YOU WERE LOOKING FOR: Locker Problem Answers
This is my magnum opus: a fully documented version of the infamous "Locker Problem. It is a magnificent problem which asks the following: if you had 1, lockers and 1, students, and the students took turns reversing the position of the locker that...
But wait, there's more: they then extend their thinking to classify numbers as "abundant," "deficient" or "perfect" and use them to play a factor game, both against an opponent, or against the computer links to the computer program are included....
However, if you are the first to find the pattern and solve the problem without going through all of the leg work, you will get the entire inheritance all to yourself. Good luck. He explains: Every relative is assigned a number from 1 to Heir 1 will open every locker. Heir 2 will then close every second locker. Heir 3 will change the status of every third locker, specifically if it's open, she'll close it, but if it's closed, she'll open it.
This pattern will continue until all of you have gone. The words in the lockers that remain open at the end will help you crack the code for the safe. Before cousin Thaddeus can even start down the line, you step forward and tell the lawyer you know which lockers will remain open. But how? Pause the video now if you want to figure it out for yourself! Answer in: 3 Answer in: 2 Answer in: 1 The key is realizing that the number of times a locker is touched is the same as the number of factors in the locker number. For example, in locker 6, Person 1 will open it, Person 2 will close it, Person 3 will open it, and Person 6 will close it. The numbers 1, 2, 3, and 6 are the factors of 6. So when a locker has an even number of factors it will remain closed, and when it has an odd number of factors, it will remain open. Most of the lockers have an even number of factors, which makes sense because factors naturally pair up. In fact, the only lockers that have an odd number of factors are perfect squares because those have one factor that when multiplied by itself equals the number.
For Locker 9, 1 will open it, 3 will close, and 9 will open it. Therefore, every locker that is a perfect square will remain open. You know that these ten lockers are the solution, so you open them immediately and read the words inside: "The code is the first five lockers touched only twice. So the code is The lawyer brings you to the safe, and you claim your inheritance.
Too bad your relatives were always too busy being nasty to each other to pay attention to your eccentric uncle's riddles.
I have often told people that, believe it or not, they could find the answer by searching the Ask Dr. But I prefer to give them a reference to one of the answers in which we gave only hints, because this is a fun problem to discover the answer for yourself. Tiny hints Here is a question from , which asked about two problem, the first of which is our subject: Word Problem Hints 1 There are lockers numbered 1 - Suppose you open all of the lockers, then close every other locker. Then, for every third locker, you close each opened locker and open each closed locker. You follow the same pattern for every fourth locker, every fifth locker, and so on up to every thousandth locker. Which locker doors will be open when the process is complete? Doctor Jodi gave only a hint: Our office is overflowing with patients at the moment, so let me just try to put a band-aid on these problems for you So every other locker means every locker whose number has what as a factor?
And how many times would a switch have to be flipped to be on at the end? Person 1 starts at locker 1 and opens every locker. Person 2 starts at locker 2 and closes every 2nd locker. Person 3 starts at locker 3 and changes every 3rd locker. Person 4 starts at locker 4 and changes every 4th locker. Person x starts at locker x and changes every xth locker. I need to figure out which lockers are left open in a row of 25, , and a row of lockers. I have been trying to figure this out for 4 days and my parents can not figure it out either. I don't know what number person x is. My parents say this has nothing to do with math. Can you help? Clearly this is intended to be solved by trying a small example and extending it, rather than by seeing it all at once. There were already several complete solutions in the archive, but I chose to offer some suggestions to help Michael discover a solution himself, rather than just give a link: It has a lot to do with math!
But I'm not sure whether everyone your age can be expected to figure out the complete answer on his own. You may be expected only to recognize a pattern, but there is a lot of very interesting math if you look deep enough. It sounds like a lot of your confusion is over the 'x' part, so maybe the problem wasn't made fully clear. Usually in this problem it's a classic, by the way , the number of people is the same as the number of lockers in the hallway.
So what they mean by 'person x' is all the people from person 1 up to the last person. In other words, if there are 10 lockers there are 10 people, and the pattern continues from person 1 up through person If there are lockers, there are people and each of the goes through the hallway turning lockers that are multiples of their own number. Does that help? Michael may not yet be fully accustomed to using variables, or may think x must be a specific number to be solved for. If I were you, I would first try "playing" with the problem with a small number of lockers, like 25 so you can see what the whole thing means.
