YA/Juvenile book (may be a series, I don't remember) about a boarding school for royals/high status people, plot centers around the quirkiness of the school and the people who attend it

[Anna]

Read around late 2000s/early 2010s.

It was a boarding school for like royals or like magical people/beings.

To be frank, I do not remember much about this book other than a few specific details.
The first is that they use the word ample to describe the headmistress figure.
Another is that they play sports at some point in a tournament of sorts (which I believe involved polo or water polo).
The final and most defining detail in my mind is that they don't use/don't believe in (??) the number 12 (it's like the number 12 is unlucky but the number 13 is lucky) so their clocks go 10, 11, 13.

I will add more if I remember.
(edit 1: added more that I remembered)

TIA

[Anna]

I believe the narrator was first person and I know it was from a young girl's perspective. I think there may have been a nearby school for boys that they half rival with as well.

may be in a diary format