Is there a rule or pointer explaining where to put '_even'_ in a sentence?

You don't even have a chance.
You don't have even a chance.
You even don't have a chance.
You had no chance. (where?)

Solution 1:

Even is a scalar focus particle. It can have under its scope a noun phrase, verb, verb phrase, or some modifier of a noun phrase. The element under the scope of "even"-focus requires a certain intonation for the meaning to be properly understood.

Take a simple example sentence like:

Even you don't care.

We break the sentence down into subject/predicate form, so that P(...) stands for "don't care", and x stands for "you". P(x) is equivalent to "You don't care".

The meaning of "even" in the example is that, if you were to consider a variety of entities, a, b, c, ... of which it might potentially be said that P(a), P(b), P(c), ... (i.e., "Tom doesn't care", "Dick doesn't care", "Harry doesn't care"), and then we were to rank all such expressions in order of their likelihood/noteworthiness, or some salient property of interest, then P(x) would rank near the bottom of the scale.

In other words, "even you don't care" has two separate meanings:

1. You don't care.
2. Other people that we know about (e.g., Tom, Dick and Harry) might also not care, but that is not nearly as noteworthy as you not caring.

The explanation can be generalized for categories other than the subject.

Some examples of sentences with even and paraphrases (bolding indicates some suitable intonation contour that gives the intended meaning):

• I even saw a canary/I saw even a canary. (I saw a canary AND there may have been other birds that I saw, but seeing a canary was more noteworthy than these)
• I even saw a canary (I saw a canary AND there may have been other things that I did, but seeing a canary was more noteworthy than these)
• I even saw a canary (I saw a canary AND though I may have perceived a canary in different ways (e.g., smelling, hearing), to see with my eyes was more noteworthy)

As for placement of even in a simple, positive declarative sentence, some general guidelines are:

1. When even is used to put the subject in focus, it comes at the start of the sentence.
2. When even is used to put some non-subject constituent in focus, it may come right after the subject, but intonation (and also context) is needed to disambiguate what exactly is under focus.
3. In writing and colloquial speech, even can go right before the non-subject constituent being focused to avoid ambiguity (e.g., you can say "I saw even a canary" rather than "I even saw a canary")

In negative clauses, it matters whether even is placed to the left ("outside the scope of negation") or the right ("inside the scope of negation") of the negative word. In the first case, the negation is automatically interpreted as being part of the focus. In the second case the negation is not part of the focus. But the negation inverts the scale, as it were.

I didn't even see a canary.

means that "I didn't see a canary" and that there are few things less noteworthy to see than canaries (i.e., I saw little if anything of interest).

Solution 2:

The word "even" here suggests that what comes after is one of a group, and more specifically, the last or least of that group. So if I "don't even have" a chance, it means that I don't have a chance, and I don't have a certainty, and I don't have anything in between. How terrible.

On the other hand, if I "even don't have a chance"... it means that I don't have a chance, and furthermore, that I have (or am, or do) several other things as well. This is an unusual turn of phrase, so it's hard to come up with a good example, but to me the feel of it is optimistic, or at least ironic.

"You don't know what's coming, don't care, and even don't have a chance of stopping it! Isn't blind fatalism great?"