"Make strong paper" or "Make paper strong?" Are these sentences both correct?

thank you for taking the time to take a look at my question! I am currently a teacher of English and an ex-in-company translator in startups. Now I am having a question about the correctness of a question in a test. Here is the question as follows.

Some people think that paper is too soft to make a box and that it is better to use plastic or metal. Actually, these things are stronger than paper, but we can make paper stong when we put many pieces of paper together.

The last bolded sentence is the issue.

I think we can rewrite this like....

we can make strong paper when we put many pieces of paper together.

However, the answer booklet says (in Japanese),

We cannot use the phrase make strong paper because this usage weakens the implication that paper is originally weak until it is fortified by putting many pieces of it together.

However, I believe that it is such a tunnel-visioned and poor explanation.

Here is my idea. We can safely say,

We can make a strong team when we work together.

instead of

We can make the team strong when we work together.

And above two sentences mean something totally identical.

Please give me your idea or advice regarding the aforementioned problem.

we can make paper strong when we put many pieces of paper together.

and

we can make strong paper when we put many pieces of paper together.

Are both of them grammatically correct?

'make paper strong' - this implies that it was weak, but that is not the same as saying the paper had zero strength. The paper was too weak for the task, which we already know.

However, the statement is rather vague, but if this were part of a real discussion we would know which paper was meant, and would probably be referring to it as 'the paper'. If this were so, the phrasing would be understandable.

An alternative would be to try something like 'make the paper stronger'.

'make strong paper' - this is vague as well, because it is just an open statement (what paper?) in what should be part of a discussion. We would likely be stating something like 'make strong enough paper'.

Other issues include that if we bond several layers of paper together we will produce card, which would lead the text down a different road: 'we can make a sufficiently strong card by bonding several layers of the paper together'.

"Making strong paper" suggests what you are starting with isn't paper. So when you encounter "put many pieces of paper together" we have to go back and reparse. It's not wrong per se, but it is a bit of a garden path situation.

"Making Paper strong" suggests that you are transforming something that is already paper into a stronger version. That is just better suited to what is happening here.

The word make is a bit overworked in English, and this is a way to begin sorting out which sense of make is being used.

And strangely, the verb strengthen doesn't work very well here either, because it tends to suggest an isotropic improvement to all the little pieces rather than creating a non-isotropic composite material optimizing some desirable trait.

'Make strong paper' and 'make paper strong' are both perfectly standard idiomatic English (i.e. what I think you mean by 'grammatical'. They look extremely similar but are parsed very differently. Let's compare minimal but complete sentences.

We make strong paper.

is a simple Subject Verb Object sentence:

NP(we) VP( V(make) NP(ADJ(strong) N(paper) ) ).

The verb 'make' is synonymous with construct, manufacture, create, produce, taking a single direct object which is what is produced, strong paper. What kind of paper do we make? Strong paper. There is nothing fancy going on here, the meaning of 'make' is the canonical one. The sentence is saying that we are fashioning paper that is strong.

We make paper strong.

is a special (but common) grammatical construction of 'make' with two objects:

NP(we) VP(V(make) NP(N(paper)) NP(ADJ(strong) ) ).

Here, 'make' means 'to cause to happen'. We are causing the first object to become or take on the properties of the second object. We 'make' the paper take on the property of strength. It's the same 'make' as in 'Don't tell mom I broke the vase'...'Make me (not tell mom)'. It is similar in syntax pattern to the version of 'call' in 'Call me Ishmael': it has two objects.

This pattern is learned as a special usage in intermediate language studies, where the first one is accessible to beginners with the available vocabulary.

Make paper strong means that it it is almost a fortification to the paper, which would be correct in this case.

Make strong paper would really work in the case of a paper mill that is physically creating paper.