# This is another number sequence puzzle!

Can you find the next number in the following sequence:

3, 6, 9, 8, 13, 20, 17, 14, 21, ?

If you're operating differently you probably prefer to solve the following sequence:

2, 0, 0, 12, 35, 384, 135, 32, 315, ?

Hint 1

The missing numbers are not the same.

Hint 2

Both sequences are finite

• Thanks @Rubio. I've changed the puzzle and added a sequence I wanted to use as a hint later. Do you think this one's better or should I try it in the sandbox first? – swit Apr 13 '18 at 12:23
• This seems better now, yes; thanks for being open to feedback :) – Rubio Apr 13 '18 at 18:54
• Might be useful to someone: desmos.com/calculator/m17pq62ero – Just a browsing guest Apr 13 '18 at 21:44
• Would it always be the same if it starts with the same number? Or could there be multiple sequences that start with 3, using this pattern? – Lily Potter Apr 15 '18 at 5:56
• This sequence is neither arithmetic or geometric?? – CR241 Jul 27 '18 at 23:29

Split the sequence into $$(3, 6, 9, 8)$$ - three increasing one decreasing, $$(13, 20, 17, 14)$$ - two increasing two decreasing and $$(21, ?, \cdots,\cdots)$$ - one increasing three decreasing. So the question mark must be less than $$21$$.
The consecutive differences are $$3,3,3,1,5,7,3,3,7$$. They are no more than $$10$$ and $$1$$ occurs once, $$3$$ occurs five times, $$5$$ occurs once and $$7$$ occurs twice. Except $$7$$, the number of occurrences are odd, so the difference between $$21$$ and the question mark is either $$7$$ or $$17$$. If it is the latter, that introduces an extra $$1$$ which breaks the odd rule.
Therefore, the missing number is $$21-7=14$$.