You don't have a missing file extension.
What you have is that you are trying to open a file that Windows does not know anything about. A file with the extension .PUZ is not a standard Windows file. To use these files before you rebuilt your system, you would have needed to download the specific software that understands PUZ files. Maybe you did this so long ago that you do not remember doing it.
Now that you have re-installed Windows, you will need to re-install the software or program that knows how to open and use PUZ files.
Did you try to Google for 'what are PUZ files' or 'how to open PUZ files'?
I just did a quick look on Google and found several sites describing and a few showing software downloads to open these PUZ puzzle files.
This one is from the creator of the crossword puzzle software and it's free:
There are even some browser plugins tha make it so your web browser understands how to open these files.
The PUZ format is even used by the New York Times online paper.