Мы нашли другой способ.

Нашли is the past tense plural form of the verb найти, which means to find.

Here is the breakdown:

• найти = to find
• нашёл = he found
• нашла = she found
• нашло = it found
• нашли = they found

Because the subject is мы (we), Russian uses the plural past form: нашли.

In Russian, past tense verbs do not change for person the way present tense verbs do. Instead, they change for:

• gender in the singular
• number in the plural

So:

• я нашёл / нашла = I found
• ты нашёл / нашла = you found
• он нашёл = he found
• она нашла = she found
• мы нашли = we found
• они нашли = they found

That is why нашли can mean we found or they found depending on context. In this sentence, мы makes it clear that it means we found.

Найти is the perfective verb, while находить is the imperfective partner.

• найти = to find, to succeed in finding, to discover
• находить = to be finding, to find repeatedly/habitually, or sometimes used in process-focused contexts

In Мы нашли другой способ, the speaker is talking about a completed result: we found another way. That is exactly why Russian uses the perfective verb найти.

Compare:

• Мы нашли другой способ. = We found another way. (result)
• Мы находили другой способ. = We used to find another way / we were finding another way / we found another way on repeated occasions, depending on context

Because другой is an adjective modifying способ, and it must agree with the noun in:

• gender
• number
• case

Here, способ is:

• masculine
• singular
• accusative (direct object)

Since способ is an inanimate masculine noun, its accusative form is the same as its nominative form. So:

• другой способ = another way / another method

Agreement:

• другой = masculine singular form
• способ = masculine singular noun

Because it is the direct object of the verb нашли.

In the sentence:

• Мы = subject
• нашли = verb
• другой способ = the thing that was found

Russian marks direct objects with the accusative case.

However, with inanimate masculine nouns, the accusative looks exactly like the nominative. So способ does not visibly change here.

Compare:

• nominative: способ
• accusative: способ

But with a feminine noun, you would often see a clearer change:

• Мы нашли другую возможность.
• Here другую возможность is clearly accusative.

Yes, very often.

Russian often drops subject pronouns when the meaning is clear from context. So:

• Мы нашли другой способ.
• Нашли другой способ.

Both can mean We found another way, depending on context.

However, мы may be included:

• for clarity
• for emphasis
• to contrast with someone else

For example:

• Мы нашли другой способ, а они нет. = We found another way, but they didn’t.

Yes. Russian word order is fairly flexible, although the neutral order here is:

• Мы нашли другой способ.

Other orders are possible, but they change the emphasis:

• Другой способ мы нашли.
  Emphasis on another way
• Нашли мы другой способ.
  More marked; can sound expressive, contrastive, or stylistically special
• Мы другой способ нашли.
  Emphasis more on the object before the verb

So the original version is the most neutral and standard for a learner.

It can mean either, depending on context.

The noun способ often means:

• way
• method
• means

And другой can mean:

• another
• different

So другой способ could be understood as:

• another way
• a different way
• another method
• a different method

The exact English translation depends on the situation, but the Russian phrase itself is very natural.

The usual stress is:

• Мы́ нашли́ друго́й спо́соб.

More specifically:

• нашли́ → stress on the last syllable
• друго́й → stress on the last syllable
• спо́соб → stress on the first syllable

If you want to sound natural, those stresses matter quite a lot.

Yes, способ is a common word, but it usually means method or means, not a literal road.

• способ = way/method of doing something
• путь = path, route, way, course
• дорога = road

So in this sentence, способ suggests:

• a solution
• a method
• an approach

If you are talking about solving a problem, способ is a very natural choice:

• Мы нашли другой способ решить проблему. = We found another way to solve the problem.

Yes, but they are not exactly the same in tone.

• другой способ = the most neutral and common
• иной способ = another/different way, but often more formal or literary
• альтернативный способ = alternative method; more specific and a bit more formal or technical

So:

• Мы нашли другой способ. = the most everyday, natural phrasing
• Мы нашли альтернативный способ. = We found an alternative method
• Мы нашли иной способ. = We found a different way, with a slightly more formal tone

For most everyday situations, другой способ is the best basic phrase to learn.