I wrote:

Is there any kind of virtual environment in which one can play with puzzles like this?

Here's an example of the kind of puzzle I'd like to play with in a non-physical way: the states of the puzzle are the permutations of the numbers 1 through n=6 (though you can use other values of n if you like), and a legal move swaps two numbers if there is a number that is between them positionally and between them numerically (a "see-saw move"). E.g., in the permutation 1 4 2 5 3 6, you can't swap the 2 and the 3, because the only number that's between them positionally (5) isn't between them numerically, but you can swap the 2 and the 6 because the 5 is between them both positionally and numerically. (Note that the inverse of a see-saw move is again a see-saw move, so this puzzle is reversible.) Steve Linton wrote a computer program that checked that all the permutations of 1 through 6 can be obtained from one another via see-saw moves, but I'd like to come up with a non-brute force proof, and the most natural way to do that is to develop some intuition by playing around. If Conway were still posting to math-fun, he'd probably reply that I'd be better off just making numbered counters and moving them around. His early work on the Game of Life was done by hand, and he's of the opinion that some of the insights he gained from this couldn't have been gained if he'd just had a computer do the simulations. But I don't want to lose track of how I got to where I am. At the same time, I want the ability to backtrack. And I want to be able to mess around at the speed of thought, without having to write stuff down. Jim Propp