Does this have a colloquial meaning in german as it could in english?
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Why not, "Sie suchen für Holz."
There is no need for "für" because "suchen" = to look for, to search for
This is why I would have translated it to "seek" which doesn't need "for" after it.
Is there a difference between "Sie suchen Holz" and "Sie suchen nach Holz"? I think they are using both in the excercises and I haven't found out what the difference between the two of them is (if there is a difference, that is).
In general, there is no difference. But when using 'nach', the objects case changes. Without: accusative, with: dative.
As a native English speaker, I find this worded oddly. Would 'They are looking for wood' or 'They are trying to find wood' be accepted?
"They are looking for wood" ought to be; try it, and report it if it isn't.
"They seek wood." Accepted too
How does it different if you say nach after the verb
looking for wood is not accepted