This is a follow up to Is every shape possible with a snake? .
Imagine a 2d snake formed by drawing a horizontal line of length $n$. At integer points along its body, this snake can rotate its body by $90$ degrees either clockwise or counter clockwise. If we define the front of the snake to be on the far left to start with, the rotation will move the back part of the snake and the front part will stay put. By doing repeated rotations it can make a lot of different snake body shapes.
Now let us define a valid shape. A shape is valid if it can be formed from a straight line snake by applying at most one $90$ degree bend at each one of the integer points along its body and no two parts of the resulting shape intersect or touch each other.
We now apply some further rules to say a shape is reachable. A shape is reachable if it is valid and it is possible to reach the orientation without any parts of the snake's body intersecting or touching in between. This includes during the rotations needed to bend a part at right angles.
Here are some examples thanks to Martin Büttner.
We start with the horizontal snake.
Now we rotate from position 4.
We end up after the rotation in this orientation.
Now let us consider this orientation of a different snake.
We can now see an illegal move where there would be an overlap caused during the rotation.
When a rotation happens it will move one half of the snake with it. We do have to worry about whether any of this part which is rotated might overlap a part of the snake during the rotation. For simplicity we can assume the snake has width zero. You can only rotate at a particular point in snake up to 90 degrees clockwise of counter clockwise. For, you can never full fold the snake in two as that would have involved two rotations at the same point in the same direction.
Shapes that can't be reached
A shape that can't be reached is
(Thank you to Harald Hanche-Olsen for this example)
In this example all the adjacent horizontal lines are 1 apart as are the vertical ones. There is therefore no legal move from this position and as the problem is reversible there is therefore no way to get there from the starting position.
We say that the length of a valid shape is simply the length of a snake that could form it. For example, the example shape above which can't be reached has length $59$.
What is the shortest possible unreachable valid shape?
Current upper bound
David K gave a shape of length 31 which is unreachable. Is this the shortest possible?