"Fun" Run

“On your marks, get set,” is all I heard before the loud roar of a gun went off, startling me and leaving a ringing in my ears. I started jogging at a normal pace, and I was in the middle of a huge crowd of about 500 kids. I thought about this morning, when my mom woke me at about 6:00 A.M. on a Saturday. She asked me if I wanted to go on a fun run because my brother was doing it. I grudgingly went along, because it was better then going to work with my dad.

Soon, I noticed I was near the front of the crowd. I was running right near a group of kid athletes wearing matching shorts and sweatbands. At this point in the race, my feet, with incredibly heavy skate shoes that could have caused an earthquake measuring 5.0 on the Richter scale with every step, started to hurt badly. At this point, I was thinking, “I should have just gone to work with dad, at least it would be better then the 20 years of foot therapy I would need after this, heh.”

My feet were hurting so badly, I just wanted to get it over with, so I started running as fast as I could. I noticed I was passing groups of matching-short-and-sweatbands people. As I was running, I suddenly became really hungry. I thought of that barley-breakfast consisting of half a cinnamon-toast waffle and tiny pouch of OJ in the car on the way to this race. I was also thinking to myself,“Hey, I’m running pretty fast!”

I was near the head of the pack, turning a corner, and I saw the blue mat that indicated the finish line. Sweat was dripping from my face as I was making the final push for the finish, passing a few people on my way there. I glanced at the scoreboard. 25TH PLACE! Dang, I never knew I had all this potential! Suddenly my feet had stopped hurting as my mom embarrassed me with victory screams. Her face was glowing with joy and she was jumping up and down with joy. Thankfully, I felt more pride at what I just did then embarrassment. And to think that just that morning I had planned on being a couch potato all day.

The Locker Problem

In this problem, we had to find out how many lockers would be open if there were 1000 lockers, and one person opens all of the lockers, the 2nd person closes every other locker, the 3rd person changes the state of every 3rd locker, and so on.

First, we tried to find a pattern that we could build from. We messed with bottle caps to see if we could find one, and lo and behold, we did. We found out that there was one open locker, then 2 closed lockers, then another open locker, then 4 closed lockers, then another open locker, then 6 closed lockers and on and on. From this, we found yet another pattern that would help us even more.

We continued going through numbers, seeing if there was a better way to do this, and we found out that all the open lockers are perfect squares, being the square root of the number would be a whole number without and decimals (ex. 9, 16, and 25). There are 31 If you look at the attached diagram, you’ll see why that the perfect squares are the only lockers that are open. The diagram also almost explains why perfect squares have an odd number of factors, but not quite.

Perfect squares have an odd number of factors simply because one of the factors is there twice, but only counts once. For example, in 47, you can multiply 7 by 7, but only one 7 would count as a factor.