Solution 1:

As one of the nicest series of (often) relaxing articles I found the following of Ed. Sandifer about tools, tricks and considerations of L. Euler : http://www.maa.org/news/howeulerdidit.html

Solution 2:

Raymond Smullyan has some awesome puzzle books. You might want to check out "What is the Name of This Book" or "This Book Needs No Title".

Solution 3:

Almost any article from Princeton Companion to Mathematics.

Solution 4:

Look at the Mathematical Intelligencer. Some of what they publish fits this description; other things don't.