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This puzzle is based off the What is a Word™ and What is a Phrase™ series started by JLee and their spin-off What is a Number™ series started by smriti.

If a number conforms to a certain rule, I call it a Well-toned Number™. Use the following examples to find the rule:

Well-toned Numbers™ Not Well-toned Numbers™
36 49
343 729
2000 1000
13597 13579
254708 125678
807452 876521
5551205 5552106
8675190 8675309
138076942564 138076942546
942564138076 942546138076

Here is a CSV version:

Well-toned Numbers™,Not Well-toned Numbers™
36,49
343,729
2000,1000
13597,13579
254708,125678
807452,876521
5551205,5552106
8675190,8675309
138076942564,138076942546
942564138076,942546138076

The puzzle relies on the series' inbuilt assumption, that each number can be tested for whether it is a Well-toned Number™ on its own. In particular, a number's relationship to other numbers in the sequence is irrelevant.

These are not the only examples of Well-toned Numbers™ (or not Well-toned Numbers™), more can be found.

Hints:

Maybe @DarfNader is onto something...

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    $\begingroup$ While 8675309 may not be a well-toned number, it is a well-tutoned number! :-D $\endgroup$
    – Darf Nader
    Feb 1, 2018 at 11:53
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    $\begingroup$ Should I phone a friend for the answer? $\endgroup$
    – James Webster
    Feb 8, 2018 at 7:11

1 Answer 1

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Edit: A lot of credit is due to @Levieux for helping with the reasoning.

I think the answer relates to the following image

enter image description here

Explanation

Well-toned numbers, when read left to right, have their $n$th digit in the same column as if they were read right to left. The implication is that they resolve the column melody in a symmetric way in terms of DTMF frequencies.

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    $\begingroup$ Not just the first and last digit, otherwise 942546138076 would also be well-toned. Every nth digit starting from the left is in the same column as the nth digit starting from the right. So the numbers would be symmetric if you replace each digit by its column number. $\endgroup$
    – Levieux
    Feb 8, 2018 at 9:59
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    $\begingroup$ Did my comment work as a hint? $\endgroup$
    – James Webster
    Feb 8, 2018 at 10:09
  • $\begingroup$ Happy to help. I'd figured it was something to do with dial-tones and the phone pad, but not what. I was trying to link it with music. I spent quite a while dialling those numbers to see if they sounded musical. $\endgroup$
    – James Webster
    Feb 8, 2018 at 10:14
  • $\begingroup$ Congratulations! $\endgroup$
    – boboquack
    Feb 8, 2018 at 20:21

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