Do you follow what I did, and understand how the problem works? The idea is that each person opens or closes only the lockers that are a multiple of his number: 2 changes the multiples of 2, 3 changes the multiples of 3, and so on up to person x, the last one to go through. There are many ways you might write out your work; I chose a way that requires less writing than some, while keeping all the information visible. Each column represents what that person does. The first person opened them all; the second closed 2, 4, 6, 8, and 10; the third opened 3, closed 6, and opened 9; and so on. The only doors left open with 10 lockers are 1, 4, and 9. One way to work the problem is to do this with more lockers and look for a pattern in the numbers of the lockers left open; a better way is to look for a REASON why there should be a pattern.
What is it that makes one locker end up open and another end up closed? I always emphasize reasons over patterns , because a pattern you see may not be real, and may not continue for larger numbers. I think the first time I solved the problem I saw the pattern very quickly, but had to stop and think in order to convince myself it was real. So now we think about what it takes to leave a locker open: Notice that each time a locker is "touched" it changes from open to closed or vice versa.
So in order to end up open, it has to be touched an odd number of times. Now, what might make that happen? A key is to realize that the whole problem is about multiples and divisors. Do you see why? That's where the math comes in! If you have any further questions, feel free to write back. Good luck! We never heard back to see whether this was enough to help Michael. We could describe my plan to attack this problem as Play, Pattern, Prove. A little more of a hint … This question from will take us further: Lockers There are lockers in a high school with students.
The problem begins with the first student opening all lockers; next the second student closes lockers 2,4,6,8,10 and so on to locker ; the third student changes the state opens lockers closed, closes lockers open on lockers 3,6,9,12,15 and so on; the fourth student changes the state of lockers 4,8,12,16 and so on. This goes on until every student has had a turn. How many lockers will be open at the end? What is the formula? I can't figure out the pattern. Kate Note the slightly different way of saying the same thing; using example numbers is helpful. Doctor Bruce carried out parts of my plan, in effect taking Kate partway through the process: I enjoyed thinking about this problem when I first heard it some years ago. The students who come after them are not going to touch lockers , so we can see which ones in that first batch are still open and try to guess the pattern. When we do that, we find that lockers 1, 4, and 9 are open and the others are closed.
Now, that isn't much to go on, so maybe you could let the next 10 students go do their thing. Then the first 20 lockers are through being touched, and we find that lockers 1, 4, 9, and 16 are the only ones in the first 20 that are still open. So what is the pattern? Do the numbers 1, 4, 9, 16 look familiar? Now we reverse the experimentation, picking a single locker and thinking about what happens to it, in order to answer my question about what it takes for a locker to end up open: Let's take any old locker, like 48 for example. It gets its state altered once for every student whose number in line is an exact divisor of Here is a chart of what I mean: this Student leaves locker 48 1 open 2 shut 3 open 4 shut 6 open 8 shut 12 open 16 shut 24 open 48 shut Notice that 48 has an even number ten of divisors, namely 1,2,3,4,6,8,12,16,24, So the locker goes open-shut-open-shut Any locker number that has an even number of divisors will end up shut.
So the lockers that are open must have an odd number of divisors. We saw something about this last week … Which numbers have an odd number of divisors? That's the answer to this problem. Just to help you along, here are the locker numbers up to that are left open: 1,4,9,16,25,36,49,64,81, See if you can describe these numbers in a different way from "having an odd number of divisors. When you understand how to describe them, you will see that 31 of the lockers are still open without having to work it all out!
With more numbers, it should be clear: the answer is, all perfect squares. We just have to convince ourselves that this makes sense, and count how many of them there are. A full answer Another question was given a complete answer: Opening and Closing Lockers There are closed lockers. There are students. The first student comes in and opens every locker. The second student comes in and closes every other locker. The third student comes in and opens every third locker. The pattern continues until all students have done what they're supposed to do. At the end, how many lockers are still open?
I need to know what track I have to be on at the very beginning. Doctor Anthony started with a correction, assuming this is meant to be the usual problem: I think you have made a mistake in your description of the problem. In this situation it is easy to see that every locker whose number is a perfect square will be open at the end of the exercise, and all other lockers will be closed. Is that easy to see?
Only when you see it the right way. Now all numbers with an even number of factors will end up closed. We conclude that all the lockers whose numbers are perfect squares will be open at the completion of the exercise. But why? We saw this last week, and he briefly explains it here, using prime factors: To show that perfect squares have an odd number of factors we express the number in its prime factors. If it is a perfect square the power of each prime factor must be even, e. The number 2 could be chosen 0,1,2 times,i. Note that taking none of 2, 3 or 5 as factors gives the 1 which we require as a factor.
