This is in the spirit of the What is a Word/Phrase™ series started by JLee with Number version puzzles.

If a number conforms to a special rule, I call it a Finale Number™.
Use the following examples below to find the rule.

$$ % set Title text. (spaces around the text ARE important; do not remove.) % increase Pad value only if your entries are longer than the title bar. % \def\Pad{\P{0.0}} \def\Title{\textbf{ Finale }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Numbers}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ }\displaystyle \smash{\lower{29px}\bbox[yellow]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ 28 }&\text{ 30 }\\ \hline \text{ 55 }&\text{ 53 }\\ \hline \text{ 388 }&\text{ 776 }\\ \hline \text{ 514 }&\text{ 207 }\\ \hline \text{ 982 }&\text{ 984 }\\ \hline \text{ 1,765 }&\text{ 1,763 }\\ \hline \text{ 4,978 }&\text{ 9,956 }\\ \hline \text{ 6,040 }&\text{ 3,020 }\\ \hline \text{ 8,110 }&\text{ 8,112 }\\ \hline \text{ 19,999 }&\text{ 19,997 }\\ \hline \end{array}$$

And, if you want to analyze, here is a CSV version:

Finale Numbers™,Not Finale Numbers™

A Finale Number™ is a number such that,

When you add its digits, add the digits of that result, and add the digits of that result and so on, you finish with $1$. This is called the digital root of a number. (Finale numbers™ have digital root $1$)

Finale Numbers™:

$19999\rightarrow 1+9+9+9+9=37$
$37\rightarrow 3+7=10$
$10\rightarrow 1+0=1$

$8110\rightarrow 8+1+1+0=10$
$10\rightarrow 1+0=1$

$4978\rightarrow 4+9+7+8=28$
$28\rightarrow 2+8=10$
$10\rightarrow 1+0=1$

Not Finale Numbers™:

$19997\rightarrow 1+9+9+9+7=35$
$35\rightarrow 3+5=8$

$3020\rightarrow 3+0+2+0=5$

$8112\rightarrow 8+1+1+2=12$
$12\rightarrow 1+2=3$

This is equivalent to: (thanks @athin!)

A Finale Number™ has a remainder of $1$ when divided by $9$.

  • 1
    $\begingroup$ Actually, it is equivalent to: "jura gur ahzore vf qvivqrq ol avar, gur erzvaqre jvyy or bar". :) $\endgroup$ – athin Sep 6 at 3:51
  • $\begingroup$ Looks so! rot13(Vs lbh trg a ol qbvat guvf gur ahzore zhfg or a zbq avar evtug?) $\endgroup$ – user47134 Sep 6 at 4:15
  • 1
    $\begingroup$ yep, exactly ^^ $\endgroup$ – athin Sep 6 at 4:37
  • $\begingroup$ You might want to include the mathematical terms: rot13(qvtvgny ebbg & zbqhyb 9), which aren't equivalent methods but do give equivalent results for this puzzle. $\endgroup$ – amI Sep 6 at 5:33
  • 1
    $\begingroup$ Nice, I didn't notice that there has a math property here:P, Good job! $\endgroup$ – Conifers Sep 6 at 7:20

To be precise, my answer is also in the same lines as that of

supersonic, but with a generalization at sum of the digits of the given number level. It is of the form ( (9*n) +1)- for Finale numbers and we cannot do so for their counterparts


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