1. Imagine you are at a school with 100 lockers, all shut. • • • Suppose the first student goes along and opens every locker. The second student then shuts every other locker, beginning with locker number 2. The third student changes the state of every third locker, beginning with locker number 3. (If the locker is open, the student shuts it, and if the locker is closed, the student opens it.) The fourth student changes the state of every fourth locker, beginning with number 4. Imagine that this continues until the 100 students have followed the pattern with the 100 lockers. At the end, which lockers will be open and which will be closed? Which lockers have been switched the most often? How many lockers, and which ones, were touched exactly five times? Hint: You might initially think about the problem with a small number of lockers and then generalize if possible
1. Imagine you are at a school with 100 lockers, all shut. • • • Suppose the first student goes along and opens every locker. The second student then shuts every other locker, beginning with locker number 2. The third student changes the state of every third locker, beginning with locker number 3. (If the locker is open, the student shuts it, and if the locker is closed, the student opens it.) The fourth student changes the state of every fourth locker, beginning with number 4. Imagine that this continues until the 100 students have followed the pattern with the 100 lockers. At the end, which lockers will be open and which will be closed? Which lockers have been switched the most often? How many lockers, and which ones, were touched exactly five times? Hint: You might initially think about the problem with a small number of lockers and then generalize if possible
Chapter9: Sequences, Probability And Counting Theory
Section9.5: Counting Principles
Problem 40SE: A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in...
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Transcribed Image Text:1. Imagine you are at a school with 100 lockers, all shut.
•
•
•
Suppose the first student goes along and opens every locker.
The second student then shuts every other locker, beginning with locker
number 2.
The third student changes the state of every third locker, beginning with
locker number 3. (If the locker is open, the student shuts it, and if the locker
is closed, the student opens it.)
The fourth student changes the state of every fourth locker, beginning with
number 4.
Imagine that this continues until the 100 students have followed the pattern with
the 100 lockers. At the end, which lockers will be open and which will be closed?
Which lockers have been switched the most often? How many lockers, and which
ones, were touched exactly five times?
Hint: You might initially think about the problem with a small number of lockers and
then generalize if possible
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