The Locker Problem

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The Locker Problem
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Imagine you are at a school that has student
lockers. There are 1000 lockers, all shut and
unlocked, and 1000 students.
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Suppose the first student goes along the row and
opens every locker.
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The second student then goes along and shuts
every other locker beginning with number 2.
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The third student changes the state of every
third locker beginning with number 3. (If the
locker is open the student shuts it, and if the
locker is closed the student opens it.)
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The fourth student changes the state of every
fourth locker beginning with number 4.
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Imagine that this continues until the thousand
students have followed the pattern with the
thousand lockers. At the end, which lockers will
be open and which will be closed? Why?
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Click here to simulate this problem.http//hydra.
educ.queensu.ca/java/Lockers/
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SOLUTION
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These numbers are the square numbers 12, 22, 32,
and 42. So, there is an open locker door at every
square number. How many square numbers are there
between 1 and 1000? Through a little trial and
error, you'll find that 312 is the last square
number less than 1000. So, there are 31 open
doors (the last one occurring on the door
numbered 312 or 961).
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Looking at FactorsA special property of square
numbers is that they always have an odd number of
factors. A factor is a number that divides
another number evenly (with no remainder). For
example, 8 has an even number of factors, namely,
1, 2, 4, and 8. But, 9 has an odd number of
factors, namely, 1, 3, and 9. In fact, all
numbers except the square numbers have an even
number of factors.
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You can use this fact to solve the locker
problem. Take any locker number, 40, for example.
Its state (open or closed) is changed for every
student whose number in line is a factor of the
locker number. So, write out all the factors of
40, like this
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Like all other lockers numbered with non-square
numbers, it ends up closed after all the students
have gone through the line because it has an even
number of factors.
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Here's the factor pattern for a square number, 16.
Locker 16 remains open because it has an odd
number of factors.
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You can now conclude that all the doors with
square numbers on them will remain open because
all square numbers have an odd number of
factors. You can now conclude that all the
doors with non-square numbers on them will remain
closed because only square numbers have an odd
number of factors.
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There are 31 open doors. The numbers of the open
doors are listed below.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, 961