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Lousy Lockers
One hundred students are assigned lockers 1 through 100. The student assigned to locker number 1 opens all 100 lockers. The student assigned to locker number 2 then closes all the lockers whose numbers are multiples of 2. The student assigned to locker number 3 changes the status of all lockers whose numbers are multiples of 3 (think of it this way - locker number 3, which is open, gets closed, while locker number 6, which is closed, is now opened by student 3.) The student assigned to locker number 4 changes the status of all lockers whose numbers are multiples of 4, and so on for all 100 students. When student 100 is finished changing the state of the locker doors:
1. Which lockers will be left open?
2. How many lockers, and which ones, will be touched exactly twice?
Tips:
1. Look for patterns so you do not have to go through all 100 state changes.
2. Maybe organizing the state changes in some manner may help you make some predictions that you could test.
Extra Credit: Which locker(s) was (were) switched the most times?
Thursday, February 25, 2010
ReplyDeleteLousy Lockers
One hundred students are assigned lockers 1 through 100. The student assigned to locker number 1 opens all 100 lockers. The student assigned to locker number 2 then closes all the lockers whose numbers are multiples of 2. The student assigned to locker number 3 changes the status of all lockers whose numbers are multiples of 3 (think of it this way - locker number 3, which is open, gets closed, while locker number 6, which is closed, is now opened by student 3.) The student assigned to locker number 4 changes the status of all lockers whose numbers are multiples of 4, and so on for all 100 students. When student 100 is finished changing the state of the locker doors:
1. Which lockers will be left open?
2. How many lockers, and which ones, will be touched exactly twice?
all prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. 25 lockers
1. Look for patterns so you do not have to go through all 100 state changes.
2. Maybe organizing the state changes in some manner may help you make some predictions that you could test.
Answer to number 2 is all square numbers so 4,9,12,16,25,36,49,64,81,100
Extra Credit: Which locker(s) was (were) switched the most times? 60, 72, 84, 90, 96
S.A
H.H Thursday, February 25, 2010
ReplyDeleteLousy Lockers
One hundred students are assigned lockers 1 through 100. The student assigned to locker number 1 opens all 100 lockers. The student assigned to locker number 2 then closes all the lockers whose numbers are multiples of 2. The student assigned to locker number 3 changes the status of all lockers whose numbers are multiples of 3 (think of it this way - locker number 3, which is open, gets closed, while locker number 6, which is closed, is now opened by student 3.) The student assigned to locker number 4 changes the status of all lockers whose numbers are multiples of 4, and so on for all 100 students. When student 100 is finished changing the state of the locker doors:
1. Which lockers will be left open? 4 9 16 25 36 49 64 81100
2. How many lockers, and which ones, will be touched exactly twice?
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97, 25 touched twice.
Tips:
1. Look for patterns so you do not have to go through all 100 state changes.
2. Maybe organizing the state changes in some manner may help you make some predictions that you could test.
Extra Credit: Which locker(s) was (were) switched the most times? 60, 72, 84, 90, 96. H.H
1. Which lockers will be left open?
ReplyDeleteNone of the prime numbers, not even two.
2. How many lockers, and which ones, will be touched exactly twice?
Any of the prime numbers like 2, 3, 5, 7, etc.
E.I.D.
ReplyDelete1. Lockers 1, 4, 9, 16, 23, 25, 37, 39, 44, 45, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 67, 70, 71, 72, 73, 74, 75, 76, 77, 80, 81, 83, 84, 86, 87, 89, 90, 92, 95, 96, 97, 98, 99, and 100 will be left open.
2. Lockers 2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 65, 69, 79, 82, and 94 will be touched exactly twice.
EXTRA CREDIT
Locker60 will be touched the most times, 12 times.
bs
ReplyDelete1. The lockers that are left open are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. All these numbers are squared numbers.
2. There are 25 lockers that are touched twice. Those lockers are numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. All these numbers are prime numbers.
Extra Credit – The lockers that are touched the most are lockers 60, 72, 84, 90, 96. All these lockers were touched 12 times. All these locker numbers except 90 are multiples of 12.
(excel table would not post correctly)
S.D
ReplyDeleteI knew that to find the locker numbers that would be left open I would need to find all the numbers between 1 and 100 that had a odd number of factors. These are all the numbers between 1 and 100 that have an odd number of factors, 1, 4, 9, 25, 36, 49, 64, 81, and 100.I also knew that only prime numbers would be touched twice, so here are all the prime numbers between 1 and 100, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
P.M.
ReplyDelete1. All prime numbered lockers. This is because they don’t have any numbers that go into them, so no one would open them after the person who has that same numbered lockers.
