Rubber band puzzle.
 

This puzzle is inspired by something that I saw recently.

You have a board with 25 pins in it, arranged in a 5 by 5, evenly spaced grid (illustration below).
These pins are the hypothetical ones of zero diameter and infinitely strong that occur in all such puzzles.
You are also in possession of a hypothetical elastic band (also known as a [URL="http://en.wikipedia.org/wiki/Rubber_band"]rubber band[/URL], etc.).[INDENT]This band when stretched [U]just enough[/U] to take the slack out and hold on to the pins is [B]4[/B] units long. This is illustrated by the red square at the lower right, which is exactly 4.000 units.
It is 3.999 units long unstretched and will not hold onto the pins.
When fully stretched to its maximum, it is [B]10[/B] units long.
At 10.001 units it ruptures.[/INDENT]You may place the band onto the pins in any arrangement that is possible within those limitations. The green triangle illustrates one of these.

The puzzle is: [FONT="Garamond"][SIZE="4"][COLOR="Sienna"][B]How many unique shapes (patterns) can you make?[/B][/COLOR][/SIZE][/FONT]

Different sized squares (1x1 vs 5x5) would be counted as unique, same with triangles and rectangles. Rotationally equal shapes are not unique, position within the 5x5 grid is similarly not unique.
Remember that the pins have zero diameter, therefore routing the band around either side of the pin achieves the same shape.
Each arrangement is considered one shape or pattern. If you make a square and a triangle in a single arrangement that is one shape, not 2.
This is a completely 2D puzzle, no angling of the band on the pins is possible and twisting the band is not an option.

For those that want to document their shapes, use row & column system to describe the shape (following the band). (I would suggest starting at 1-1 if possible, to make things easier to track.) For example the shapes shown below would be documented as:
[COLOR="Red"]Unit square: [FONT="Fixedsys"]4-4, 4-5, 5-5, 5-4[/FONT][/COLOR]
[COLOR="Blue"]1x3 rectangle: [FONT="Fixedsys"]2-1, 2-2, 3-2, 4-2, 5-2, 5-1, 4-1, 3-1[/FONT][/COLOR]
[COLOR="Green"]2x2 right triangle: [FONT="Fixedsys"]1-3, 2-4, 3-5, 3-4, 3-3, 2-3[/FONT][/COLOR]


I suspect that there are at least 50 unique patterns that can be produced and possibly over 100.

I see the blue rectangle as 8.000 not 10.001. What am I missing?

Also why is the Red Square 3.999 and not 4.000?

[QUOTE=petrw1;368181]Also why is the Red Square 3.999 and not 4.000?[/QUOTE]

The red square [I]is[/I] 4.000. The unstretched rubber band is 3.999. i.e. A shape must have a perimeter of at least 4.0 to qualify, else the rubber band won't "hold onto the pins".

Are zero-width "shapes" valid? In general, are shapes where the band double backs on itself, valid?

[QUOTE=axn;368183]Are zero-width "shapes" valid? In general, are shapes where the band double backs on itself, valid?[/QUOTE]Yes. Lines are shapes. Just think about this as if it were in real life.

What is the width of the rubber band?

If I wrap the band twice around the red shape is that different from it going only once around?

[QUOTE=retina;368187]What is the width of the rubber band?[/QUOTE]
It, like most hypothetical rubber bands, has no width.

[QUOTE=retina;368187]If I wrap the band twice around the red shape is that different from it going only once around?[/QUOTE]Since this is only a 2D puzzle, with a band that has no width, [U]no[/U], they both look exactly the same.

Are the following two band routes considered the same?

1-1, 1-3, 2-2, 1-1
1-1, 1-3, 2-2, 1-2, 2-2, 1-1

[QUOTE=retina;368193]Are the following two band routes considered the same?

1-1, 1-3, 2-2, 1-1
1-1, 1-3, 2-2, 1-2, 2-2, 1-1[/QUOTE]
The second one adds a line segment (2-2, 1-2) that is not in the first. So that is a different shape (a right triangle with a 2 unit hypotenuse vs. a right triangle with a 2 unit hypotenuse and a bisector).

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=Uncwilly;368180]I suspect that there are at least 50 unique patterns that can be produced and possibly over 100.[/QUOTE]
Huh??? If "lines" with different length and "corners" are considered different shapes, you have way over 100 shapes which are only "lines", i.e. figures with void interior. For example, there are 14 "straight lines", with respective lengths 1, 2, 3, 4, s2, s5, s10, s17, s8, s13, s20, s18, s25=5, s32, where by sx I denoted sqrt(x) - this if you connect the band between any two random pins, and if it is too short, then you re-wind it again to the same pins.

Then you have the "trinix", "tetrix", "pentix" shapes in "billions" of ways/angles (like "L"-shape, "N"-shape, "V"-shape, "K"-shape, "F"-shape, etc, if any diagonal, then you can not have more than 4 pins, otherwise you can not "come back", not enough band, so the "pentix" are only with orthogonal rubber lines, and you can't have "hexix" with void interiors, but yet, there are much over 100 shapes!)

Edit: I am only talking about shapes with void interior

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.