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  "description": "Introduction\n\nIn this inquiry, we build a sequence from a single 2.\n\nThe first rule of this sequence is that it has to describe itself.\n\n\nStarting with Two\n\nHere is a 2.\n\n2\n\nIt says, \"There are two here.\"\n\nThe 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!\n\n2 2\n\n\nWhen it's not a two, it's a one\n\nHere comes the second rule of our sequence. When it's not a two, it must be a one.\n\nWhat comes next?\n\nTo start, our sequence says there ",
  "path": "/inquiries-week-10-self-descriptive/",
  "publishedAt": "2026-06-21T03:17:56.000Z",
  "site": "https://www.fractalkitty.com",
  "tags": [
    "here",
    "InquiriesInteractive math exploration toolsInquiries",
    "toy here",
    "tree toy here",
    "OEIS",
    "OEIS 042942",
    "Desmos",
    "Tree toy",
    "Integer sequence on OEIS",
    "Numberphile video",
    "Here is mine",
    "Recurse Center"
  ],
  "textContent": "## Introduction\n\nIn this inquiry, we build a sequence from a single **2**.\n\nThe first rule of this sequence is that **it has to describe itself.**\n\n### Starting with Two\n\nHere is a **2**.\n\n**2**\n\nIt says, \"There are two here.\"\n\nThe 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!\n\n**2 2**\n\n### When it's not a two, it's a one\n\nHere comes the second rule of our sequence. **When it's not a two, it must be a one.**\n\nWhat comes next?\n\nTo start, our sequence says there are two of one thing and then two of another thing.\n\n**2 2**\n\nWe know the first two are **2's** , so the next two must be **1** 's:\n\n**2 2 1 1**\n\nWe couldn't use twos, because then there would have been four twos, and the sequence wouldn't describe itself.\n\n### Continue to Build the Sequence\n\nSo far we have:\n\n\n**2**\n\n\n**2 2**\n\n\n**2 2 1 1**\n\nWhat 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.\n\n#### Reveal\n\n****2 2 1 1 2 1****\n\nAnd what about the next three numbers?\n\n#### Reveal\n\n****2 2 1 1 2 1 2 2 1****\n\n### Activity - Be Creative\n\nBuild out more of the sequence and then dive in.\n\n  * What do you notice?\n  * Do you have any conjectures about the sequence?\n  * Do you think it repeats?\n  * How many 1's versus 2's are there?\n  * Can you draw this as a tree? A spiral? A cake with layers?\nWhat would this sequence sound like?\n  * Can you construct a sequence like this? What rules would you give it?\n\n\n\nAfter playing, you might want to investigate this sequence as a tree here:\n\nInquiriesInteractive math exploration toolsInquiries\n\n## Educator Resources\n\nSpoiler alert - go play before proceeding (this means you too).\n\n##  Activity Structure\n\nThis is a 30-60 minute activity exploring a self-descriptive sequence.\n\n#### **Exploration (10–15 minutes)**\n\n#### Option 1\n\nBuild the sequence as described previously by starting with a 2 and letting it unfold. Work together to find what comes next.\n\n  * How can you keep track of what comes next?\n  * Is there an algorithm we can use?\n  * Does it repeat?\n\n\n\nYou can do this with numbers, toys, or symbolic representation. It can be inline, or in other forms that learners come up with.\n\n#### Option 2\n\nStart with the sequence:\n\n**2 2 1 1 2 1 2 2 1 2 2 1 1 2 ... ?**\n\n  * How is this sequence built?\n  * What comes next?\n\n\n\nCheck your conjectures to see if they are right:\n\n**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,**\n\n  * Is there an algorithm we can use?\n  * Does it repeat?\n  * What happens if you put a 1 on the front of it?\n\n\n\n#### **Diagramming and Conjectures (15–20 minutes)**\n\nContinue to conjecture, share, and test.\n\nWhat 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.)\n\n  * Hexagons?\n  * Trees?\n  * Spirals?\n  * Can you define a vocabulary of hierarchy and/or connectivity? (parent/child, up/down, levels, etc.)\n\n\n\nPlay with finding different ways and see if anything interesting is revealed.\n\nDoes anything repeat? If so – how?\n\n**Example Conjectures:**\n\n> Example: \"As the sequence goes on and on, the number of 1's will be about 50%.\"\n\n> Example: \"There are patterns of connected numbers that spiral\"\n\n> Example: \"This sequence doesn't ever repeat.\"\n\n> Example: \"This sequence repeats eventually.\"\n\n> Example: \"This sequence is a fractal.\"\n\n#### **Optional Tree Toy + Discussion (10–15 minutes)**\n\nPlay with the tree toy here. Look for interesting patterns.\n\n  * Share thoughts.\n  * Share conjectures.\n  * What does it mean to be self-descriptive?\n  * If you start with a 3 and always go in the order of 3,2,1 - can you make a self-descriptive sequence?\n\n\n\nCan you play the sequence with a drum beat? Does it sound like it repeats?\n\n### Going Deeper (optional)\n\nWhether there are 50% 1's and 2's remains open at the time of this post. OEIS has references to dig in more.\n\nSome other questions:\n\n  * Is there a formula for the nth term? Is it a closed-form?\n  * For any sequence within, does it repeat?\n  * Is there a way to predict the frequency of any repeating patterns?\n  * 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,...\n\n\n\n#### Tools and Supplies\n\n  * Paper and pencil or whiteboard.\n  * Manipulatives or other toys to build a physical representation.\n  * Tree toy (optional).\n\n\n\n#### Vocabulary\n\n  * **Term** – a single number in the sequence.\n  * **Run/run-length** – a block of equal neighbors, and how long it is.\n  * **Run-length encoding (RLE)** – describing a sequence by its run-lengths.\n  * **Self-describing** – it carries the instructions that build it.\n  * **Prefix** – the first _n_ terms.\n  * **Parent/child** – number and the numbers it produces.\n  * **Density** – the long-run fraction of a symbol.\n  * **Conjecture** – A statement believed to be true but not yet proven.\n  * **Fractal** – echoing itself across scales.\n\n\n\n#### Extensions, What-Ifs, and Resources\n\n  * Integer sequence on OEIS (Kolakoski).\n  * What if you start with {1 3} with only 1's and 3's – is there different behavior (see OEIS)?\n  * Related sequences.\n  * Numberphile video.\n  * Start with a **1** instead of a **2** – what changes?\n  * Make music – drum the 1s and 2s. Do you hear a repeat?\n  * Invent your own self-describing rule. What's the smallest one that works?\n  * Program this sequence, make it art. Here is mine.\n\n\n\nI 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.\n\nBehind the scenes – I tried hexagon spirals, paint pouring, and other visuals:\n\n\n",
  "title": "Inquiries-Week 10: Self-Descriptive",
  "updatedAt": "2026-06-21T03:17:57.196Z"
}