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2 Oops, 3x3x3 not 3x3

I'm surprised this example hasn't been mentioned already:

The 3x3 3x3x3 Rubik's cube forms a group. The 15-puzzle forms a groupoid.

The reason is that any move that can be applied to a Rubik's cube can be applied at any time, regardless of the current state of the cube.

This is not true of the 15-puzzle. The legal moves available to you depend on where the hole is. So you can only compose move B after move A if A leaves the puzzle in a state where move B can be applied. This is what characterises a groupoid.

There's more to be found here.

1

I'm surprised this example hasn't been mentioned already:

The 3x3 Rubik's cube forms a group. The 15-puzzle forms a groupoid.

The reason is that any move that can be applied to a Rubik's cube can be applied at any time, regardless of the current state of the cube.

This is not true of the 15-puzzle. The legal moves available to you depend on where the hole is. So you can only compose move B after move A if A leaves the puzzle in a state where move B can be applied. This is what characterises a groupoid.

There's more to be found here.