The Locker Problem

 

I started with 10 lockers to start on a smaller scale. I drew out 10 ‘lockers’ and every time their state changed the colour changed. Blue was closed and pink was open. 

• When I reached the 10th state change I noticed...

• If the number of the locker had an even number of factors it was open - going through an even number of state changes

• If the number of the locker had an odd number of factors it was closed - going through an odd number of state changes

• The lockers that remained closed after 10 state changes were perfect squares (1, 4, 9)

• So the number of the three closed lockers are all perfect squares and there are 3 meaning there are 3‘perfect square’ lockers between lockers 1 and 10

• To determine the number of closed lockers out of 1000

• Since the lockers that remained closed after 10 state changes went through an odd number of state changes, they have an odd number of factors, and they were also perfect squares

• Since perfect squares have an odd number of factors, to find the number of closed lockers out of 1000 lockers, the number of ‘perfect square’ lockers between 1 and 1000 will need to be determined 

• Thus taking the square root of 1000 gives approximately 31. 

• So there are 31 closed lockers after all 1000 students have had their turn.