If you want to know more about how to solve the puzzles then read Google Doodle - A Turing Machine Puzzle - Update: Play it Now.
The code itself is available under an Apache 2 license, which means you can use it for commercial purposes, but the graphics are under a Creative Commons license, which means you can't. The reason for the difference is that Google wants to protect its logo.
To quote the Google Open Source blog :
Our doodle for Turing's 100th birthday showed a live action Turing Machine with twelve interactive programming puzzles. Turing Machines are theoretical objects in formal logic, not physical things, so we walked a fine line between technical accuracy and accessibility. We focused on finding a good representation for programs and choosing puzzles of appropriate complexity. We did considerable user testing and iteration, more than for any past doodle.
A quick look at the source shows it to be nicely structured and easy to understand. If you want to start to work on improving the code Google, has even been nice enough to post some starter issues that you might like to improve on.
It is fairly clear that there are a lot more puzzles that could be set and one improvement that is obvious (and its on the initial list) is an editor to allow puzzles to be created.
The easy-to-use but quite tricky puzzles, made the Doodle a great hit and it clearly has both recreational and educational uses.