fantasy fiction - female main character, sees everything in numbers.

Read this over 8 years ago. It is a fantasy fiction book (medieval setting) about a girl that was found beaten up, possibly dying, nursed back to health and sees everything in numbers. She is the main character, she ends up as part of a group of people who study geography (or some sort of intellectual group) they travel, she can use her 'power' to see the future or predict events or something along those lines. She does not speak much, is considered traumatised by whatever happened when she was found. I recall a chapter where the group visits an emperor (i think) and he is known for being cruel (beheadings?) and he has a 'hall of mirrors/glass'. or something with mirrors/glass that reflects colourful light in a way to make the hall memorable. I recall the main character's female teacher/guardian/leader of the group getting killed, possibly in a carriage. Long time ago - any suggestions would be appreciated. Cheers.