At Millock Highschool, there are 1,000 lockers and 1,000 students. The first day of school, all the students line up outside of the school and enter one at a time. The first student touches evry locker,the second student touches evry locker that is a multiple of two, the third student touches evry locker that is a multiple of three, the fourth student touches evry locker that is a multiple of four, and the pattern continues. when they start all the lockers are closed. read on for it to make sence.
definition:When a student touches a locker, if it is closed they open it and if it open they close it.
1.After all the students have gone, which lockers are open and which are closed?
{15points}
2. What pattern, if any, do you notice?
{5points}
3.Explainwhy these are left open.{20points}
http://www.braingle.com/brainteasers/teaser.php?op=2;id=7824;comm=0
There is quite a bit of documentation on this problem online. Try googling "first student touches all lockers" without the quotes. I would give you some specific websites, but I'm unable to post links. Good luck!
Lets look at locker number "n". All the lockers are locked to begin with. Since all the lockers are opened on the 1st pass, locker "n" is now open. For locker "n" to be closed on the 2nd pass, n must be divisible by 2. For the locker to be opened on the 3rd pass, it must be divisible by 3. For the locker to be closed on the 4th pass, it must be divisible by 4. Clearly, the locker is either opened or closed as long as the locker number is divisible by each successive divisor of "n". After the one thousandth student has made his contribution to the celebration, locker "n" will only be open if it was acted upon an odd number of times. We can therefore conclude that locker "n" will be open if, and only if, the number "n" has an odd number of factors or divisors. But the only numbers that have an odd number of factors/divisors are the perfect squares. Thus, if "n" is open, it is one of the perfect squares. The lockers that remain open are therefore numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, on up to 961.
For each locker number, find all of the exact divisors including 1 and the number itself. If the number of divisors is odd, then the number of people who reversed the locker is odd, and the locker is open. If the number of divisors is even, then the number of people who reversed the locker is even, and it is closed. The only numbers with an odd number of divisors are the perfect squares. Therefore, the lockers that remain open are those identified by the perfect squares.
To solve this problem, we can approach it step by step to determine which lockers are open or closed. Here's how:
1. Start by imagining the lockers as being closed initially.
2. As the first student touches every locker, they will open all the lockers.
3. The second student will then go through and touch every second locker starting from the second locker. This means they will close every even-numbered locker (2, 4, 6, ...) and open every odd-numbered locker (1, 3, 5, ...).
4. The third student will touch every third locker starting from the third locker. Here, patterns begin to emerge. For example, the third student will open locker 3 if it is currently closed, and close it if it is currently open. They will proceed to locker 6, which had been opened by the second student, and will close it. The third student will open locker 9 if it is closed and close it if it is open. This pattern continues.
5. From this point onward, each student touches lockers corresponding to multiples of their respective student numbers. For example, the fourth student touches lockers 4, 8, 12, ... and so on.
Now, let's answer the questions:
1. After all the students have gone, the lockers that will be left open are the ones that have been touched an odd number of times. In other words, the lockers that have an odd number of factors. This includes numbers that are perfect squares, as they have an odd number of factors (e.g., 1, 4, 9, 16, ...).
2. The pattern I notice is that the lockers that will be left open are the ones with numbers that are perfect squares.
3. These lockers are left open because they are the only numbers that have an odd number of factors. Any other number will have an even number of factors since factors come in pairs (e.g., factors of 12: 1 and 12, 2 and 6, 3 and 4). Therefore, any locker that has an odd number of factors will be touched an odd number of times, resulting in it being left open at the end.
I hope this explanation helps you understand the problem better! Let me know if you have any further questions.