Mazes and puzzles are wonderful for teaching kids critical thinking, problem solving, strategy development, etc. All critical skills needed for success in math.
With my four year old, we have been playing with a ton of print out mazes. To decrease paper waste and increase re-usability I slide the sheets into page protectors and she uses markers to solve her puzzles. Wipe off and re-use. I love that I can see her brain in action as she goes down one path only to find a dead end and steer back and try another path. Awesome lesson in resilience and trial and error. A quick google search will come up with tons of free mazes.
Here is a pretty comprehensive list of online mazes and puzzles for kids: http://puzzles.about.com/od/familyfun/qt/KidsMath.htm
A game we love is Rush Hour Jr. "After setting up their game board according to pictures on various challenge cards, players steer their ice cream truck through a jumble of blocking cars, trucks and buses". Highly recommend this game-it's leveled, it's cars, it's travel sized, it's perfection.
Remember these little teasers from old school Sesame Street:
We have been playing with the concept of " what doesn't belong" with my two year old. It's another great way to work on critical thinking in addition to categorizing, describing, and divergent thinking. We keep it simple with obvious differences, but as she gets used to the concept, we can up the complexity so that there may be many different correct answers. Like in the video above, the boot is an obvious choice, but you can also say the sneaker since it's the only one with laces.
I'd love to hear from others about brain teasers and other math puzzles their kids enjoy!