Start with a longish rectangle of paper (cut a 1-inch strip off the edge of a piece of notebook paper), give one end (A) a half-twist, and tape it to the opposite end (B). Voilà! You have created a Mobius band, or strip. Make sure it's only a half twist, not a full twist.

              ____________________________________________
              |                          ^               |
              |                          |               |
             A|   <- length ->         width             |B
              |                          |               |
              |__________________________v_______________|


                             ****************************
                            *                            *
                           *                         *
              --->>         * ****************** *       *
                              *               *        *
                           *     *        *         *
                            *        *           *
                              *    *     *    *
                               ***         **

Okay, some questions for you to explore and explain. Be sure to explain and describe what you get for each step.

Cut the Mobius strip into two by cutting down the middle lengthwise. What happens? (Why do you think this is?)

What happens if you do more cutting?

Extra: What is the length of the shortest rectangle of width = 1 that can be bent into a Mobius band as described above? We are assuming that the paper can be bent, but not stretched or torn (no fair using crepe paper). Answers that are well explained and thought out, even if they aren't exactly right, will be given credit, but you have to be in the ball park. (Hint: build one and see how small it will go.)

Hoo boy, this was a tough one! I think part of that was my fault, and we'll talk about why in a second. Fully 109 people got this one wrong, and only 7 folks got it right. One person got the bonus.

What makes this a tough problem is not that it's "hard" to do, but that it's hard to know how to describe what you got. I should have been more clear about that, and will be if I ever use it in the future. Being able to describe physical objects is an important skill, and possibly one that you don't have to do enough of as part of your class work. I didn't give you specific things to investigate, so let's look at what they are. Since the things that are interesting about the Mobius strip are that it has one side and one half twist, those are certainly things you might want to talk about when you describe what you get after the first and second cuts.

I would guess that about half of the folks told me (correctly) how many things they got after each cut. Some of them talked about how many sides their objects had, and a lot talked about how many twists. However, a LOT of people said that their objects had one full twist - cut something with a half twist, and you double the length and the number of twists, getting 1 full twist. Well, you don't. You get something with TWO full twists. As I read through the solutions, I kept counting and recounting the twists in the two once-cut Mobius strips on my desk. I even asked other people to count them, to be sure I was right. And gosh darn it, there ARE two full twists. Weird, huh?

A number of people were far enough off base that I wondered if they even tried it, or if they just guessed. It's fun to play around with stuff like this, so always give yourself a chance by building it. And when you're done with it (or even before you're done) your cats will be happy to do some exploring as well!

As for the reasons why you get what you do, those were interesting, though most were based on incorrect counting of twists. One good thing that three folks pointed out is that your new object has two edges, which means it's two-sided. (More folks said that it's two-sided, but didn't mention the edges.) Mike Bockus of John Shero Junior High School points out that when you cut the strip, you create a second edge, and that's why it's two-sided. Jason Chiu of Laramie Junior High School takes that one step further and says that you double the edges each time - one edge becomes two, and two edges become four... though since you can't have a four-edged strip of paper, it instead becomes two two-edged pieces. Mike and Jason's answers are included below.

I have actually highlighted all of the correct solutions this week. Emily Castor of Granada High School points out that the pieces you get after the second cut are identical to those you get after the first cut in terms of length and twists. Kevin Palmer of Granada High School says that cutting an object with two full twists is like cutting an object with no twists - you get two objects.

I like Matt Smawfield and Theo Talbot's solutions because they talk about what they expected to get - two pieces after the first cut. Surprise! Both Math and Theo attend the American Embassy School in New Delhi.

Only one person got the very hard bonus. That was Tiffanie Lam of Pleasant Hill Middle School. Give it a good read, because she explains it pretty well.

A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.

From:   Mike Bockus
        
Grade:  9
School: John Shero Junior High, Wilburton, Oklahoma

A Mobieus Strip (MS) has only one edge also. If you were to put your pencil 
close the the edge and trace along the MS you would return to your starting 
point and there would be a line appearing along all edge(s).
By cutting the MS down the middle you give it a second edge and you get a Strip 
that is twice as long as the original strip.
The new strip has two twists and if you cut it again your strip does not get 
longer but you get two interlocking strips.
I cut it a third time and got four strips, two interlocked with the other two.
I think that once you get two strips, that after that just produces more strips.

