Earn the equation.

No equation yet

Start with one dot. Wrap three more around it. Then five. Then seven. Look at what shape you get each time.

Each new layer is an L shape. Each L adds the next odd number. And every time, you get a square.

Your turn

Using only the picture: what are the first 100 odd numbers, added up?

Watch the clip

The clip builds the same squares, then writes the notation, then proves it. Watch the list of five steps on the side.

The name for it

Now the notation means something

1 + 3 + 5 + ... + (2n - 1) = n²

Each L has n + (n - 1) = 2n - 1 dots.

If this line had come first, it would have been a fact to memorize. Coming last, it's a short way to write down something you already saw.

And why it always works

1. Add the next L: 2n + 1 dots.
2. n² + 2n + 1 = (n + 1)²
3. A square plus the next L is the next square. Always.

The order

1. Intuition. Start with something you can see or hold.
2. Transformation. Change it, one step at a time.
3. Pattern. Let the viewer notice what stays the same.
4. Notation. Write the pattern down, now that it means something.
5. Proof. Show why it can't fail.

Most textbooks run this in reverse: notation, proof, then an example. The style is named after a YouTube channel, but the idea is older. The picture of odd numbers making squares goes back to ancient Greek mathematics.

Ask your agent

Explain the idea below in the 3Blue1Brown style.
Don't write any formula or formal definition until step 4.
1. Intuition: a concrete picture or small example the reader can see.
2. Transformation: change it one step at a time, and describe what changes.
3. Pattern: ask the reader what stays the same, then name it.
4. Notation: now write the formula, and map every symbol back to the picture.
5. Proof: explain why it always works, using the picture if you can.
If you can draw it, describe the drawing or produce it as SVG.

Idea:
[paste here]

What good output looks like: a picture-first explanation where the formula shows up late and every symbol points back to something you've seen.