Subject: Paradox in modified Young's double slit experiment?

My first post got pretty misunderstood, so i shall throughly explain my 
question.

First, to start off, here's sample question :"The diagram shows a Young's 
Double Slit experimental set-up. A thin piece of mica with refractive index 
0.xx is place over one of the slits. Find the new central fringe."

This problem is solve by using the fact that the path difference of both rays 
must be the same in the central fringe.

However, here's the paradox: If placing a piece of mica in the slits does not 
diminish the interferable nature of light, can't we place 2 photo counters on 
both slits yet still get an interference pattern? This way, we can know which 
photon pass through which slit, yet still observe an interference pattern. But 
the problem is that isn't this in direct disagreement with the wave-particle 
duality? 

So, is the question flawed? Rather than getting an interference pattern with a 
shifted central fringe, do we actually only get two bright dots on the screen
(since the piece of mica disturbs the photon like an observer, causing its 
particle-nature to appear.)? Thanks!

Re: Paradox in modified Young's double slit experiment?