Inquiries-Week 10: Self-Descriptive

Introduction

In this inquiry, we build a sequence from a single 2.

The first rule of this sequence is that it has to describe itself.

Starting with Two

Here is a 2.

2

It says, "There are two here."

The first number is a 2, so the next number has to be a 2 as well, so that there are "two here." We now have two 2's!

2 2

When it's not a two, it's a one

Here comes the second rule of our sequence. When it's not a two, it must be a one.

What comes next?

To start, our sequence says there are two of one thing and then two of another thing.

2 2

We know the first two are 2's, so the next two must be 1's:

2 2 1 1

We couldn't use twos, because then there would have been four twos, and the sequence wouldn't describe itself.

Continue to Build the Sequence

So far we have:

2

What are the next two numbers in the sequence after the 1 1? We know there is two 2's, then 1's, then one of something, then one something.

And what about the next three numbers?

Activity - Be Creative

Build out more of the sequence and then dive in.

• What do you notice?
• Do you have any conjectures about the sequence?
• Do you think it repeats?
• How many 1's versus 2's are there?
• Can you draw this as a tree? A spiral? A cake with layers?
• Can you construct a sequence like this? What rules would you give it?

After playing, you might want to investigate this sequence as a tree here:

Activity Structure

This is a 30-60 minute activity exploring a self-descriptive sequence.

Exploration (10–15 minutes)

Option 1

Build the sequence as described previously by starting with a 2 and letting it unfold. Work together to find what comes next.

• How can you keep track of what comes next?
• Is there an algorithm we can use?
• Does it repeat?

You can do this with numbers, toys, or symbolic representation. It can be inline, or in other forms that learners come up with.

Option 2

Start with the sequence:

2 2 1 1 2 1 2 2 1 2 2 1 1 2 ... ?

• How is this sequence built?
• What comes next?

Check your conjectures to see if they are right:

2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1,

• Is there an algorithm we can use?
• Does it repeat?
• What happens if you put a 1 on the front of it?

Diagramming and Conjectures (15–20 minutes)

Continue to conjecture, share, and test.

What are some different ways to show this sequence? (You can use a toy here to see an example, but only if it doesn't interfere with discovery.)

• Hexagons?
• Trees?
• Spirals?
• Can you define a vocabulary of hierarchy and/or connectivity? (parent/child, up/down, levels, etc.)

Play with finding different ways and see if anything interesting is revealed.

Does anything repeat? If so – how?

Example Conjectures:

Example: "As the sequence goes on and on, the number of 1's will be about 50%."

Example: "There are patterns of connected numbers that spiral"

Example: "This sequence doesn't ever repeat."

Example: "This sequence repeats eventually."

Example: "This sequence is a fractal."

Optional Tree Toy + Discussion (10–15 minutes)

Play with the tree toy here. Look for interesting patterns.

• Share thoughts.
• Share conjectures.
• What does it mean to be self-descriptive?
• If you start with a 3 and always go in the order of 3,2,1 - can you make a self-descriptive sequence?

Can you play the sequence with a drum beat? Does it sound like it repeats?

Going Deeper (optional)

Whether there are 50% 1's and 2's remains open at the time of this post. OEIS has references to dig in more.

Some other questions:

• Is there a formula for the nth term? Is it a closed-form?
• For any sequence within, does it repeat?
• Is there a way to predict the frequency of any repeating patterns?
• Investigate lengths of each iteration in the sequence (See OEIS 042942) (in Desmos): 1, 2, 4, 6, 9, 14, 22, 33, 49, 74, 112, 169, 254, 381, 573, 862,...

Tools and Supplies

• Paper and pencil or whiteboard.
• Manipulatives or other toys to build a physical representation.
• Tree toy (optional).

Vocabulary

• Term – a single number in the sequence.
• Run/run-length – a block of equal neighbors, and how long it is.
• Run-length encoding (RLE) – describing a sequence by its run-lengths.
• Self-describing – it carries the instructions that build it.
• Prefix – the first n terms.
• Parent/child – number and the numbers it produces.
• Density – the long-run fraction of a symbol.
• Conjecture – A statement believed to be true but not yet proven.
• Fractal – echoing itself across scales.

Extensions, What-Ifs, and Resources

• Integer sequence on OEIS (Kolakoski).
• What if you start with {1 3} with only 1's and 3's – is there different behavior (see OEIS)?
• Related sequences.
• Numberphile video.
• Start with a 1 instead of a 2 – what changes?
• Make music – drum the 1s and 2s. Do you hear a repeat?
• Invent your own self-describing rule. What's the smallest one that works?
• Program this sequence, make it art. Here is mine.

I was introduced to this sequence during a pairing session at the Recurse Center, and then proceeded to go down the rabbit hole for a week. I am grateful to have partners in learning.

Behind the scenes – I tried hexagon spirals, paint pouring, and other visuals: