Cutting The Paper

Since no cuts of the same length can cross each other, we can consider the following cases: $\begin{array}{rl} \text{1 cut} &\Rightarrow \text{2 = (1 + 1) pieces}\\ \text{2 cuts} &\Rightarrow \text{3 = (2 + 1) pieces}\\ \text{3 cuts} &\Rightarrow \text{4 = (3 + 1) pieces}\\ &\vdots\\ (k-1)\text{ cuts}&\Rightarrow k=((k - 1) + 1)\text{ pieces}\\ k\text{ cuts} &\Rightarrow k + 1\text{ pieces} \end{array}$ For each number of cuts, you increase that value by 1, which gives the number of pieces. From this relationship,

• To make $k$ pieces, we cut $(k - 1)$ times.

• To make $4k$ pieces, we cut $(4k - 1)$ times.

Then, $\dfrac{s}{k-1}=\dfrac{x}{4k-1} = \dfrac{\text{time elapsed}}{\text{number of cuts}}$ where $x$ is the answer to the question. Thus, $\boxed{x = \dfrac{s(4k-1)}{k-1}}$ seconds for cutting into $4k$ pieces.

I took, $k=2$ , and $\boxed{\frac{s(4k-1)}{k-1}}$ was the only answer that gave $7s$ , which I knew to be the answer for this extremely simple case. Since it was the only one that worked for the case it had to be the answer.

However, the correct solution is as follows:

$k$ pieces requires $k-1$ cuts. This took $s$ seconds.

And for $4k$ pieces you will require $4k-1$ cuts.

So,

Time $= \frac{\text{number of cuts}}{\text{cuts per second}} = \frac{4k-1}{\frac{k-1}{s}} = \frac{(4k-1)s}{k-1}$