Imagine there are 1000 lockers and 1000 students at school. The 1st student who comes through the school opens every locker as they enter the school. The 2nd student walks in and closes every other locker begining with locker number two. The 3rd student walks in and changes the state of every third locker begining with locker number three. The next student walks through and changes the state of every 4th locker begining with locker number four. Imagine this continues until all the students have followed the pattern with the 1000 lockers. At the end, which lockers will be open and which lockers will be closed ? Why?
Take fifty cards, labeling each 1,2,3,...50.
Then follow the rules above, removing ards. See what happens.
all the squares since they have an odd number of factors
To determine which lockers will be open and which will be closed, we need to analyze the pattern created by the students.
Let's break it down step by step:
- The 1st student opens all the lockers.
- The 2nd student closes every other locker, starting from locker number two. So all the even-numbered lockers are closed.
- The 3rd student changes the state of every third locker, starting from locker number three. Since none of the lockers have been toggled yet, this means that the 3rd student opens every third locker.
- The 4th student changes the state of every fourth locker, starting from locker number four. At this point, the even-numbered lockers are closed, and the odd-numbered lockers are open. Therefore, the 4th student closes locker number four.
- The 5th student changes the state of every fifth locker, starting from locker number five. The first five lockers are now: open, closed, open, closed, closed.
- This pattern continues with each subsequent student changing the state of every n-th locker, where n is the number of the student.
Now, if we observe the pattern, we can see that the number of times a locker is toggled depends on the number of factors it has. To understand this, imagine a locker number X. It will be toggled every time a student comes through the school where X is a divisor (or factor) of the student's number.
For example, let's take locker number 10:
Student #1 will open it.
Student #2 will close it.
Student #5 will open it.
Student #10 will close it.
From this, we can conclude that a locker number X will be toggled an odd number of times if it has an odd number of factors, and an even number of times if it has an even number of factors.
Now, to determine which lockers will remain open, we need to find the lockers with an odd number of factors. These lockers only have a factor when they are multiplied by a number which gives them a perfect square value, like 1, 4, 9, 16, etc.
So, the lockers that will remain open are the lockers which have a perfect square number as their number. In this case, since we have 1000 lockers, there will be 31 open lockers (which are perfect squares between 1 and 1000).
To find out which lockers exactly, you can calculate the square root of each number from 1 to 1000, and if it is a whole number, then that locker will remain open.
In summary, the lockers that will be open after all the students have followed the pattern are the lockers with perfect square numbers, i.e., locker numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961. The rest of the lockers will be closed.