## What’s the pattern?

by

1, 11, 21, 1211, 111221, 312211, 13112221, …

This sequence was the favorite pattern of my former department chair, Nelson Rich.  I thought he invented it, but a quick search on the internet reveals that it’s pretty easy to find if you search for the first few terms.

But don’t do that! It’s a sneaky one, but fun to try and figure out.

Try saying the digits out loud.  This is an aural/visual sequence rather than a numeric one.

And yes, I did have to look up aural to make sure I was using it correctly.

### 17 Responses to “What’s the pattern?”

1. Dave Richeson Says:

That’s great one. Here’s another: o, t, t, f, f, s, s, e,…

(Same hint as yours—say it.)

2. mbork Says:

I also knew this. I agree – it’s great!

And for the one with letters – no idea. But it’s 0:30 AM here in Poland, so I’ll get back to it in the morning;).

3. Dave Richeson Says:

Oops!!! I just realized I gave the wrong hint for o, t, t, f, f, s, s, e,…

Don’s say read the sequence, count the number of terms in the sequence. Sorry!

4. Ξ Says:

Dave, here’s a similar one to yours. What number comes next:

3, 3, 5, 4, 4, 3, 5, 5,

[I think I first heard this on Car Talk, and what I think is cool is that there’s also a numeric pattern that this follows, though that is arbitrary].

5. TwoPi Says:

The two examples (so far) in the comments remind me of the student dialog that Ξ describes in this post from last August.

6. Dave Richeson Says:

First of all, clearly I need a copy editor. Typos in both comments. Sorry about that.

These are fun. When I teach sequences, I give the class a collection of sequences that end in “…”. They are supposed to find the next term and give a justification. Most of them are typical—squares, fractions, alternating signs, etc., but the last few are strange ones. The point of these last few exercises are to show them that:

1. not every sequence can be expressed as a formula (like Ξ’s sequence), and
2. we can never tell for certain what the n+1st term is, given only the first n.

My favorite example for (2) above is:

What is the next term of the sequence: 1,2,3,4,5,6,7,8,9,10,11,12,… (hint: the answer is not 13).

I tell them there is no right answer, but they must come up with the next term, with a justification. Some answers I get are:
1 (hours on a clock)
1 (months of the year)
14 (floor numbers in a hotel)
14 (numbers with prime factors 2,3,5,7,11)

7. mbork Says:

And now it’s morning in Poland, and I’VE GOT IT! (I mean, the o, t, t, f, …). And without the second hint!:)

And I have to say that the dea with exercises in sequences is great. I’ll most probably use it (I’ll be teaching Analysis next semester, I’m pretty sure that sequences are covered in the syllabus…)

Thanks!

8. TwoPi Says:

Exercise: For the original sequence, describe an algorithm for producing the predecessor of any given term, and prove that your algorithm is correct. [correctness isn’t obvious…]

Related exercise: show that the largest digit that occurs in the sequence is “3”.

Another twist on the original sequence:

0, 10, 1110, 3110, 132110, 1113122110, etc….

[I like starting this pattern with 0, as that strikes me as being consistent with the overall process.]

9. What’s the (Geometric) Pattern? « 360 Says:

[…] 360 12 tables, 24 chairs, and plenty of chalk « What’s the pattern? […]

10. David Petersen Says:

Another “creative” pattern:

1, 4, 9, 61, 52, 63, 94, 46, 18, …

Hint: Your intuition on the first 3 numbers is partially correct.

11. Ξ Says:

That’s neat! I’m a little uncertain how to write the next number in the sequence, though (or how to define it formally, unless I knew how many digits were being used).

12. David Petersen Says:

Most of my students (when given this sequence for extra credit at the end of a quiz) will write 001. I ask them to write out the reasoning as well, so 1 or something else that makes sense will also work if their reasoning is correct and consistent.

13. TMA Says:

The same sequence (starting with a “3”, that is 3, 13, 1113, 3113, etc.) is known as Conway’s sequence and some interesting results have been proven about it!
If I remember correctly, you can find an interesting summary in Ilan Vardi’s book “Computer Recreations with Mathematics”

14. Ξ Says:

TMA, that’s really interesting — I hadn’t heard of that before. Even Wikipedia has some info on it.

15. james Says:

lets see u can answer this f,s,t,f,f,s

16. TwoPi Says:

s,e,n,t… of courth

17. Laura Says:

I am looking for, and seeming missing the pattern of Venus coming in front of the sun, which happens on June 5th.
The recorded dates when this was observed – 1639 – 1761-1769-1874-1882-2004-2012- and predicted in 2014
Why the decrease from 8 years to 2?