Today’s entry is short, but sweet. I wanted to write something longer, but I’m very busy at work, so this is what you get. I think it’s worth posting despite its brevity.

When we look at groups, one of the problems that we can notice is that there are things

that seem to be symmetric, but which don’t work as groups. What that means is that despite the

claim that group theory defines symmetry, that’s not really entirely true. My favorite example of this is the fifteen puzzle.

The fifteen puzzle is a four-by-four grid filled with 15 tiles, numbered from 1 to 15, and one empty space. You can make a move in the puzzle by sliding a tile adjacent to the empty

space into the empty. In the puzzle, you scramble up the tiles, and then try to move them back so that they’re in numerical order. The puzzle, in its initial configuration, is shown to the right.

If you look at the 15 puzzle in terms of *configurations* – that is, assignments of the pieces to different positions in the grid – so that each member of the group describes a single tile-move in a configuration, you can see some very clear symmetries. For example, the moves that are possible when the empty is in any corner are equivalent to the moves that are possible when the empty is in any other corner. The possible moves when the space is in any given position are the same except for the labeling of the tiles around them. There’s definitely a kind of symmetry there. There are also loops – sequences of moves which end in exactly the same state as the one in which they began. Those are clearly symmetries.

But it’s not a group. In a group, the group operation most be total – given any pair of values x and y in the group, it must be possible to combine x and y via x+y. But with the 15 puzzle, there moves that can’t be combined with other moves. If x = “move the ‘3’ tile from square 2 to square 6”, and y = “move the ‘7’ tile from square 10 to square 11”, then there’s no meaningful value for “x+y”; the two moves can’t be combined.

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