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May 21st

May 21st

# Rational Tangles

1 Introduction

This is originally from John Conway but came to me through Tom Davis (in fact, must of this
writeup is “stolen” from Tom’s work: [1]). The idea is to associate a rational number to a tangle
of two ropes by performing a sequence of two simple operations. Similarly, we can untangle these
ropes using these same two simple operations. Hopefully along the way the students will even get
some practice with fractions .

2 Getting started

Materials:
1. Nice heavy ropes about 10 feet long are about right (I use climbing straps purchased at a
store like REI).

2. 2-3 plastic grocery bags.
Have four students (A, B, C, D) and have them hold two ropes as in Figure 1.

Figure 1: Initial State

Every one needs to hold the ropes firmly. Students like the shake the ropes and everything will
have to be redone if a student drops a rope. Do not al low the kids jerk on the ropes. Generally, try
to keep a handle on the sil liness that will result from the ropes. It is a good idea to swap kids out
periodically from time to time as well.

3 The basic operations

There are two basic operations: Twist and Rotate. To twist, student D walks under the rope that
student C is holding. This is the only twisting move that is allowed. There is no “untwist” move
(that would undo the twist). See Figure 2 to see the result of 0, 1, 2, and 3 twists.
To rotate, students all rotate one position clockwise, as in Figure 3.

Figure 2: Twisting

Figure 3: Rotating

We do not actually care about what position the students are in. What we care about is the
position of the ropes. So, in Figure 3 the first and third positions are the same (even though the
ropes have actually changed places ). Similarly, the second and fourth positions are considered the
same.

In describing a sequence of moves, we will write “T” for twist and “R” for rotate.

Finally, there is the display ope ration where the 4 students hold the twisted rope up for all to
see.

We will write a sequence of moves by writing something like TTRT to mean twist, twist, rotate
and then twist, in that order .

4 Activities

One goal is to associate a number to each tangle. Here are a couple “ rules ” to get us started to
de termine how to do this.

• The starting position is given the number 0.
• Each time a twist is done, the number increases by 1 (so the number is an attempt to measure
the number of twists made).

1. What mathematical operation is R ?
Start at 1 (by doing ). Then perform two rotates and end back at 1:

What mathematical operations can do this? Try the following and determine what numbers
belong at the question marks:

Determine what mathematical operation is re presented by a rotate .

2. How do you get back to zero ?
Here our goal is to start with a tangle and get it back to the 0-tangle. So, start with a tangle
and try to untangle it. See if you can find a way to do this. Along the way you should learn
that doing two rotates in a row is not productive .

A good starting point is the tangle TTRTTTRT represented by 3/5 . This is good because the
numerator and denominator are relatively small but still complicated enough. Note:

• Doing a twist to 3/5 only moves the tangle further away from 0, so perhaps a rotate is
better:

• At this point the only reasonable move is a twist since another rotate will undo the
previous rotate:

• Now, a RT moves you further from 0, so it makes sense to twist:

• Now, we just keep doing this

So, what is the procedure and is it guaranteed to work?

3. Infinity
Try this:

What number must belong at the question mark?

4. GCD: Greatest Common Divisor
There is the Euclidean algorithm for computing GCD. Here it is for the GCD of 4004 and
700

4004 =700 × 5 + 504
700 =504 × 1 + 196
504 =196 × 2 + 112
196 =112 × 1 + 84
112 =84 × 1 + 28
84 =28 × 3

which shows GCD(4004, 700) = 3.

Note that the Euclidean algorithm still works if we use negative numbers in our calculations .
Here it is for GCD(5, 17):

5 =17 × 1 − 12
17 =12 × 1 + 5 = 12 × 2 − 7
12 =7 × 1 + 5 = 7 × 2 − 2
7 =2 × 1 + 5 = 2 × 2 + 3 = 2 × 4 − 1
2 =1 × 1 + 1 = 1 × 2 + 0

Watch this:

What is going on? Why are these two operations so similar?

5. What tangle numbers are possible?
Lets start easy. Can you start with 0 and get to −3?

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