Q:

"you didn't even start" or "you don't even start", when to use which? 

My sister lost one of my books somewhere in the house a few days ago.  Then I asked her to get it back.  
I asked her just now about the book.  She answered she didn't find it.  I guess she didn't even start searching.  Should I say

you didn't even start searching, right? 

or

you don't even start searching, right? 

I guess the first one is, if so, when should I use the last one? 

A:

You are asking if she has already started something.  If she had, that start would already have taken place, so would be in the past tense. 
Of your two examples, the correct one would be:

You didn't even start searching, right? 

It sounds natural, but I would prefer:

You haven't even started searching, right?