I actually asked this on another forum but it's proving to be quite difficult to find out so I thought I would ask here too.

Recently I found a kanji puzzle where you have to add together all of the different parts to come up with the correct compound for the answer.

For example:
Q: 口 斤 口 口 貝 斤
A: 品質

I wanted to know if there was a specific name for these types of puzzles or not, because I quite enjoyed it.

Any help would be appreciated

I second this! This sounds very interesting!

I'm not sure if this is specific enough but 漢字パズル maybe?

I remember seeing this type of puzzle on one of the DS kanji games. Just checked なぞって覚える（完璧版）and it's not in that...maybe it was in the first one...

Check this one out. Kanji's drop like tetris and then assemble together into more complicated ones.

http://www.grkt.com/flash/dbk.html