# What is the next in this ordinary sequence?

3 4 9 8 9 8 13 14 13 ?

So what is the next number?

1. Hint

the actual difference between numbers is just 1 and the sequence actually starts with 1.

2. Hint

Spelling is the key!

• Your sequence is too short, and/or consists of not enough numbers, for anyone to guess it without hints. (In fact, given no context at all, ? might be any number, as pointed out by Fimpellizieri’s answer.)
– Lynn
Feb 13, 2016 at 22:39
• it is not that complicated @Lynn . just need different point of view than just numbers in a sequence... i will give a hint soon... this is regular sequence with a little tweak in it.
– Oray
Feb 13, 2016 at 22:42
• you say spelling is key. just for clarity the numbers present are: three four nine eight nine eight thirteen fourteen thirteen ?
– Bort
Feb 15, 2016 at 14:10
• @Bort check hint 1 and hint 2 together before spelling the sequence...
– Oray
Feb 15, 2016 at 14:18
• I have an idea that almost works but it requires the 14 being 12 instead. This isn't the case by any chance, is it? Feb 15, 2016 at 18:08

Going for the obvious here.... (because someone has to..)

14.

The whole thing is very arbitrary. Goes from 3 -> 4 then jumps up 6 and sequences 1 down 9 -> 8 -> 9 -> 8 and then jumps up 6 again (from 8 to 14: just wrote this for the sake of some sort of pattern) and sequences up 13->14->13->14.

Very disappointed if this is the answer :-{

– Oray
Feb 14, 2016 at 10:50
• @Oray ...phew :-]
– Ben
Feb 14, 2016 at 10:53

12

The formula for generating the $n$th term in the sequence is

$n$ + 2*(number of letters in English spelling of $n$) - 4

For example the first term in the sequence is

1 + 2*3 - 4 = 3

The 9th term is

9 + 2*4 - 4 = 13

And so the 10th term is

10 + 2*3 - 4 = 12

In relation to the hints

The underlying sequence is the positive integers beginning at 1 and generation of the sequence depends on the spelling of the numbers.

• this is very close, actually the answer is right but the methodology is not... luckily i believe u got the right answer to be honest :) u are not supposed to extract 4... it is so weird u got it because of the number of vowels and consonants in the spellings :p
– Oray
Feb 15, 2016 at 23:01
• Interesting. I was pursuing a line of thought which involved numbers of vowels and consonants but couldn't get it to fit without some cumbersome rules. Can you tell me this, what is the smallest value of n such that the nth term in your sequence is different to that in mine. This may help me with the reasoning. Feb 15, 2016 at 23:50
• ok number+#ofvowels*#ofconsonant try it...
– Oray
Feb 17, 2016 at 8:04
• Yeah, that makes sense. Working it out, our two sequences match up if either there are two consonants or two vowels in the word. So the first time there is a difference will be 11 (yours is 20, mine 19). Although, we should have the same result for n = 12, 22, 26, 32, 36, 42, 46, 52, 56, 62, 66, 70, 80 and 90. Feb 17, 2016 at 10:13

Consider the following polynomial:

$$P(x) = 898-k-\frac{507793}{210}\cdot x-\frac{5}{192}\cdot kx^6+\frac{3013}{17280}\cdot kx^5-\frac{95}{128}\cdot kx^4+\frac{4523}{2268}\cdot kx^3-\frac{6515}{2016}\cdot kx^2+\frac{7129}{2520}\cdot kx+\frac{1092977}{420}\cdot x^2+\frac{1}{362880}\cdot kx^9-\frac{1}{8064}\cdot kx^8+\frac{29}{12096}\cdot kx^7-\frac{5}{4032}\cdot x^9-\frac{7552313}{5040}\cdot x^3+\frac{41377}{80}\cdot x^4-\frac{107677}{960}\cdot x^5+\frac{77}{5}\cdot x^6-\frac{4363}{3360}\cdot x^7+\frac{103}{1680}\cdot x^8$$

It is a straightforward calculation to check that:

• $P(1) = 3$
• $P(2) = 4$
• $P(3) = 9$
• $P(4) = 8$
• $P(5) = 9$
• $P(6) = 8$
• $P(7) = 13$
• $P(8) = 14$
• $P(9) = 13$
• $P(10) = k$

So it might be that there are multiple solutions?

• it is actually much easier than you think... but a little different. i will give a hint tomorrow if cannot be solved.
– Oray
Feb 13, 2016 at 22:41
• And I thought that working with such a long polynomial is illegal...:p Feb 14, 2016 at 7:43

It's not 100% convincing, but worth a try:

3 4 9 8 9 8 13 14 13 ?
Numbers 3, 6, 9: 9, 8, 13 (no idea. Typo? :)
Numbers 2, 5, 8: 4, 9, 14 (+5)
Numbers 1, 4, 7: 3, 8, 13 (+5)
Number 10: 13 + 5 = 18

I thought the question would be solved after giving Hint 1 and Hint 2 but I could not get the right one.

Methodology

Since the sequence starts with 1 and increases by 1 (Hint 1) and spelling was important (Hint 2), You needed to check the spelling of 1, which is one. There are two vowels and one consonant and the generalization is found from here;

Equation

Number + (# of Vowels)x(# of Consonants)