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A non-Jew does a melacha for himself, a Jew may benefit on shabbos.

A non-Jew asks to borrow a book from a Jew. The Jew fumbles around in a dark room looking for the book, and the non Jew turns on the light.

The non-Jew did the melacha for the Jew, but really he did it for himself so that the Jew can find the book faster.

What happens?

Background info: There are 2 possible prohibitions that can exist separately: Asking and benefitting. At the end of this article are mnemonics for when amira laakum is permitted. http://www.theyeshivaworld.com/news/headlines-breaking-stories/483672/the-laws-of-amirah-lakum-the-shabbos-goy.html

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In your case, it seems that the non-Jew is turning on the light to aid the Jew in his task. Although the task is being done for the non-Jew, the work (searching for the book) is being done by the Jew. Turning on the light is being done to aid the Jew in his work and thus only indirectly benefiting the non-Jew. It is primarily benefiting the Jew. In this case it would be likely prohibited for the Jew to continue to search in the room for the book as the light is prohibited for him to use.

See The Sanctity of Shabbos by Rabbi Simcha Bonim Cohen (published by Artscroll).

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