# A purely numerical sequence with a hint of the visual

What is the missing number in this cyclic sequence? It would require a notation convention not seen here to fit into the scheme here.

$$\phantom{ \textrm{(repeat)} } \quad\textbf{?}~~ \,,~~ 2.4 \,,~~ 1 \,,~~ 36.0 \,,~~ 1 \,,~~ 8 \,,~~ 1 \,,~~ 15 \,,~~ 2.4 \,,~~ 1 \,,~~ 8.0 \,,~~ \frac{~9~}2 , \phantom{ \,,~ \dots } \\ \textrm{(repeat)} \quad\textbf{?}~~ \,,~~ 2.4 \,,~~ 1 \,,~~ 36.0 \,,~ \dots \phantom{ \,,~~ 1 \,,~~ 8 \,,~~ 1 \,,~~ 15 \,,~~ 2.4 \,,~~ 1 \,,~~ 8.0 \,,~~ \frac{~9~}2 , }$$

$8$ vs $8.0\,$?   $2.4$ vs $\frac92$?   All relevant properties, including the visual one, are mathematical.

Half$/$ hint:

The visible clue: $2.4$, $36.0$, $2.4$ and $8.0$ are not represented as reduced fractions.

• Is it significant that one number is $8$ while another is $8.0$? – GentlePurpleRain Feb 3 '16 at 22:23
• I think 'visual' is a hint, but I can't figure out how it helps. I've been looking at ways to measure visual acuity, but can't see how any of the numbers fit. – jhabbott Feb 4 '16 at 1:42
• Is the sequence cyclic? – 2012rcampion Feb 4 '16 at 6:00
• The above comments have prompted a more-periodic layout and a "Hintified from comments" fortune cookie – humn Feb 4 '16 at 7:29
• Your hints are very... hinty. Any more? – question_asker Feb 15 '16 at 20:38

$3.\overline{3}$ or $3.333...$

Who knows!

Well, then how did I get the answer?

Brute force! Kind of!

(record scratch) lol what

Yeah, look, I went to college for like 1.5 semesters before dropping out, and during the semester I liked, I took Calculus I, and while I enjoyed it, it was the last of my formal math education.

Anyway, the kind (non-, as per their profile) doctor had at one point left a hint on this question: "This sequence rolled out in reverse order as reciprocals for a probability problem here." In a brief chat with Pangloss, they mentioned that the sequence was only of importance with regard to games where randomness was necessary (emphasis mine). From this I could narrow down that the problem I was looking for probably involved dice or a coin flip. Skimming those problems, I found that, lo and behold, there was a problem that involved both, and specifically an answer that had the exact thing mentioned in the hint. I "checked" the "math" briefly and realized that it was, in fact, the reciprocals of the elements of the sequence in reverse. From this I could conclude that the missing value was $10/3$, and since it was mentioned that the missing element required a notation convention not already seen in the puzzle, I realized it must be $3.\overline{3}$ or $3.333...$

As such, with Pangloss's blessing, I am posting this answer and awaiting my parade.

• Cheers to a solution that marshals so many resources! The purely mathematical approach had its chance for half a month before you revived this puzzle. – humn Feb 18 '16 at 21:36
• @Pangloss I may not be educated, but I'm resourceful, dammit – question_asker Feb 18 '16 at 21:37
• The old chat room is open again if you'd like to add to this solution – humn Jul 3 '16 at 12:03

Possibly halfway to a solution, here's my worksheet so far.

\begin{matrix} & 9&~~?~~& 12 & 1 & 36 & 1 & 8 & 1 & 15 & 12 & 1 & 8 & 9 &~~?~~& 12 & \\[-1ex] \cdots & -& &\cdot& - &\cdot& - & - & - &-\!\!-&\cdot& - &\cdot& - & &\cdot& \cdots \\[-1ex] & 2& ? & 5 & 1 & 1 & 1 & 1 & 1 & 1 & 5 & 1 & 1 & 2 & ? & 5 & \end{matrix} Think I could almost stagger the rest of the way,

without fully figuring out the rule. Then again, what was that about notation convention?

• Pangloss, I hate you. – question_asker Feb 16 '16 at 12:27
• For the record, that's my upvote you're seeing on this answer, as well as that on your comment. – question_asker Feb 16 '16 at 12:33
• When you refer to a "notation convention", are you referring to this thing above with the dots and dashes, or something entirely separate? – question_asker Feb 17 '16 at 13:19
• I thought that might be the case, I just wanted to get every possibility I had considered onto the page, so to speak, in case it jumpstarted anybody else's thought processes. – question_asker Feb 17 '16 at 19:16