# 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.

## Game Components

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

• Toys: The toys in this game consist of a list of the numbers $1,2,3,\ldots,n$ for some small number $n$, listed in a specific order from left to right. (Such a list is called a permutation of the numbers $1,2,...,n$ in mathematics.) You can also view the list as a sequence of colors or shaded numbered boxes by pressing V
• A Goal: The goal is a randomly selected ordering of the numbers. The object of game is to produce this list by scrambling the starting list.
• The Starting Position: The starting position in this game is always the list consisting of the numbers $1,2,3,\ldots,n$ in increasing order from least to greatest. This is the only toy on your list when you begin the game.
• The Rules: The rules you can use to change the order of the numbers in the current list are different in each level of the game, and are listed under the Moves menu. It is up to you to determine what the various moves do for each level by experimentation.
• Inputs: The input for both Rules consists of the current list of numbers. The current list of numbers is always the last number list on the table. You can undo a move and go back to a previously constructed number by pressing U
• Output: The output is the list of numbers produced by applying one of the Rules to the current number. This number is added to the bottom of the list and becomes the current number. If it matches the goal, you win!

## 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!