Friday Puzzler -- Sequence

Re: Friday Puzzler -- Sequence

Henry Higginbotham

The sequence part is, I think, a rerun from a year or two back (or maybe I saw it elsewhere), so I'll leave it to someone who hasn't already seen it.

As for the bonus, there may be more than one but I stopped when I found forty.

Yes, and Yes!

Alex Y

I haven't kept track, but I believe you are right about the first being a rerun. Thanks for leaving it, though, for someone to solve who did not see it the first time!

And I don't know if there are any answers to the bonus other than 40. Nice job!

Re: Moving right along

Henry Higginbotham

Apparently everyone else also remembered this one and chose to leave it for someone who hadn't already seen it. But it's sitting here holding up the show, so:

1

11

21

1211

111221 (one 1, one 2, two 1s)

312211 (three 1s, two 2s, one 1)

13112221 (one 3, one 1, two 2s, two 1s)

...and so on. Each number just describes the previous one.

I didn't take it much further than this. Would it ever produce a 4 or 5?

4 or 5

Alex Y

Would it ever produce a 4 or 5?

Great question, Henry! I assumed so, and carried it six more places, but did not get a 4 or 5, I wondered if a computer whiz here could program it to go out further. But driving back from some errands today, the answer came to me:

You will always get numbers made up of 1's, 2's, and 3's only. Assume there was a larger digit in a number in this sequence, and consider the first such number. Then the previous number had at least 4 1's, 4 2's, or 4 3's. 1111 means that the number before that had one 1 followed by one 1, but that would be represented by 21. Similarly for 2's and 3's.