100 Closed Lockers
Math brain teasers require computations to solve.
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?
Answer
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: open-closed-open-closed-open-closed-open-closed.
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): open-closed-open.
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Comments
dragon6786
Oct 29, 2002
| Maybe this should of gone in math but i liked it |
speedyg1000 
Dec 12, 2002
| i was thinking all of the primes but you closed all of them in the beghinning. oops. |
zangel3000
Jun 21, 2003
| WOW! I would have had to write it all out and everything...I'm just not that smart. |
jimbo  
Aug 08, 2003
| Great puzzle but surely belongs in the Mathematics category? |
cnmne
Jun 27, 2004
| Excellent. I worked through the first 20 lockers, noticed that only the perfect squares were open, and realized that those were the only numbers with an odd quantity of factors. I need to eat more fish. |
knbrain
Dec 27, 2004
| Very Good. It took me a long time to get that one.  |
(user deleted)
Dec 27, 2004
| that made my brain hurt   |
besbball917
Dec 27, 2004
| Wow!! I finally got it but now I think I have a headache!!! Great teaser! I agree, though, it should be in the math category!!  |
joeschmmoo
Dec 27, 2004
| this teaser was really hard. it got me goooooood.  |
smar_tass
Dec 27, 2004
| The teaser sounded fun at first till i remebered the same exact problem my sophmore son brought home for algebra homework  |
surfgrl999
Dec 27, 2004
| That really made you think!! alot!! but it was still kewl! |
babygirl35
Dec 27, 2004
| This was an interesting problem although i do agree that it belongs in the math category. |
stewartfreak20
Dec 27, 2004
| interesting.....  unfortunetly I didn't even bother to figure it out!!!!1  How classic of me!!! |
ahurley
Dec 28, 2004
| very interesting! and a lot of fun  |
sweetpeao4o3
Dec 28, 2004
| took to long to figure out the answer.... i didnt work it out  should go in the math |
brandynicole89
Feb 22, 2005
| this is a really great teaser because if you dont have any patients then you will not sit down and work this out. and it really makes you think! i liked it and i got it!!! |
thepianistalex
Feb 08, 2006
| I love this teaser. Only problem is, I did this as homework for my math class and it was up to 1,00 lockers... |
thepianistalex
Feb 08, 2006
| Whoops. 1000 lockers |
Smudge 
Jun 22, 2006
| LOVE THIS! This will be the first entry into my 'favorite teasers' list.
Easy to figure out when you look for the pattern.
Start with 1, then 2, then 3, then 4, then 5, etc.
You will see the pattern...
using the numbers:
4-1 = 3
9-4 = 5
16-9 = 7
25-16 = 9
36-25 = 11
The pattern, therefore, increases steadily by 2...uncovering the squares |
eyenowhour
Dec 18, 2006
| Very hard, but extremely fun! |
Ildhund 
Aug 21, 2007
| The result depends on what you mean by "every second" locker. I assumed this meant the first, the third, the fifth and so on. "Every third" means no. 1, no. 4, no. 7 and so on. In which case there are only 9 open after 100 passes, their numbers being n^2 + 1 (0 |
javaguru
Jan 03, 2009
| Very easy, but also elegant teaser. The odd number of factors was obvious, and after thinking a moment it was clear that to have an odd number of factors that one of the numbers must be multiplied by itself. Good job! |
(user deleted)
May 10, 2009
| I am unable to understand the logic that why would ALL other numbers have even factors only. I must admit I tried several numbers and came up with even numbers only, but am unable to figure out a logic behind that. |
fairygrrrl
Sep 28, 2011
| please help me to understand what this line means,
" Continue toggling every nth locker on pass number n. After 100 passes, where you toggle only locker #100,"
so that I can continue working on this one!
thanks |
jporter892
Oct 12, 2012
| Love this brain teaser - one of my favorites to ask during interviews just to see how people think.
Found another great answer here too:
http://www.programmerinterview.com
/index.php/
puzzles/lockers-puzzle/ |
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