. A factor is a number that divides another number evenly (with no remainder). For example, 8 has an even number of factors, namely, 1, 2, 4, and 8. But, 9 has an odd number of factors, namely, 1, 3, and 9. In fact, all numbers except the square numbers have an even number of factors.

Student Leaves locker 40 1 Open 2 Closed 4 5 8 10 20 40 You can use this fact to solve the locker problem. Take any locker number, 40, for example. Its state (open or closed) is changed for every student whose number in line is a factor of the locker number. So, write out all the factors of 40, like this: Student Leaves locker 40 1 Open 2 Closed 4 5 8 10 20 40

Like all other lockers numbered with non-square numbers, it ends up closed after all the students have gone through the line because it has an even number of factors.

Here's the factor pattern for a square number, 16. Student Leaves locker 16 1 Open 2 Closed 4 8 16 Locker 16 remains open because it has an odd number of factors.

You can now conclude that all the doors with square numbers on them will remain open because all square numbers have an odd number of factors. You can now conclude that all the doors with non-square numbers on them will remain closed because only square numbers have an odd number of factors.

There are 31 open doors. The numbers of the open doors are listed below. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961