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In fact, if you exchange any column of a given snake with any other column of the same

snake, you have an intermixed rotation of the snake, and it is really the same snake as

before even though the list of points is very different.

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Columns

Horizontally inter-mixed Columns

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Old Snake

New Snake

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This is the big intuitive leap. As long as you start out with a snake, you can interchange any of the columns and you still have a snake. The same snake.