## Discover the Rule

Found the following cute puzzle on Facebook.

Puzzle. Discover the rule governing the following sequence to find the next term of the sequence: 8, 3, 4, 9, 3, 9, 8, 2, 4, 3.

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6,7

2. #### Blaine:

…4,4,2,3,8

That was fun!

3. #### gabriella:

…4,5,3,5,4,2,6,3,4,7,3,7,6,2,8,3,4,9…

…,1,9,5,2,4…

5. #### Robert:

… 4, 6, 3

but I’m not sure .. what is the reasoning behind?

6. #### tanyakh:

One of the five answers above is correct. Hint: The rule is simple. If you are not sure, you didn’t discover it.

7. #### Michael:

Blaine has it. Unfortunately I don’t see a way to determine additional terms… unless they are 1, 1, 1…?

8. #### Blaine:

Michael, perhaps it could continue as 5, 5, 4, 4, 5, 4, 5, 3… and would it eventually devolve into 4, 4, 4… ?

9. #### Richard:

Mmmh, I don’t follow

10. #### tanyakh:

Hint: This is a self-referencing puzzle.

11. #### Leo B.:

I got it; however, my very simple (and human language-independent!) program says 1,9,5,2,4.

Indeed, the rational number with the lowest denominator, which decimal form starts with .8349398243 is 138681/166097 = 0.83493982431952413348…

12. #### Stefano:

…4,4,2,3,9(or 8) ?

13. #### Cristóbal Camarero:

I see the mirror rule a(i)=a(13-i)+1 for i<13-i, i!=4. This would continue with two terms: 2 and 7. But this is different to all given ones and Tanya said the good is already given.

14. #### Blaine:

Hint: ignore the word “Puzzle” and proceed from there.

15. #### Cristóbal Camarero:

Oh! I see now, nice. Not as much a self-referencing puzzle as a prelude-referencing one, I would say.

16. #### Marnix:

…8

And perhaps a couple of 1’s if you’re so inclined 🙂

17. #### Marnix:

..4, 4, 2, 3, 8.

And perhaps a couple of 1’s if you’re so inclined… 🙂

18. #### Per:

Meh, its not in the OEIS, so most likely not given by some easy formula…

19. #### Taposik:

This is fun, I am sure the next few terms according to this rule would be 4, 2, 3, 1, 2, 4, 3 , 4, 3, 5, 9

20. #### Yanick:

I saw this puzzle at work so I wrote down the sequence on a post-it to solve it over the weekend. Here is my hint: I could only solve it on monday.