Bonus: I tried several lengths and almost could do it with a length of 5 but the 
paper kept creasing, so I think it might be 6 which is also close to 2PI *1. But 
I don't know for sure

Gordon is my brother.  He left this problem on the table last night and I took 
it, he is real busy this week.  I asked Dad what a MB was and after he showed me 
I tried out your problem.

From:   Tiffanie Lam
        
Grade:  8
School: Sequoia Middle School, Pleasant Hill, California

A Mobius Strip has only one side as a result of a half twist.
If you label the strip on both the left and the right side of
the strip with numbers and an arrow that points to the outside
(ie. the numbers on the left will have an arrows pointed to the 
left and the numbers on the right will have arrows pointed to the
right) and then cut the strip along the center line, then you
would get one long strip with two twists and the
it would have two sides with one side contains the numbers on
the right side of the Mobius Strip and the other side contains
numbers on the left (the arrows are used to tell me which are 
the left numbers and which are the right numbers).

If you cut this long strip along the center line again, you would
get two pieces of strips that are linked together each has two twists. 

The shortest strip you can used to make a Mobius Strip would be
SQRT(3) inches long.  In order not to have double folds, the
shortest Mobius Strip would have three equilateral triangles,
each has an altitude of length 1 inch. Remember, because the half
twist, the two sides of the original strip of paper would become
the one and only one side of the Mobius Strip. So the length of
the Mobius Strip is three times the base of any one of the
equilateral triangles.  Equilateral triangles of altitude 1 have
side equal to 2/SQRT(3) (Consider the 30-60-90 right triangle
with lengths  1/SQRT(3), 1, and 2/SQRT(3)). Therefore, the side
of the Mobius strip has total length of 3*2/SQRT(3) = 6/SQRT(3)
= 2SQRT(3) inches  and one side of the original piece of paper
has a lenght of half of that or  SQRT(3) inches. 
strip of paper would be  

From:   Emily Castor
        
Grade:  9
School: Granada High School, Livermore, California

 When I cut the strip lengthwise, it became one larger strip.  It appeared to 
have two full twists instead of one half-twist, and when traced on by a pencil 
as before, it was proven to have two sides.  When I cut it lengthwise yet 
another time, it turned into two separate strips linked together with twists.  
Each of these strips had two full twists, identical to the strip formed after 
one cutting.  When cut again, the four new strips formed two pairs.  The members 
of each pair linked with each other, and the two pairs were linked together as 
well.  What a jumble! If I continued to make cuts the number of strips would 
double each time, faoming more and more pairs interlinked with the others.

 Extra: To make the shortest possible Mobius strip, one can take a normal-sized 
Mobius strip, untaped, and slide the ends over and away from each other.  Do 
this until the fold of the strip's kink forms the circumference of the top of a 
cone, and the ends cannot be slid any further.  The length of this shortest 
strip is one unit, making the width and length of the strip equal.

From:   Kevin Palmer
        
Grade:  9
School: Granada High School, Livermore, California

A single ring was formed when I cut the Mobius strip into two by
cutting it down the middle lengthwise.  I think this happened
because the Mobius strip has only one surface and one side.
Therefore, cutting the Mobius strip would not create two separate
pieces.  When I cut the new single ring down the middle, two more
rings were created that were linked together.  Two separate rings
were created because the ring with the whole twist is like a ring
with no twists, with two edges and two sides.

After the first cut when a single ring was formed, the single ring had four
half twists or two whole twists. When that ring was cut then
two rings linked together were formed, and each ring had four half twists.

		-Kevin

From:   Matt Smawfield
        
Grade:  
School: American Embassy School, New Delhi, India

Subject: The Mobius Strip

When the band is started, there is only one half twist in it. When I cut
the band down the middle, I expected that two smaller bands would come
from it, but I was wrong. When I cut the band, instead of creating two
new individual bands, it turned into one long thin band. When I untwisted
it, I found that there were 2 complete twists.  When I cut this band
again, it split into two individual bands, but they were not quite
individual because they were linked like a chain. I cut them up and
untwisted them to find that both of these bands had 2 complete twists.
This happened because when there was only one half twist, there was only
one side, and it kept going forever. So when the half twist band is cut,
it cannot form two separate bands because there is only one side. When
the new band is formed, it has 2 complete twists; so therefore, it has
two different sides. This is the reason that when it was cut again, it
formed two separate bands rather than forming a longer and thinner band
with yet more twists. 

