Yes, another 6th grade math problem:
The locker problem goes like this:
A school has 1,000 lockers and 1,000 students.
The first student runs through the school and
opens all of the locker doors.
The second student runs through the school and
opens every second locker.
The third student runs through the school and
"changes the state" of every third locker. (This
means she opens closed lockers, and closes open
The fourth student runs through the school and
changes the state of every fourth locker.
And so on, until the 1,000th student goes through
the school and changes the state of the 1,000th locker.
How many locker doors are open? Why?
What do you think? Scared? Intrigued? Curious?
The book recommended taking a small sample
and charting what happens as the students move
through the school. Using a 25 block grid, I went
a step further and wrote down the number of the
student on each square as they touched a given locker.
If you do the experiment, you find that the
first locker is open (only the first student ever
touches it), the second and third lockers are closed,
with two students having touched each. The fourth
locker is open. The fifth through 8th lockers are
closed. The 9th locker is open. This continues.
Analyzing the grid, I find that only lockers 16 and 25
are open. All of the others are closed.
So. Lockers 1, 4, 9, 16, and 25 are open. They are
all perfect squares. What makes a square number
special in this case? It's the number of factors.
A non-square number has an even number of factors.
(For example, 12 has the factors of 1, 2, 3, 4, 6, and
12.) A square number has an odd number of factors.
(25 has factors of 1, 5, 25).
Not only can you determine how many lockers are open,
you can figure out how many students touched each
locker. If you choose a locker, say number 100, you
can determine how many people touched it by counting
its factors. 100 has factors of 1, 2, 4, 5, 10, 20, 25,
50, and 100, so 9 students touched that locker, and
the locker is open at the end of the experiment.
I would love to say that I figured this problem out
in no time flat, but I didn't. I got stuck.
My first idea was to use prime factors, and that seemed
to work, a bit. Until I realized it didn't work at all.
This is where having a complete model, one with
not just open and closed door symbols, but the student
number that touched the door written on the chart,
really helped. Still, even with my misstep, I really
enjoyed the time I spent charting and figuring out
this problem. So. Much. Fun.