What is the least number of cuts that should be made to obtain the maximum pieces possible in a parallelogram?

Question

The problem is as follows:

The figure from below represents a piece of wood that has the shape of a parallelogram. whose sides are of equal length. If we want to cut piece of wood in the maximum number of equilateral triangles possible with a side of $5$ cm. How many straight cuts at least are necessary to make using a circular saw to obtain these pieces?

The alternatives which do appear in my book are:

$\begin{array}{ll} 1.&\textrm{3 cuts}\\ 2.&\textrm{4 cuts}\\ 3.&\textrm{5 cuts}\\ 4.&\textrm{6 cuts}\\ \end{array}$

Gee for this problem I have no idea where to begin. What I thought was to construct equilateral triangles inside the figure. But this did not helped that much.

In other words I've attempted to fold the figure by its corners but this approach did not yielded the desired results and I ended up with no answer.

Thus what should be the way to solve this problem?. Is there any trick here or what?. It would help me a lot an answer which would indicate with some arrows or lines where should the cuts be made or the justification for these cuts because I'm lost with this.

Answer 1

It’s not hard to check that the parallelogram can be divided into $32$ equilateral triangles with $5$ cm sides. Clearly you will have to stack the pieces after each cut if you are to get anything like that number. Note that each cut can at most double the number of pieces that you currently have, and $32=2^5$, so it will take at least $5$ cuts to get the maximum possible number of triangles. And it can in fact be done in $5$ cuts.

HINT: Make the first cut parallel to the base, producing two identical parallelograms, and stack them in the natural way before making your next cut.

Answer 2

I get $5$ cuts:

1. Cut along the middle, parallel to the $x$ axis. Then stack the top half exactly atop the bottom half.

2. Cut along the middle, parallel (again) to the $x$ axis, giving four "strips." Stack them atop one another, perfectly.

3. Cut the stack in half (by the only way possible). Restack the pieces to be in a $4 \times 1$ parallelogram.

4. Cut this stack in half (by the only way possible). Restack the pieces to be in a $2 \times 1$ parallelogram.

5. Cut this stack in half (by the only way possible).

As @Brian M. Scott shows: $5$ cuts is the minimum. So this must be an optimal solution (likely not unique).