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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.

Finale Numbers™ Not Finale Numbers™
28 30
55 53
388 776
514 207
982 984
1765 1763
4978 9956
6040 3020
8110 8112
19999 19997

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

Finale Numbers™,Not Finale Numbers™
28,30
55,53
388,776
514,207
982,984
1765,1763
4978,9956
6040,3020
8110,8112
19999,19997
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2 Answers 2

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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 $9$.

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  • 1
    $\begingroup$ Actually, it is equivalent to: "jura gur ahzore vf qvivqrq ol avar, gur erzvaqre jvyy or bar". :) $\endgroup$
    – athin
    Sep 6, 2019 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, 2019 at 4:15
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    $\begingroup$ yep, exactly ^^ $\endgroup$
    – athin
    Sep 6, 2019 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, 2019 at 5:33
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    $\begingroup$ Nice, I didn't notice that there has a math property here:P, Good job! $\endgroup$
    – Conifers
    Sep 6, 2019 at 7:20
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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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