A Great Day for a Pattern

When I first saw today’s date in mm/dd/yy format — 11/12/21 — I thought, “Well, that’s pretty cool. It’s all 1s and 2s.” And then I thought, as I’m sure you did, “In ternary, that’d be 376,” because everyone thinks in ternary, right?

But then I looked at the number again, and I thought, “Ah, hello, old friend. Good to see you again.”

Those six digits form the fifth term of a famous pattern:

1

11

21

1211

111221

So your first question is, what’s the next term?

If you’ve never seen this pattern before, it’s worth a little of your time to try to figure it out before reading more about it at MathWorld.

Your second question — if you’re still reading — is, what’s the greatest digit that will ever appear in this sequence? As you can see above, the first five terms only contain 1s and 2s. What digits are in the sixth term? What digits will appear beyond the sixth term? How do you know?