Problem 1 To start with, we can simply begin walking through process by hand. Locker 1 will begin closed, then the 1st person will come in an open it. After that, no one touches the 1st locker so we know it stays open. Locker 2 is opened by the 1st person, closed by the 2nd and it stays closed. Locker 3 is opened by the 1st person, left alone by the 2nd and then closed by the 3rd. This is going to get tedious if I keep explaining in words. Lets use a table. This is also going to get tedious if I want to do this for lockers and students. We definitely need to find a pattern. My next step was put together a lazy little Matlab script to do the same exact thing as my table. See here. There is definitely a pattern! Notice those open lockers form the bands you see in the picture.
So why do some end up open and some end up closed? The easiest way to see why is to consider what happens to a single locker. For example, think about locker When does it change state? Obviously, person 1 opens the locker, person 2 closes it, person 3 opens it, person 4 closes, person 5 does nothing, etc. Notice that if the locker number, 24, is divisible by the person number, then the state changes: The number 24 has 8 factors, that is, 8 numbers that divide evenly into it.
So the key is that any locker number with an even number of factors will end up with closed and any with an odd number will end up open. So what numbers have an odd number of factors?
Gear-obsessed editors choose every product we review. We may earn commission if you buy from a link. How we test gear. Feb 28, Kory Kennedy using Illustration Copyright csaimages. Student 1 will open it, since student 1 opens every locker. Student 2 will then close it, since student 2 closes every even locker and 24 is even. Student 3 will open it and then student 4 will shut it, since 24 is a multiple of both 3 and 4. Student 5 will pass it by since 24 is not a multiple of 5. Thus, locker 24 will have its status changed by students 1, 2, 3, 4, 6, 8, 12, and Now, how does this lead us to figure out which lockers are opened at the end?
Locker 1, which has one factor, will clearly be open at the end, since the only student who touches it is the first student, who opens it. Locker 2, with two factors, will be closed, since the only two students to touch it are student 1, who opens it, and then student 2, who closes it. Locker 3, also with two factors, will also be closed at the end. On the other hand, locker 4, which has three factors 1, 2, and 4 , will be open, shut, and then open again. This line of thinking leads us to the conclusion: Only those lockers with an odd number of factors will be left open at the end of the prank. So which numbers have an odd number of factors? Consider that a factor is an integer that, when multiplied by another integer, produces the number of interest. Since factors come in pairs, most numbers have an even number of factors. The only exception occurs when factors are paired with themselves. In this case, 4 is paired with itself to produce the number Thus, 16 has an odd number of factors: 1, 2, 4, 8, Factors are only ever paired with themselves in the case of perfect squares, which means that perfect squares are the only numbers with an odd number of factors!
Student View Task The 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in a row. The first student starts with the first locker and goes down the hallway and opens all the lockers. The second student starts with the second locker and goes down the hallway and shuts every other locker. The third student stops at every third locker and opens the locker if it is closed or closes the locker if it is open. The fourth student stops at every fourth locker and opens the locker if it is closed or closes the locker if it is open. This process continues until all 20 students in the class have passed through the hallway. Which lockers are still open at the end of the game? Explain your reasoning. Which lockers were touched by only two students? Which lockers were touched by only three students? Which lockers were touched the most? IM Commentary The purpose of this instructional task is for students to deepen their understanding of factors and multiples of whole numbers.
This is a classic mathematical puzzle; often it is stated in terms of lockers, so students have to make the connection to factors and multiples to solve it. In this version, students can just go through all the rounds of opening and closing locker doors and observe after the fact that there is a relationship between the factors a locker number has and whether it is open or closed at the end. This task provides students with an excellent opportunity to engage in MP7, Look for and make use of structure if they see early on that there is a relationship with factors and multiples or MP8, Look for and express regularity in repeated reasoning if they start to see and describe the pattern as they imagine students opening and closing the lockers.
Because the total number of lockers is only 20, students might answer the questions without thinking about the underlying reasons for their answers. In the first question for example, a student might say "1, 4, 9, 16 are all still open because I tried it out and those were the ones that were left open. Asking students to try a larger number of lockers say 50 or and repeating the game can also help students look for a pattern since going through all the rounds of the game becomes less and less feasible as the number of lockers increases.
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