And if we cut the band then we can tie each end to a pin with an infinitely small knot and have different starting and finishing points. That would add quite a number of routes.

[QUOTE=retina;368201]And if we cut the band then we can tie each end to a pin with an infinitely small knot and have different starting and finishing points. That would add quite a number of routes.[/QUOTE]
Haha, correct. Are we allowed to do that? :razz:

[QUOTE=retina;368197]Are the following two band routes considered the same?

1-1, 1-3, 2-2, 1-2, 2-2, 1-1
1-1, 1-2, 2-2, 1-2, 1-3, 2-2, 1-1[/QUOTE]Yes, the produce the same shape of the same size.

[QUOTE=retina;368201]And if we cut the band then we can tie each end to a pin with an infinitely small knot and have different starting and finishing points. That would add quite a number of routes.[/QUOTE]
[QUOTE=LaurV;368202]Haha, correct. Are we allowed to do that? :razz:[/QUOTE]No, that is not permitted.

[QUOTE=Uncwilly;368180][COLOR="Red"]Unit square: [FONT="Fixedsys"]4-4, 4-5, 5-5, 5-4[/FONT][/COLOR][/QUOTE]
Here are some more that a unit square of smaller.
Unit line segment: 1-1, 1-2, 1-1 (requires a twist at 1-2)
Unit L: 1-1, 1-2, 2-2
Unit C: 1-2, 1-1, 2-1, 2-2
Unit hourglass: 1-1, 1-2, 2-1, 2-2, 1-1
Unit right triangle: 1-1, 1-2, 2-1 (twist and repeat)

I think you completely missed my former post (well, the risk of the end of the page...)

One point that should be clarified is if two shapes are mirror images of each other, do they count as being the same shape?

As LaurV indicated, it appears there are so many possible shapes it is probably not practical to work this out by hand. I've started with looking at a smaller problem (in which I still used computer to solve). In my smaller problem, it is not allowed to have any pin touch the rubber band at more than one point along its length. This means that most void interior shapes aren't valid. The only such shapes (void interior) are ones where the rubber band forms a straight line between two pins at least 2 units apart and no more than 5 units apart, and where there are no other pins that lie directly in between.

If my computer-based results are correct, I claim that there are 3525 unique shapes satisfying the problem with my additional restriction, and counting mirrored shapes as the same shape. I have broken these down according to how many segments are present in each of the shapes. I define a segment as a region of the rubber band between two successive pin contact points. It doesn't matter if the rubber band bends at the contact point or not.

[code]
Solutions not allowing any pin to touch the rubber band in more than one place

segments shapes
-------- ------
2 5
3 11
4 130
5 462
6 1203
7 1463
8 226
9 19
10 6
-- ----
total 3525
[/code]

[QUOTE=cuBerBruce;368457]One point that should be clarified is if two shapes are mirror images of each other, do they count as being the same shape?[/QUOTE]A 2 unit right triangle is a 2 unit right triangle, no matter the orientation.
So, yes they are the same.