DeCav’s Intellectual Corner #3: Dang that’s a lot!

[DeCav]

Well, this week I’m just gonna share some interesting math problems that demonstrate the counter intuitiveness of exponential progression.

The first one I told to my coworkers after reading about it and they got up in arms about the answer I gave them, insisting that I was crazy. We had to get out the calculator before they’d believe me.

Basically the question goes like this...

Take a piece of paper. Let’s say for the sake of imagination that it’s quite a large sheet of paper.
Now first you cut the paper in half and stack the two pieces on top of each other. That’s the first step or iteration. Next cut the two pieces of paper in half again and stack the four pieces on top of each other. That’s the second step. Now cut the four pieces of paper in half (long ways or short ways) and stack the eight resulting pieces on top of each other. That’s the third step.

OK, now you have a stack of paper eight sheets high. Cut and stack these papers 50 more times.

Now, I don’t recommend actually trying to do this because you’ll find this quite an impractical exercise. Eventually these pieces of paper will be quite small
and difficult to cut.

The question is...
How high will the stack of paper be when you have finished cutting and stacking the paper as explained?

I am always quite amused by the answers I get from people. Some of the guesses are quite large and some quite small but none are ever quite right.

But before we talk about how high the stack of paper would be let’s tackle another similar hypothetical situation...

Take a chessboard and place a grain of rice on one of the corner squares. Then place 2 grains of rice on the square next to it. Place 4 on the third and 8 on the fifth. Double the grains of rice each time until you’ve done every square. How many grains would be on the chessboard?(assuming you could fit that many grains of rice on a chessboard which you can’t)

Well, lets count them and see. Now, let’s say you started at zero and counted one grain of rice a second until you’d counted all of them. How long do you think it’d take to count them all?

Well while you think about that let’s go back to the paper stacking problem. The question is how high would a stack of papers be if you started with one big sheet and cut it in half and stacked the resulting pieces on top of each other 53 times.

I swear when I asked my friends this one of them actually said two or three feet high. The actual answer is that the stack would reach from here to the sun.

If you don’t believe it get a calculator or better yet create a spreadsheet that takes .003″ which is just slightly thinner than notebook paper and double that number 53 times and see how many miles that is.

As for the problem about how long it would take to count the grains of rice, well if you’d started counting these grains of rice when the universe was created (estimated to be about 13 billion years ago) and you counted one grain every second you would not EVEN be a tenth of the way through counting all the rice today.

Peace Out,
DeCav