# Solving exponential equations

A ruler is not necessary. You need to cut off a piece that's

$$\dfrac{2}{3} - \dfrac{1}{2} = \dfrac{4}{6} - \dfrac{3}{6} = \dfrac{1}{6}$$meters long. The only cuts you can make with accuracy, will involve folding in half. Thus, on your first fold, you can obtain 1/3 and 2/3. Now there is a crease down the middle, forming two sections. Each of these sections can be folded in half. Thus, you can obtain 1/6 and 3/6. But 3/6 = 1/2, so the desired length can be found in just two folds. In general, making n folds will give you 2^n + 1 measurements. If your initial length is \(x,\) then \(m\) folds yields measurements from \(0\) to \(x,\) in increments of \(x/2^m.\) Further, if we define the number of tick marks as \(n \le 2^m,\) then all numbers of the form \(xn/2^m\) are obtainable. Thus, our original problem can be seen as searching for a pair (m, n), as follows

$$\begin{align} & \dfrac{2}{3} \cdot \dfrac{n}{2^m} = \dfrac{1}{2} \\[1em] & \dfrac{2n}{3 \cdot 2^m} = \dfrac{1}{2} \\[1em] & 4n = 3 \cdot 2^m \\[1em] & n = 3 \cdot 2^{m - 2} \end{align}$$Because \(n\) must be a whole number, \(m\) must be greater than or equal to \(2.\) Thus, \(1/2\) can be achieved by making \(2\) or more folds.