## Archive for April 18, 2011

### An Open and Shut Case

Most mathy folks are familiar with the locker problem:

Every day, 1000 students enter a school that has 1000 lockers. All of the lockers are closed when they arrive. Student 1 opens every locker. Student 2 closes every other locker. Student 3 then “changes the state” of every third locker – that is, he opens it if it’s closed, and he closes it if it’s open. Student 4 then changes the state of every fourth locker, Student 5 changes the state of every fifth locker, and so on, so that Student

nchanges the state of everynth locker.Which lockers are open after all 1000 students have finished opening and closing lockers?

At the 2011 NCTM Annual Meeting, the following variation of the locker problem appeared in the Daily Puzzle Challenge on Thursday:

Every day, 30 students enter a room with 30 lockers. All of the lockers are closed when they arrive. Student 1 opens every locker. Student 2 closes every locker. Student 3 then “changes the state” of every third locker — that is, he opens it if it’s closed, and he closes it if it’s open. Student 4 then changes the state of every fourth locker, Student 5 changes the state of every fifth locker, and so on, so that Student

nchanges the state of everynth locker.One day, some students are out sick. Regardless, those present repeat the process and just skip the students who are absent — for instance, if Student 3 were absent, then no one would change the state of every third locker.

When they finish, only Locker #1 is open, and the other 29 lockers are all closed. How many students were absent?

The following applets can be used to investigate the original problem or to solve the variation.