# F-LE, A-CED Paper Folding

Alignments to Content Standards: A-CED.A.1 F-LE.A.2

A common claim is that it is impossible to fold a single piece of paper in half more than 7 times (try it!).

Among other attempts, the challenge was taken up on an episode of the TV show Mythbusters, trying to avoid the physical restrictions by beginning with an exceptionally large sheet of paper. Watch the quick summary of their efforts below:

As you can see, the height of the folded sheet of paper increases dramatically as you continue folding. Assuming you started with a large enough sheet of paper, how many folds would it take for the stack of paper to reach the moon?

## IM Commentary

This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth. While implementations will vary (as discussed below), the core idea is that each fold of the piece of paper doubles the height of the stack. Combined with an estimate of the original thickness of the paper and the distance to the moon, this is enough information to deduce the minimum number of folds to get there. The solution uses the estimate of 0.1 mm for the thickness of paper and 385,000 km for the distance to the moon. Teachers may want to include a discussion of reasonable levels of numerical accuracy (MP 6) as students come up with their own values for these quantities through reference materials.

Instructors should exercise discretion in how much scaffolding they provide for their students. A completely open-ended version would have students recognize the doubling pattern on their own (perhaps with some hands-on experimentation), and also recognize the need for the two physical quantities described above. The distance to the moon will probably require some reference material, but the thickness of a piece of paper could either be estimated from the video or measured experimentally in the classroom (perhaps after being folded a few times!). At the other extreme, a very direct approach would explicitly ask students to produce an expression for the height of the folded stack (either in sheets or a standard metric measurement of length) after $n$ folds, e.g., $2^n$ sheets tall, so $.1(2^n)$ millimeters tall. Then either by algebraic manipulation (for students with such preparation) or numerical experimentation, students can find that the first value for which $.1\times 2^n \gt 385,000,000,000$ is given by $n=42$.

Instructors may find it valuable to ask students to make advance predictions for the number of folds, especially if the task is being used as an introduction to exponential growth. If presented with the distance to the moon up front, students' predictions may be significantly larger than the computed value, providing a good discussion point about the striking growth rate. Also reasonable here would be a discussion of the realism of the physical scenario -- it turns out that to achieve the moon-height paper stack in the problem with a final stack the dimensions of a regular sheet of paper, you'd have to start with a sheet of paper roughly the size of Colorado.

As some further interesting reading for motivated students, we note that (as students might notice if actually folding paper) as the paper gets thicker, the geometry of the folded sheet looks less and less like $2^n$ individual sheets stacked on top of each other, but has semi-circular rounded parts near the folds. This puts a more realistic physical constraint on how many folds you can do in any given direction with a given length of paper. This problem was analyzed and solved in 2001 by a high school student by the name of Brittney Gallivan (http://mathworld.wolfram.com/Folding.html), which in and of itself might be an inspiration to current high school students.

## Solution

Solutions will vary depending on the physical constants found.  The current answer uses a paper width of 0.1mm for the thickness of a sheet of paper, and a distance of 385,000km (or 385,000,000,000 mm) from the earth to the moon.

Each time the paper is folded in half, the number of sheets in the stack doubles.  So after one fold, the stack is two sheets thick (0.2mm), after two folds, the stack is four sheets thick (0.4mm), and so on.  Since it doubles each time, after $n$ folds the thickness of the stack in millimeters will be $$.1\times \underbrace{2\times 2\times \cdots\times 2}_{n\text{ times}}=.1\times 2^n\quad \text{mm}$$

To find how many folds it takes to get to the moon, we want to know how many folds it takes until the height of the stack is at least 385,000km.  Converting to millimeters, this is the inequality

$$.1\times 2^n> 385,000,000,000$$

Using a calculator, we find that $n=42$ is the first value of $n$ for which that inequality is satisfied.  We conclude that it would take 42 folds to get to the moon.