# 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.