the 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in row. the 1 student starts with the first locker and goes down the hallways and opens all lockers. the 2 student starts with the second locker and goes down the hallway and shuts every others lockers. the third student starts with the third locker and changes every third locker: if the locker is open the student closes it, and if it is closed the student open it. the fourth student starts with the fourth locker and changes every fourth locker: if the locker is open the student closes it, and if it is closes the student opens it. THis process continues until all 20 students in the class have walked down the hallway. a.) which lockers are still open at the end of the game?. b.) which lockers were touched by the only two student?. c.) which lockers were touched by only three students?.d.) which lockers were touched the most?
Haven't devised an algebraic solution. Maybe you can work on that after taking a look at the table below, where row #n is the state of the lockers (0=closed, 1=open) after the nth student has gone by. The last line is the number of times each locker was touched.
01: 11111111111111111111
02: 10101010101010101010
03: 10001110001110001110
04: 10011111001010011111
05: 10010111011010111110
06: 10010011011110111010
07: 10010001011111111010
08: 10010000011111101010
09: 10010000111111101110
10: 10010000101111101111
11: 10010000100111101111
12: 10010000100011101111
13: 10010000100001101111
14: 10010000100000101111
15: 10010000100000001111
16: 10010000100000011111
17: 10010000100000010111
18: 10010000100000010011
19: 10010000100000010001
20: 10010000100000010000
1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6
thanks Steve And Happy new year and God bless you
To find the answers to the given questions, let's go through each step of the game.
Step 1: The first student opens all the lockers.
At this point, all the lockers are open.
Step 2: The second student starts shutting every other locker.
The second student closes the even-numbered lockers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Step 3: The third student starts changing every third locker.
The third student closes the lockers that were previously open (odd multiples of 3): 3, 9, 15, 21 (not present).
Step 4: The fourth student starts changing every fourth locker.
The fourth student closes the lockers that were previously open (multiples of 4): 4, 8, 12, 16, 20.
This process continues until all 20 students have walked down the hallway.
a) Which lockers are still open at the end of the game?
The lockers that will remain open at the end are the ones that have been toggled an odd number of times. In this case, it means their number of divisors is odd. The lockers with an odd number of divisors are perfect squares.
Therefore, the open lockers at the end of the game are: 1, 4, 9, 16.
b) Which lockers were touched by only two students?
The lockers that were touched by only two students are the ones that remain changed. We can determine this by finding the lockers with an even number of factors. The only numbers that have an even number of divisors are perfect squares.
Therefore, the lockers touched by exactly two students are: 4, 9, 16.
c) Which lockers were touched by only three students?
The lockers touched by exactly three students are the ones that have an odd number of factors but are not perfect squares. These can be found by excluding the perfect squares from the list of numbers with an odd number of factors.
Therefore, the lockers touched by exactly three students are: 3.
d) Which lockers were touched the most?
The lockers that were touched the most are the ones with an odd number of divisors. These can be found by excluding the perfect squares from the list of numbers with an odd number of factors.
Therefore, the lockers touched the most are: 3, 9, 15 (assuming there is a locker numbered 15).
To find the answers to the questions, let's go through each step of the game and track which lockers are affected.
a) To determine which lockers are still open at the end of the game, we need to analyze the pattern.
The first student opens all the lockers. So, all the lockers are open.
The second student starts with the second locker and shuts every other locker. This means that all even-numbered lockers (2, 4, 6, ...) are closed.
Then, the third student starts with the third locker and changes every third locker. This means that the multiples of three will be toggled. The lockers that were closed by the second student (even numbers) will be opened, and vice versa. So the lockers numbered 3, 6, 9, 12, 15, 18 will be closed, while the rest will be open.
Next, the fourth student starts with the fourth locker and changes every fourth locker. This means that the multiples of four will be toggled. The lockers that were closed by the second student and then reopened by the third student (multiples of four) will now be closed, and vice versa. So the lockers numbered 4, 8, 12, 16, 20 will be open, while the rest will be closed.
This pattern continues as each subsequent student toggles the lockers with numbers representing their multiples. At the end of the game, the lockers that have an odd number of factors (meaning they were toggled an odd number of times) will remain open, and the lockers with an even number of factors will be closed.
So, the lockers that remain open at the end of the game are 1, 4, 9, 16.
b) The lockers that were touched by only two students are the prime number lockers. A prime number has only two factors (1 and itself). In the given scenario, the prime number lockers are those that were toggled by the first and the second student and not touched by any subsequent student.
So, the lockers that were touched by only two students are 2, 3, 5, 7, 11, 13, 17, 19.
c) The lockers that were touched by only three students are the lockers that were toggled by the first, second, and third students but not touched by any subsequent students.
To find these lockers, we look for numbers that have three factors. Only square numbers have three factors because they have two equal prime factors. Therefore, the lockers that were touched by only three students are the square number lockers.
So, the lockers that were touched by only three students are 1, 4, 9, 16.
d) To find the lockers that were touched the most, we need to determine which numbers have the most factors. These numbers would have been toggled multiple times throughout the game.
The lockers that have the most factors are the perfect square numbers since they have an odd number of factors. This means all lockers that are perfect square numbers will be toggled odd number of times (except for the first student).
So, the lockers that were touched the most are 1, 4, 9, 16. (These are the same lockers that remain open at the end of the game.)
Therefore, the answers to the questions are:
a) Lockers that remain open: 1, 4, 9, 16.
b) Lockers touched by two students: 2, 3, 5, 7, 11, 13, 17, 19.
c) Lockers touched by three students: 1, 4, 9, 16.
d) Lockers touched the most: 1, 4, 9, 16.