# What is the least amount of straight cuts that should be made to get squares from a rectangle?

The problem is as follows:

Rachel has a rectangular piece of cloth which measures $$2$$ meters large and $$0.2$$ meters wide. She is to use a guillotin which can only makes cuts of $$60$$ cm of maximum in length. Another constrain in this guillotin is that it can only cut one layer of this cloth. Using this information, how many straight cuts minimum can Rachel make in order to get from such fabric $$10$$ square pieces of $$20\,cm$$ from each side?.

The alternatives given in my book are as follows:

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

I'm not sure how to solve this question regarding cuts. But I believe that a way to reduce the number of cuts will require to fold the piece. But in this case this seems not to be possible because there is a condition in the problem which is forbidding to do such approach.

My initial approach was to make the necessary cuts to make some sort of grid which could I guess reduce the number of cuts but 10 is not a perfect square. Therefore what sort of logic should be here?.

I've went in circles for long on this question so I will really appreciate someone could give me a hand with this.

Therefore what should be the way to go here?. Can someone help me here please?. I'm confused.

• I think the question allows folding, but a guillotin cut cannot be through multiple layers. – peterwhy Mar 3 at 2:28
• I brute forced a 4 cut solution. If this is the answer, there must be a proof as to why this is the minimum required. – Andrew Chin Mar 3 at 2:33

## 1 Answer

To cut the strip of cloth, it only takes 9 cuts of 20 cm, and each cut of the guillotin cuts through 60 cm at most, so at least 3 cuts are necessary.

Number the cuts $$1$$ to $$9$$ sequentially, and the resultant squares $$0$$ to $$9$$.

Align cut $$1$$ under the guillotin, then fold two right angled triangles on the strip around cuts $$2$$ and $$3$$ so that the strip goes back under the guillotin, and align cut $$4$$ under the guillotin next to cut $$1$$.

(Fold twice like (B) in this random image found in Google: )

Do the same thing to align cut $$7$$ under the guillotin. With cut $$1, 4, 7$$ all aligned side by side, one guillotin cut of 60 cm would do all the 3 20-cm cuts.

Now squares 1-3, squares 4-6, squares 7-9 are on separated pieces of cloth, and they can be aligned freely to make the remaining 6 cloth-cuts in 2 guillotin cuts.

The optimal answer is 3 guillotin cuts.
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