1. The Monty Hall Problem

http://www.youtube.com/watch?v=mhlc7peGlGg

A demonstration of the Monty Hall problem which is a fine example of probabilistic reasoning based on a problem first outlined in the American gameshow Let's Make a Deal. The problem is named after the presenter of the show, Monty Hall. Although the problem is perhaps not classic probabilistic reasoning, it nevertheless illustrates how we can be fooled by probabilities.

2. The Tower of Hanoi Puzzle

http://www.cut-the-knot.org/recurrence/hanoi.shtml
 

The Tower of Hanoi puzzle – one of the most studied puzzles in cognitive psychology. The object of the problem is to move a tower of disks from one peg to another, with certain conditions upon moves. The problem is well studied because whilst most people can solve it, they often do not manage to do so in the fewest moves.

3. Cryptarithmetics

https://www.youtube.com/watch?v=lbT5qMHMtJ0

More information on cryptarithmetics together with help on how to solve these types of problem. There are a variety of problems that can be devised in cryptarithmetic and this website gives a few examples, together with tips on solving such problems.

4. The Phenomenon of Insight 

http://faculty.mercer.edu/spears_a/studentpages/insight/insightwebpage.htm

A useful set of resources giving a description of insight plus further information. There are examples of insight problems and also useful links to further research on insight.