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In this sequence:

1, 11, 21, 22211, 33221, 433332211, 44444444332221, ?

Can you guess the next number? What is this sequence?

Hint 1:

Every number always ends with 1 or 11

Hint 2:

If the number ends with 11, Then the next number ends with 1111 but this is longer, So ends with 1

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  • $\begingroup$ aw it reminds me of oeis.org/A005150 $\endgroup$ Dec 30, 2021 at 0:12
  • $\begingroup$ @BenjaminWang this is not look and say sequence $\endgroup$
    – Fmbalbuena
    Dec 30, 2021 at 0:18

1 Answer 1

5
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The next number is:

5555554444433333211

Here is the process to transform a number to find the next one in the sequence :

Consider each value ($1$'s,$2$'s,$3$'s,...) separately. Per construction, they will always be written in decreasing order. Then:
* First each value $k$ is repeated $k+1$ times: $1$ becomes $11$, $2$ becomes $222$, $3$ becomes $3333$, etc.
* Then if a value $k$ is appearing $k+2$ times or more, each block of $k+2$ occurences is replaced by one occurence of $k+1$

Hence:

$1$ becomes

$11$ and nothing is replaced

$11$ becomes

$1111$ and the first three $1$'s are replaced by one $2$, making $21$

$21$ becomes

$22211$ and nothing is replaced.

$22211$ becomes

$2222222221111$ ; Then the first eight $2$'s are replaced by $33$, the first three $1$'s are replaced by one $2$, making $33221$

$33221$ becomes

first $3333333322222211$ ; and then $433332211$ after replacements of five $3$'s by one $4$ and four $2$'s by one $3$.

$433332211$ becomes

first $444443333333333333333221111$ ; and then $44444444332221$ after replacements

$44444444332221$ becomes

first $44444444444444444444444444444444444444443333333322222222211$ ; and then $5555554444433333211$ after replacements.

Research process on paper:

my paper

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  • $\begingroup$ How do you found the solution? $\endgroup$
    – Fmbalbuena
    Dec 30, 2021 at 13:07
  • $\begingroup$ @Fmbalbuena : with pen and peper ;) . From the allure of the sequence I tried to understand when a higher value could appear, and then it was just try and guesses. Having so many $4$'s but no $5$ in the last number helped me understand that this was a two-steps process. $\endgroup$
    – Evargalo
    Dec 30, 2021 at 13:11
  • $\begingroup$ how many time do you found? $\endgroup$
    – Fmbalbuena
    Dec 30, 2021 at 13:12
  • $\begingroup$ If you are asking "how much time ?", I think it was between 10 and 15 minutes. I was lucky to be on the right track from the beginning. $\endgroup$
    – Evargalo
    Dec 30, 2021 at 13:14
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    $\begingroup$ a 10 per 5 centimeters piece of paper (edited in the answer...) ! $\endgroup$
    – Evargalo
    Dec 30, 2021 at 13:23

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