A cool sequence problem…


ZipperOur oldest son (nearly 10) posed the following challenge:

What comes next in this list?

1, 1, 1, 2, 2, 3, 2, 4, 3, …

Answer and rationale (his and mine) after the jump…

He spoke the challenge from memory, and after having him repeat it twice, I typed it out (since there’s no paper handy).

After a bit, I tried putting it into rows and columns.  I start with 3 columns (since there were nine data):

1  1  1

2  2  3

2  4  3

but I saw nothing obvious.

Then I tried 2 columns, and hit paydirt:

1  1

1  2

2  3

2  4

3  ? …

And decided that the second column are the natural numbers, the first column are  ” (n+1)/2 rounded up”, in effect.

I told my son the next few terms in the sequence, and his eyes got big.

He was impressed that I “got it” so quickly, and I was impressed that he was thinking about zipper sequences.

I call a sequence a zipper sequence if it can be viewed as being built up by taking alternate terms from two simple sequences.  For example,    5, 2, 5, 2, 5, 2, …   is a simple zipper sequence built from two constant sequences.  A more complex example might be 1, 2, 3, 4, 9, 8, 27, 16, …

As you might expect, I was delighted by my son’s example.  “Where did you get this from?”, I asked.

“Oh, I made it up.”

So I asked that he tell me how he created it.  Here is what he said:

I just took three groups and put tallies in each of them. I put one on left side, one in middle, one on right, and so on.  I knew that by doing that pattern, the middle would have the most but I wanted to know how many it would have.

So he was thinking about putting tallies into three piles, passing from left to center to right to center to left to center to right to center etc…; his sequence was the count of how many tallies were in the “current” pile when a new tally was being placed.

After we compared methods, we both were excited.  He and I had totally different ways of thinking about 1, 1, 1, 2, 2, 3, 2, 4, 3, ….

Being in parent (not teacher) mode,  I didn’t push further to discuss  why his construction implied my formula.  But when I contemplated whether he and I should have that discussion, he spoke up and said he thought there must be other ways to think about this pattern as well.  “Maybe you should post this!”, he said.

We look forward to any other patterns or insights you might offer!

15 Responses to “A cool sequence problem…”

  1. Sue VanHattum Says:

    Love it!

  2. Spiked Math Says:

    Interesting sequence! It doesn’t appear to be in The On-Line Encyclopedia of Integer Sequences. You can also generalize it by putting tallies into k piles (and it should be clear what the k-1 columns are for the other interpretation).

  3. joshg Says:

    Nice! One question: in your “(n+1)/2 rounded up”, are you starting with n=0?

  4. TwoPi Says:

    At that point, working with a two-column array, I was thinking of “n” being the row number starting from 1.

    If you think of “n” as being the index for the entire sequence (starting from 1), then a description for the entire sequence would be

    a(n) = ceiling( n/4 ) if n is odd, a(n) = n/2 if n is even

  5. Shecky Riemann Says:

    Cooool! I predict a long life as a number theorist… or, a tax accountant, for your son! 😉

  6. Fibonacci Retracement, Projection And Extension - Trade the E-minis Says:

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  7. wayne Says:

    I don’t get your son’s “pattern”
    I see yours [the zipper]
    But your description of his method just does not compute [for me].
    Maybe I’m misunderstanding the word tally.
    Could you spell out in greater detail [or long example]
    how he built that pattern by placing 1’s [or sums?] into consecutive 3-celled rows?


  8. wayne Says:

    nevermind. I [finally] got it.
    maybe this stuff isn’t my cup of tea.
    sad to read martin gardner passed : (
    moment of silence…..

  9. TwoPi Says:

    My son’s pattern of putting “tallies” (his word) into three piles leads to the following. Each row is the next iteration of the system; the three columns hold the total number of tallies or counters or beans in each pile.

    1 0 0
    1 1 0
    1 1 1
    1 2 1
    2 2 1
    2 3 1
    2 3 2
    2 4 2
    3 4 2
    3 5 2
    3 5 3

    His original sequence is the list of new values from each row.

  10. Evan Carroll Says:

    Can you prove through math that his real world problem can be represented by your algorithm? I understand the zipper sequence, and I understand what he is doing, but what makes his three bins compose perfectly to a mental model of two independent sequences?

    Is this necessarily true for n=409209

  11. TwoPi Says:

    My son’s pattern repeats every 4 plays: LCRC, LCRC, LCRC, etc…

    Let n be the index for the entire sequence, and a_n be the number of “tallies” in the current bin.

    If n is even, you’re in the center bin, and a_n = the number of occurrences of C so far in the sequence = the number of even numbers up to n = n/2

    If n = 1 mod 4, you’re in the left bin, and a_n = the number of numbers congruent to 1 mod 4 up to n = ceil(n/4).

    Similarly, if n=3 mod 4, you’re in the right bin, and a_n = ceil(n/4).

  12. Garret Says:

    Wow! I really need to start practicing math. My mind has been away from it so long that I think it is getting math dumb. But no worries it will be back soon, thanks to math blogs. They are great. I will learn more math online then I ever did in a classroom.

    math games for the classroom
    Learning is more than just merely filling a cup it is the lighting of a fire!

  13. Maria H. Andersen Says:

    Are you guys on a blogging break? It’s been a month. I’m getting worried. 🙂

  14. TwoPi Says:

    Hi Maria! Ξ and I have sabbaticals upcoming (so we’ve started early this summer.) This either could lead to lots of spare time and inspiration for blogging, or it could instead lead to a total math break for a while. At the moment, the latter is winning, but not necessarily by design.

  15. Top ten math posts of 2010 | 11:23 Says:

    […] A cool sequence problem… I liked this little anecdote, showing how one can arrive at the same idea from widely different […]

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