"Class, open your books TO/AT page 13!"—Is it a matter of dialectal difference?

Question

My original notion was,

• A) If there's a movement and a destination (as in the case of thumbing a book to reach a certain page), it should be to:
  Class, open your books to page 13!

• B) If there's none, then at:
  There on the counter rested my cookbook, open at page 13.
  The tome fell open right at the middle page.

Then I asked around and rummaged in corpora and found out that my reasoning was flawed. [IMO, there's either something wrong with logic or with language—or both. Somebody fix the whole mess please.] In fact, when it comes to opening or being open, the preposition is almost always to in AmE, whereas BrE is probably mixed, and (sometimes?) there is even a strong preference for at (especially in cases like B above).

I'll close with some of the search results that I mentioned, and leave the rest to others to hash out. Speakers of other dialects are more than welcome to chip in as well!

1. shut the book for a moment, then open it back up to page one and begin again – COCA
2. Students, if you could please open your math books to page two – COCA
3. Open your grammar at page fourteen – BNC
4. every time I took up the book it opened at page 92, although I have never deliberately read that page – BNC
5. I happened to open the Rome Treaty at page 89 of the English text – Hansard Corpus
6. his first act on sitting down to breakfast was to open the tabloid at page three, fold it and prop it against the sugar bowl – BNC

Answer 1

In brief, it may make sense to think of the choice of preposition as both based in dialect and based on semantic requirements.

First, I agree with your general findings that the choice of preposition in this context is based in dialect. Other informal sources attest it, like several people in a Word Reference forum post. UK dictionaries also tend to model the open ... at usage: Oxford Dictionaries (at), Oxford Learner's Dictionary (at). This pattern seems to be common enough that many American and UK speakers don't recognize it as a dialect difference until they encounter the other version. It's that subtle.

That said, there are also non-dialect explanations for the choice of preposition. Famed British linguist David Crystal wrote a blog post in 2011 explaining the difference semantically, though his choices (perhaps writing from a UK perspective) are "at" and "on":

A correspondent writes to ask if he can say both ‘Open your book on page...‘ and ‘Open your book at page...’ Is there a difference?

...

‘Opening a book’ is an interesting example of overlap between the two perspectives. In one way it’s a reference to location - so, ‘at’. Most people would open a book ‘at’ a particular page. But people have a semantic reason for asking someone to open a book at a particular point - so ‘on’ isn’t ruled out. In the first case, they’re thinking ‘where’; in the second, they’re thinking ‘what’.

If Crystal were writing from an American perspective, "to" would have a sense of location or position with "open a book," but "at" or "on" would be used in other cases. Perhaps that sense of to would be as you describe in (A), as the preposition in the phrasal verb "go to," or as something like definition 4a for the preposition "to" in the Oxford English Dictionary:

Expressing simple position: At, in (a place, also figurative a condition, etc.).

So in American dialects, there may be a further semantic breakdown between "at" as denoting a static location, "to" denoting a purposeful action toward a location, and "on" as a more literal descriptor of place.

In other words, using Crystal's language, at denotes "where (static)," to denotes "where (purpose or goal)," and on denotes "what." Because UK English already uses "open at," within that system at denotes both kinds of location ("where") without a further distinction except in a phrasal verb where to is already baked in, like "go to."

Answer 2

In this context, "Open your books at page 13." if proper, is the minority alternative. "Open your books to page 13." is certainly more common across the major dialects of English. The action of opening a book describes a movement of sorts towards a specific page, so to would be the better preposition. At, on the other hand, does not typically describe movement, rather it tends to suggest a fixed position.

The chaos becomes clearer when both usages are applied to other nouns:

1. Open your mind at new events. (Only be receptive during new events)

2. Open your mind to new events. (Be receptive to the new events)

3. Open the door at the house. (Ambiguous)

4. Open the door to the house. (The front door)