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I think the most common is "X is left as an exercise for the reader", but it looks like both are in use.

Is the "to" variant correct? If not, why?

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@WillHunting Well, it's just a question that came to my mind today. But yes, it's a frequent sentence in math parlance, although I've seen it used—maybe to make fun of mathematicians—in other contexts. I'm Italian and there's two ways to translate this sentence as well, one of which corresponds to the "for" form, and the other to the "to" one. They're both correct, even though I prefer the one that would translate to the "for" form. – s.m Nov 15 '12 at 22:54
Are you leaving the exercise as a bequest? If so, use to; otherwise, favor for. – Robusto Nov 15 '12 at 23:20
AND X is left to the reader as an exercise (to solve) AND X is left as an exercise for the reader (to solve)* AND X is left to the reader for solution &c. Worms ain't got nothin on prepositions for being better left in unopened cans. – StoneyB Nov 15 '12 at 23:21
On the arXiv, google says for the reader gives 1680 hits and to the reader (there's not enough space in this comment to give the link, but it's trivial to modify the other link) gives 1350 hits. Use whichever you prefer. – Peter Shor Nov 15 '12 at 23:27
@StoneyB: I believe that Google stops showing hits on most queries at some random spot between 500 and 1000. I suspect that for both of these phrases, there are a lot more hits out there that Google isn't showing you. – Peter Shor Nov 16 '12 at 2:06
up vote 3 down vote accepted

I would say the alternatives should be:

  • Left to the reader as an exercise
  • Left as an exercise for the reader

It seems to me that "to the reader" modifies "left" whereas "for the reader" could modify reader or it could modify exercise. When left at the end as in the question, then I'd have to suggest that "to the reader" is incorrect.

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That is basically a question about diction/meaning, not grammar: perhaps the main reason that the "basic" to has come within the last few decades to be a catchall preposition that can connect practically any two words or phrases, whether the result is awkward, sensible or not. It will not be of much use to try parsing examples. Simply look up and understand their meanings, and then use the word, to or for, that expresses the meaning you want (not wish) to convey.

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