2. Every locker that has two numbers that are the same in their prime factorization, because those only have two numbers in them, that are the same.
J.H
ReplyDeleteThe pattern I found that starting at one they would all be open and then those they were all factors of 1. Then I went on to 2 and it was every other one and they were all factors of 2. Then I went on to 3 and then I went on to 3 and it was every three and they were all factors of 3, and it keeps going on.
1. The locker that will be left open will be the numbers that have common factors and are multiples of each other.
2. All prime numbers like… 2,3,5,7,11,13,17,19,23…ect
A. A. M.
ReplyDeleteThe pattern that we found was that the factors of the number of the locker number were going to be the new locker that would either be opened or closed.
1. The lockers that will be left open are the numbers that have common factors and are multiples of each other.
2. The lockers that will be touched exactly twice are the prime numbers like 2, 3, 5, 7, 11, 13, 17, 19, 23… etc.
C.A.
ReplyDeleteI made a table in Excel and I did students 1-100. Then I put “open” by all the lockers, then I put “close” by all the multiples of 2. Then I changed the status for every multiple of 3, 4, and so on. The lockers that will be left open are the ones that have common factors and are multiples of each other. The lockers that will be touched exactly twice are 2, 3, 5, 7, 11, 13, 17, 19, 23…etc-they are all prime numbers.
S.R.
ReplyDelete1. Ever number with a prime factrivtion string of one number multiplied but its self is self open so 4 because its 2*2, 6 because its 3*3, 16 because its 2*2*2*2=4*4
2. Ever number with I prime factrivtion string of one and its self is touch twice
MNH
ReplyDelete1. All lockers with an odd number of factors.
2. Numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 were all touched only twice. There are 26 lockers touched only twice.
JD
ReplyDelete1. All lockers that have an odd number of factors will be left open.
2. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
JS
ReplyDelete1. All the lockers with a number that have a odd number of factors
2. lockers that are touched twice: 1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
There are 26 lockers that will be touched twice and they are all the prime numbers
HB
ReplyDelete1: The lockers that will be left open are all of the lockers with an odd amount of factors.
2: The lockers that were touched exactly twice are all the prime numbers because the only people that touched them are the number itself and the number one. The primes are: 1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
MJH
ReplyDeleteLousy Lockers
1. 1, 4, 9, 16, 25, 36, 49, 64, 81. I found two patterns for this: Finding squares (1x1=1, 2x2=4, etc.), and a Fibonacci series (1+3=4, 4+5=9, etc.)
2. All of the prime numbers will be touched exactly twice: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
I started this P.O.W. by making an excel table. I started off doing everyone other one open. 1= open 3=open and so on. All the even numbers would be closed. I then finished my table.
ReplyDelete1. The lockers left open would be 1, 4, 16, 25, 36, 42, 49, 52, 56, 66, 70, 75, 76, 80, 81, 83, 93, 94, 95, 98, 100
2. All the prime numbers would be touched exactly twice.
I then realized in question 1.. I completely did it wrong.. All the square numbers would be left open.
RW
M.D.
ReplyDeleteI started by making a table in excel. I made a column of 1-100 to mean the lockers 1-100. Next to that I made a new column and put “open” next to every number. On the next column I put “close” next to every multiple of two. I made another column and for every multiple of three, I changed the status of the locker. In the next column I changed the status of every multiple of four. I made a new column and changed the status of every multiple of five. I continued making my columns and changing the status of the multiple of each locker. Finally, I reached 100 and finished my table.
1. The lockers left open would be 1, 4, 16, 25, 36, 42, 49, 52, 56, 66, 70, 75, 76, 80, 81, 83, 93, 94, 95, 98, and 100
2. The lockers touched exactly twice would be:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97, which are all prime.
A.B.
ReplyDelete1. ALL lockers that are square numbers will be left open: 1, 4, 9, 16, 20, 36, 49, 64, 81 and 100.
First, I was looking to see how many factors each locker number had, and I thought that if the locker # had an odd number of factors, then it’d be open. As I was looking at the numbers I had found, I realized that they were all square numbers. So, I double checked by going through all of the times these lockers were touched and it turned out that I was correct.
2. The lockers that will be touched only TWICE are all prime numbers, which is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
This is because all prime numbers have two factors, which means that they’ll be touched exactly twice.
S.G.
ReplyDelete1.) 1, 4, 9, 16, 25, 36, 37, 49, 56, 64, 74, 81, 99, and 100.
2.) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Extra Credit:
The lockers that were switched the most times were numbers 96, 90, 84, 72, and 60.