> -----Original Message-----
> From: Paul H Christidis [mailto:[log in to unmask]]
>
> My 7th grader, who has just started working with first degree
> equations, was
> given the following 'challenge' problem.
>
> The local High School has 1000 lockers.  The students decide
> to perform the
> following experiment:
> The first student will open all the locker doors.
> The 2nd student will close every door of a locker starting
> with the 2nd locker
> and is a multiple of 2.
> The 3rd student will 'change the position' of each locker
> door, starting with
> the 3rd locker and is a
>    multiple of 3.
> The 4th ....
> The 5th ....
>  etc.  etc.
> After all 1000 students perform their task which locker doors
> will remain open.
>
> The above can be solved very easily by writing a program but
> that was not one of
> the available options.
>
> Short of manually simulating the above on a quadrille paper
> and seeing the
> developing pattern, which was the method selected, can anyone
> think of another
> method that a 7th grader could have used?
>
> BTW.  The answer is:  Every I**2th locker will remain open
> where "I" assumes the
> values of the natural numbers from 1 to INT(SQRT(numoflockers)).

I don't know if a 7th grader would think of this, but if you consider the
problem from the point of view of "given any arbitrary locker number, will
it be open or closed?" how would you solve it?  It takes some mental
gymnastics to realize that what is occuring is that each door is being
"toggled" once for each FACTOR of the number, so the question becomse "for
any arbitrary door number x, how many factors exist?"

From there, you need to make the mental back-flip of realizing that all
numbers will have an even number of factors EXCEPT "squares" of a number --
after all, a "factor" is one of TWO numbers that, when multiplied together,
return the number in question.  In the case of a "squared" number, the two
"factors" are the same "number", so the count of factors for squares is odd
and even for all other numbers [including primes]

Starting with a "closed" state, all "squares" of a number will be "toggled"
an ODD number of times, and will therefore remain "open".  All other numbers
will be toggled an EVEN number of times and thus return to a "closed" state.
[this works even for lockers 1 and 2, the wording of the initial actions is
a red herring -- it could have been worded as "starting with all lockers
closed, each student 'toggles' the state of every 'nth' locker, where n=<the
student's number from 1 to 1000>"]