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](https://www.english-corpora.org/coca/)  
 2. *Students, if you could please open your math books **to** page two*  – COCA  
 3. *Open your grammar **at** page fourteen*   – [BNC](https://www.english-corpora.org/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](http://www.hansard-corpus.org/)
 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