Monday, May 6th Problem

[Lara Hulbert]

There are 30 closed lockers in a row. Lockers are numbered from 1 to 30.  Student A walks down the line and opens every locker . Student B walks down the line and closes every locker that is a multiple of 2. Student C changes the position of lockers that are multiples of 3 (so if it was open, he closes it. If it was closed, he opens it). Student D changes the positions of lockers that are multiples of 4. And so on. What is the final position of the lockers after the 30th student has passed by?

[msshehane]

I started to do this by hand, but it was pretty messy so I decided to make it with excel.

[Gregory M. Hulbert]

All except the perfect squares are touched by an even number of students. So all lockers except numbers 1, 4, 9, 16, and 25 are closed. These 4 lockers are open.

[joshuaw.bartlett]

Perfect squares are the only ones open. They've each had an odd number of interactions because they are the only numbers with an odd occurrence of factor "pairs". For example, the number twelve has factor pairs 1*12, 2*6, 3*4 - three pairs of two distinct numbers. The number sixteen has an odd number of distinct factors: 1*16, 2*8, 4*4 where the number four is repeated, giving us only 5 distinct factors in 3 "pairs".

[Lara Hulbert]

Michael, I love the spread sheet! Josh, I love your explanation!