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We will be looking for maximal length snakes which I call

Great Snakes.

The snake shown here is valid, but is not a Great Snake because

it is not the longest snake possible in 3-space.

This is a valid 3-space snake of length 4.

It is not a maximal length (Great) snake.

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What is the longest snake possible in a 3-space cube? A 4-space cube? A 5-space cube? OK. What’s the longest snake possible in any N-space cube?