At a new junior high school, there are exactly 1000 students and 1000 lockers. Lockers are numbered in order from 1 to 1000. On April Fool’s Day the students played the following prank. The first student to enter the building opened every locker. The second student closed every locker that had an even number. The third student changed every third locker, closing those that were open and opening those that were closed. The fourth student changed every fourth locker, and so on. After all 1000 students passed through the locker room, which lockers were open?
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31 comments:
The 1000th student only changes 1 locker.
The 501st-999th only change 2 lockers each.
Locker 1 gets open/closed 1000 times
Ben
THIS COMMENT WAS LEFT FROM DANIEL G.
Daniel G. said..."Where do I start? There are so many numbers?"
Hey guys
hey theres so many numers im trying from 1-10...let me see what happens
Ben, the 1st doesn't get opened/closed 1000 times it changes once
Ya it only gets changed once and 2,3,4,5 also only get changed once
I really like all of your comments; you are really helping each other...that is the idea.
Try Camelia's suggestion of making a simpler problem out of it. Use lockers from 1 to a number smaller than 1000 and see if you see a pattern.
Prime numbers only get changed once
All prime numbers have closed lockers because they each have been touched twice. The first person and the person of the same number as the locker are the ones that touch the locker. For example, take locker number 5. The first person opened it and the fifth person closed it. No other people after person #5 touched the fifth locker.
About #1: Like Jake and Camellia said, Number 1 was only touched once.
This means that number 1 is open.
and will stay opened
Rule: The number of factors a locker number has influences whether the locker will be opened or closed after the 1000th person. If a number has an odd number of factors (for example: 9 has 3: 1, 3, 9), this locker will be open in the end. If a number has an even number of factors (Prime numbers are included in this category), they will be closed (EX.: 8 has 4: 1, 2, 4, 8).
So, my answer is that there will be 31 opened lockers in the end.
Good work, Iris!
Now...consider this...
What kinds of numbers have an odd number of factors?
9 does...1,3,9
16 does...1,2,4,8,16
What others?
(These numbers fall into a special category and have a name)
the first locker definelty STAYS OPEN
Those whole numbers are called square numbers or perfect squares because they can be produced by a whole number with an exponent of 2.
I believe that every locker was closed by the end of tyhe pran because 3 doesn't go into 1000 but 4 does.
I know that 4 has an odd number of factors because 4: 1,2,4.
I can't find any others besides 4,9,and 16.
Mrs. Cooper,
I don't understand why it posts me as "dani says" when I want it to say "danielle says" because we aren't allowed to use nicknames.
I found out that if their is an odd number of factors it is open and if there is an even number of factors it is closed, but I did around 30 lockers and couldn't find a pattern. I'm sort of lost.
every time a locker is opened the difference between the lockers is increased by 2. ex: 1+3=4,4+5=9.
What do you mean ethan?
Mrs. Cooper,
I went through all the prime number from 1-200 but gave up and looked on the internet. All the prime locker numbers are closed. There are 168 prime number from 1-1000. So 1000-168=832. My answer is 832 but I dont know if its right.
how can it be 832 if it says which lockers? Are you saying it is 832 only?
if the all were open then only odd were open then there were 500 open
what do they mean by change though?
changing them means if they are open then you close them if they are closed then u open them
y'all did a GREAT job of helping each other. The ones that remain open are 4, 9, 16,25,36,49,64,81,...
These numbers are called square numbers and all have an odd number of factors. So...you are looking for the ones that are square numbers less than 1000.
Danielle, you must have put in Dani for your display name when you first responded to the invitation to the blog.
how can the answer be 31 if it is asking which ones were open?
1 remains open
4 remains open
9 remains open
25 remains open
36 remains open
49 remains open
64
81
100
121
144
.
.
.
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