A Colored Loops Extension

This problem was sparked by a question asked by a student in the JRMF Community Math Circle featuring Colored Loops, an activity based on Gord Hamilton's Stirring Paint.  While it can be used an an extension to Colored Loops, I'll present it as a stand-alone activity.

Setup

There are a good number of manipulatives that can be used for this activity.  The only things you really need are a way to arrange numbers in a circle and a way to mark them so you can tell which have been visited.  This can be done with blank paper and a pencil in a pinch, but that can be painstaking, so here are some options to make the activity less tedious:

• If you don't mind printing, the simplest way is by printing sheets that have circular arrangements of dots in varying quantities.  This is the version we'll be looking at here.  Here are some sheets you can use.  For tracking, colored pencils or markers are great.
  

• If you want to use the JRMF Colored Loops app, you can enter the options menu, choose a custom puzzle, select the number of arrows you want and number them in make mode, and then color them in play mode.

• Another manipulative option is to have numbered tokens and colored beads.  Students could arrange the numbers in a circle and then use the colored markers to mark the numbers that have been visited.  The numbered tokens could be paper squares with handwritten numbers or number tiles.  Bingo chips or vase fillers make great colored markers.

Once you've got your manipulatives sorted, let's learn how this activity works by playing with a few examples.  

Example: 13

Taking the numbers 1-13, we'll arrange them in a circle, in increasing order clockwise.

Starting at 1, we will move 1 step clockwise to 2:

Now that we're at 2, we move 2 steps clockwise to 4:

Then 4 steps clockwise to 8:

Then 8 steps clockwise to 3:

Then 3 steps clockwise to 6:

We continue in this manner, always moving the number of steps indicated by the number we're currently on.  If we do this, we visit every number except 13, returning to where we started at 1:

Example: 14

For 14, the first few moves have us visiting 1, 2, 4, and 8:

But then the next move takes us back to 2:

This time we've visited a far more limited subset of the dots and did not end where we started!

Example: 15

For 15, the first few moves have us visiting 1, 2, 4, and 8, before returning to 1 and closing the loop:

A lot of dots haven't been used.  If we pick a new color and start from 3, we visit 3, 6, 12, and 9 before returning to 3.

Starting from 5, we visit 5 and 10 before returning to 5.

Starting from 7, we visit 7, 14, 13, and 11 before returning to 7.

If we try starting at 15, we immediately end up at 15 again.

Recap

If a loop is a set of numbers that are visited, then we see that in our example with 14 dots, 1 cannot be part of such a loop.  On the other hand, with 13 and 15 dots, every dot is part of a loop!  For 15, we have the loops [1, 2, 4, 8], [3, 6, 12, 9], [5, 10], [7, 14, 13, 11], and [15], since 15 loops back to itself immediately.

Initial Questions

The main question we want to take up is:

For which numbers of dots can we end with every dot in a loop that starts and ends in the same place?

To that end, here are some useful questions to explore:

• Does it matter which dot you start with when creating your loops?

• Is it ever possible to end with all dots in a single large loop?

• Is there anything special about the numbers of dots for which all dots can be put into loops?

• If all dots can be put into loops, is there anything interesting you can say about the dots that end up in a loop together?

Explorations

Here are some paths you can travel based on some observations that students might make and their mathematical maturity.  Modular arithmetic is always dancing in the background, so at some point students will likely interact with it, but that interaction can be pretty informal and intuition-driven.  There are a lot of things for students to latch onto with a basic understanding of divisibility and symmetry.  Throughout this discussion, \( n \) will be the number of dots in the circle.

• Single-dot loops: Students will pretty quickly see that the \( n \) th dot will always be in its own loop.  They can probably also argue why other dots cannot!

• Odds and evens: After working for a little while, students should have enough examples to see that all dots end up in loops when \( n \) is odd and do not when \( n \) is even.  

  Many might be able to give an argument for why it can't happen when \( n \) is even.  (Hint: When could leaving a dot result in visiting an odd dot and when is it possible to return to an odd-numbered dot?)  

  Seeing why it's always possible when \( n \) is odd hinges on the idea that adding a number to itself is multiplication by 2 and 2 always has a multiplicative inverse modulo \( n \) when \( n \) is odd.  This means there is a unique path into and out of each dot.  One way to interact with these inverses is to look for a multiplier that generates a loop backwards like multiplying by 2 generated them forwards.  (While this number will change with \( n \) , it will be the same number for any loop on \( n \) dots!)
  

• Familiar shapes: After working on some examples, students might see some of the same shapes showing up for different numbers of dots.  For example, here some suggestively colored loops for 3, 5, and 15:

  When else do we see things like this?  (Hint: \(3 \cdot 5 = 15\).)  

  You might also notice that the loop starting at 1 and the loop starting at 7 turned out to be mirror images.  Why?  What other numbers of dots do we see this sort of mirroring for?  (Hint: What is \( -1 \mod 15 \)?)
  

• Loop sums: What is the sum of all of the numbers in a loop?  For example, with 15 dots, the loop sums are

  1+2+4+8	=	15
  3+6+12+9	=	30
  5+10	=	15
  7+14+13+11	=	45

  What's interesting about these sums and why is it happening?  (Hint: How can you keep track of the distance traveled during the loop?  Do you ever go around the loop more than once before returning to your starting point?)

Wrap-Up Questions

What are some things we can predict about loops based on the number of dots before we start drawing?  For instance, can we predict:

• Whether each dot will be in a closed loop?

• How many loops there will be?

• The sum of the numbers in the loop?

The problem of the number of loops on \( n \) dots is actually quite tricky!  For example, students might notice that the only times you find two loops (one for n and the other for the rest of the numbers) happen to be when n is prime.  However, this is only for some of the primes:

\[ 3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, ... \]

Which primes this happens for is an active research question.  In number theorist speak, the question is, "For which primes is 2 a primitive root?"  It is not known whether this list of primes is finite or infinite!

Extensions

If we were allowed to place the numbers around the circle in any order instead of increasing clockwise, what sorts of closed loop configurations can we get?  There is a nice argument for why we still can't get every dot into a closed loop when the number of dots is even based on

\[ 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2} \]

when combined with some observations about the sums of the numbers in each loop.  However, some new configurations might be possible for odd numbers of dots!

 You might also notice that as you increase the number of dots, a persistent structure emerges, regardless of the number of loops.  Here's what we'd see with 61, 63, 65, and 67 dots:

What is this shape and why does it appear?  Since the shape is an envelope (i.e., outlined by tangent lines), you might have a suspicion it requires calculus to investigate.

This post was sponsored by the Julia Robinson Mathematics Festival