My methodology started out poorly...
1st up was to boot up excel, my tried and true numerical analysis program. Then I simply laid out a matrix of all 23 used letters, each row being an equation for one of the names. Then I attempted gaussian elimination. This is when I saw that Y can equate to anything I choose, without effecting the rest of equations. At this point, my answers were way off base, so I tried using the solver tool in excel on the first 23 names (23 becuase I only had 23 potential letters) (got the method from googling "Excel Simultaneous Equations", initial guess was 1 for all letters)
Bammo! I got my answers in seconds
So I tried the linear algebra, but failed, then used my easy tool to solve.
Given the problem definition, I'm still not convinced that there is a definate solution, or that the missing letters are of any consequence. We'll need to see if anyone can find a pattern in this.
Not sure if I can, but I can post the excel file if anyone wants, or e-mail, as that will work.