OK, here is the scenario.
I have a Linux Mint machine (maya) that has not been booted for some time.
I started it up and decided to do some housekeeping as a prelude to upgrading it to 'Nadia'
The first thing I did was do an apt-get update which failed with "hash sum mismatch" when updating the Mint list (Sorry, I didn't keep a copy of the full error)
Another machine on my LAN is already on Nadia and it apt-get updates fine.
I use apt-cacher-ng, so I disabled this and tried update again. This worked, which led me to believe there was some corrupt file in the cache somewhere. I spent hours trying to nail this and even did apt-get purge apt-cacher-ng followed by a re-install.
None of this worked.
Eventually something twigged in my brain, and I wondered if there was a compatibility problem with the version of apt on this machine. As I said, it's been some time since I updated.
Here is what I did;
1) Edited sources.list and commented out the single Mint line, leaving just the ubuntu repositories in place. These were updated from 'precise' to 'quantal'.
2) Did an apt-get update, this worked without errors.
3) Upgrade apt (apt-get install apt) this installed about three new files.
4) Edited sources.list again, removing comment from the Mint line and changed it from 'maya' to 'nadia' while I was there.
5) Another apt-get update and this time there were no errors.
From there everything worked as expected and no more hash sum mismatches!