Dividing Paper Puzzle

When I was young, I would fold a sheet of letter paper in half, for origami projects. It occurred to me that the two halves looked almost the same as the whole sheet of paper - except they were smaller. I could see they weren't exactly the same shape; they were off by a little bit. But the idea stuck in my head.

You can use a pen, instead of scissors, to halve the paper.
Those rectangles all have the same shape, but are different sizes.

One night when I was 12, I thought about my idea. I wondered if it was possible to have a sheet of paper that could be cut in half, resulting in 2 smaller versions of the same paper. That would be neat, to be able to cut a paper in half and get 2 papers that had the same exact shape. If that were possible, then you could cut those papers, too; and the resulting papers would have the same shape as all the other papers. You could keep cutting in half forever, and each paper, no matter how small, would have the same shape as all the others.

I HAD to figure it out. Was it possible, or not? I took a pen (or pencil, I don't remember) and a sheet of paper, and began writing. In a few minutes of working with math and numbers, I found that it was possible. I had the solution right in front of me.

The puzzle is this: what could the dimensions for the paper be?

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The Journey Home

We got back home from our trip to Arizona about a week ago.

Instead of driving back the way we came, we drove through New Mexico, so I didn't get to see the Grand Canyon again. But I got to see parts of New Mexico I hadn't seen before. Here are some photos, listed in chronological order:

Common Arizona Cacti

When referring to multiple cactus plants in the English language, some say "cactuses", others say "cacti", and a few don't change anything and just say "cactus". All are acceptable, but I prefer to say "cacti".

Cactus plants are very common in Arizona. They are specially adapted to the dry climate; some types can live through up to 10 years of drought. Their green, fleshy stems are often ribbed, making it easier to expand to hold water. The majority have sharp spines, instead of leaves; photosynthesis occurs in the stems. The spines only grow from areoles, which all cacti have.

The largest type of cactus in Arizona is the Carnegiea gigantea, or "saguaro" (suh-wah-ro). If you saw this type of cactus, you would remember it more easily than any others. It gets so tall when mature that none of the other cacti can even start to compare; mature saguaros dwarf even the tallest men.

There are many other types of cacti besides the Saguaro in Arizona, but here are the few you'd be most likely to see:

Arizona-Sonora Desert Museum

On Monday we visited the Arizona-Sonora Desert museum, which is just west of Tuscon. It's like a zoo, but there are a lot of things besides animals there - like meteorites, caves, and plants.

Western Screech-Owl

To the left is a Western Screech-Owl. The handler for this bird said the owl doesn't actually screech, and the name isn't accurate.

American Kestrel

On the right is a live male American Kestrel. Notice that the handler is wearing a glove to protect his hand from the sharp talons. In the wild, those talons are used for killing prey. This particular bird has never used its talons on live prey, however. It was born in captivity, and has never been fed anything alive. Because of this, the bird hasn't developed the skills it needs to survive alone in the wild.

Here's a Harris's Hawk. Notice how much larger it is than the Kestrel: