Like Rubik's cube ... only flatter

Welcome to Scrambler! The object of the game is to scramble a nice orderly pattern of numbers or colors to produce a given sequence. You always start with $1,2,3,\ldots,n$ for some small number $n$. On each of your turns you can apply one of the available rules to your current sequence to construct another sequence (which becomes your new current sequence). The list of available rules is different for each level of the game. It may require some study and investigation on your part to come up with an efficient strategy for beating this game at the higher levels of difficulty.

Play Scrambler!

Game Components

As described in the Introduction, this game illustrates the basic features of a formal proof system.

The Math Behind the Fun

This game is closely related to the branch of mathematics known as group theory. In particular, each level illustrates a the action of a different group on the numbers $1,2,\ldots,n$. Note that it is known that not all ordered lists can be obtained by applying the given rules in some order. Fortunately, the game only picks goals which can be achieved. So you can be confident that there is a way to win the game, even if you can't find it.

Like Rubik's cube, the object of this game is to use certain available moves to try to put some colors into a particular arrangement. For Rubik's cube, the available moves consist of twisting a face of the cube by 90°. In Scrambler the colors are arranged in a row, rather than on the face of a cube, and the available moves are those given by the buttons labeled Moves.

Two words of warning. First, the last level can be extremely difficult (we've only solved it in theory, not in practice!). The other levels are all doable by hand if you play with them and study them enough. Finally, this game is very addictive! You have been warned!