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Kaprekar’s constant 6174 can be
considered to be a disguised form
of the silver ratio (that is, the
reciprocal of the golden ratio).


There are 9990 4-digit numbers
having at least 2 distinct digits.
The location of Kaprekar’s
constant among them is, to
4 decimal places, the silver
ratio deep among them,
the silver ratio being the
reciprocal of the golden ratio
(not to be confused with
the silver mean, also known
as the silver ratio).

That is to say, 6174/9990 is,
to 4 decimal places, 0.6180.

So, every 4-digit number, at
the beginning of the Kaprekar
process, shouts out “Hi-Yo Silver!”

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