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

ones.)

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.