Something that always bugged me..

[dannyyyy]

while lyra was in one half of the...err.. Dæmon cutting thingy.. and Pan was in the other half, why did pan not just turn into a beetle or something small and crawl through the wire fence thing into lyras half, because then theres nothing to cut on the other side?

[aurora borealis]

Good question...
Perhaps the compartments are specially designed to prevent the dæmon from crossing to the other side. It could be made from the same alloy as the guillotine-blade.
Another option I had is that perhaps Pan is too shocked to act. Remember, this is a frightening situation, and he had just been touched by strangers, and he is going to be separated from Lyra...
Maybe it's something else though...

[CMR]

Should have turned into a dust mite.

[Aurora_Lumi]

I would have to agree with Aurora on this, after reading this post I went back to the book to see if they mentioned something about the structure of the Cutter but there was none that I could find. So I will have to say that the ones that built it should have thought of the possibility that a Dæmon would have thought of changing into something small enough for it fit through the cage. Now I would not be surprised if Pan was too in shock to even think of this possibility but I certainly hope that one child or another would have been able to think before the guillotine blade cut them. So I really think it had to be the way the Cutter was built and the material that was used to lock them in the way it did.