Extra:  I made myself a new band with a width of 1 inch and I pushed the
twist area (the part where the twist is) to the side so I had it out of
the way. With this small stretch of untwisted band, I folded a 90-degree
angle, and then I folded this part back over so that I had a triangle.
When I looked at the rest of the band, I noticed that by doing this, the
twist had gone, or at least it had moved to the fold in the triangle. I
then folded the triangle over so that it would make a crease, showing
both ends of the rectangle I was forming. I opened out the fold now, so
that the twist was back out again, and there was a rectangle/square with
a diagonal cross from vertex to vertex. I had my rectangle. When I
measured out the sides, my width was 1 inch, and the length of the
rectangle was approximately 1.1 inches. If I had accurate instruments, I
believe that it would be a perfect square, but due to human error
(inaccuracy of folds, ruler, etc), it came out just over an inch.
Therefore, I believe that if I had 100% accuracy, the correct measurement
should come out to be one inch, making a perfect square.

From:   Jason Chiu
        
Grade:  9
School: Laramie Junior High School, Laramie, Wyoming

Subject: Geo. POW (The Mobius Band, Feb. 2-6, 1998)

    When a person cuts a Mobius strip down the middle lengthwise the Mobius 
strip becomes a two-sided strip.  That two-sided strip has two complete twists 
to it (720 degrees).  When a person cuts this two-sided strip again (lengthwise) 
he or she would get two strips.  Each of the two strips are like the Mobius 
strip with one cut down the middle and cross over each other four times.  These 
individual strips are connected and cannot be separated without breaking one of 
the strips.  I think that this is because when a person cuts a Mobius strip the 
strip doubles the number of sides that it possesses.  The first time that made 
the one-sided strip a two-sided strip then cutting it again made the total of 
sides four.  Since a two-sided piece of paper cannot become four-sided, the 
strip becomes two two-sided strips when cut for a second time.
    The shortest that I can make with a piece of paper of width 1 mathematic-
ally is from the middle bottom to the right side a quarter of the way up to the 
left side three quarters of the way up and finally to the middle of the top of 
the figure combined with its reflection over the middle.  That is twice 
(reflection) the hypotenuse of a right triangle with legs 1 and 2 if this is 
laid out.  By Pythagora's Theorem that is twice the square root of five or about 
4.46.  Because two sides combine to make the circumference of the strip the 
length is the square root of five.  When I tried to make one of these, I used a 
longer length to start and rolled the excess into the other and removing the 
excess.  This brought the length to 2.35 times the width.

From:   Theo Talbot
        
Grade:  
School: American Embassy School, New Delhi, India

Subject: Mobius Strip

Dear Anne,
	The first thing you notice about the Moibus Band is its most
important quality- if you run your fingers around it twice, starting at
point x, you will always end up at point x. This means that the Moibus
band has only one side, and this helps us answer the first question.  When
I cut the Moibus Band, I was expecting to come up with two separate bands
which is logical, but instead found a single, longer band.  This band had
a total of two twists in it.  This was not a Moibus band because it did
not contain the property of having only one side- you could run your
finger across it and not return to where you began.  If you cut this
double-looped figure down further, you come up with a two double looped
figures which are entertained.  So from a half loop, to a double loop, to
two intertwined double loops.  The reason why the Moibus Band would not
cut into two separate piece of paper is  because it has only one side- you
cant separate a single side from itself, and therefore the Moibus band
remained as a whole.  The band formed from the Moibus Band had two sides
and thus separated into a figure consisting of two double looped figures
intertwined.

To do the Extra, I had to go back to 2-D.  I knew that the
minimum length would be just the half loop, so I pressed down on the loop,
until I had a flat strip of paper.  This done, I unfolded it and found the
loop had left creases in the shape of a rectangle with diagonals across
it.  My belief was that if I could find a relationship that would give me
the length of the rectangle, then I would have found the minimum Moibus
Band length.  I called the width one unit, and drew it on a piece of
paper. Using my one unit, I tried to find the length-  the length was
approximately 1.2 units.  Now given human error and the inaccuracy of my
Moibus Band, the length should be, with accurate measurement and a perfect
Moibus Band, 1 unit.  This means the half twist is a perfect square- the
minimum Moibus Band length is its width!

The following students submitted correct solutions this week: