
Every month we will give you a math/statistics brain teaser that lets you test your knowledge with a fun problem.
This Month's Challenge: Lockers
Suppose you're in a hallway lined with 100 closed lockers.
You begin by opening every locker. Then you close every second locker. Then you go to every third locker and open it (if it's closed) or close it (if it's open). Let's call this action toggling a locker. Continue toggling every nth locker on pass number n. After 100 passes, where you toggle only locker #100, how many lockers are open?
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The Answer!
10 lockers are left open.
Lockers #1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Each of these numbers are perfect squares. This problem is based on the factors of the locker number.
Each locker is toggled by each factor; for example, locker #40 is toggled on pass number 1, 2, 4, 5, 8, 10, 20, and 40. That's eight toggles: openclosedopenclosedopenclosedopenclosed.
The only way a locker could be left open is if it is toggled an odd number of times. The only numbers with an odd number of factors are the perfect
squares. Thus, the perfect squares are left open.
For example, locker #25 is toggled on pass number 1, 5, and 25 (three toggles): openclosedopen.
Many of the ideas in these Teasers come from thoughts formed by some of the great mathematical/statistical geniuses in history. To learn more about them and to take a fun quiz to find out which famous mathematician through history most interests you, try our Mathematicians' Quiz.
Or if you'd like more Math/Science fun you might want to try...
Now with both 4th and 8th grade level questions!

