What's the $\bf ?$ in $~ \bf 814576 : \, ? ~$ ?
$\require{begingroup} \begingroup
\let \SSS \tiny
\let \SS \scriptsize
\let \S \small
\def \T {\small\sf}
\def \Q #1{{ \bf ? { \S\raise2mu ( } #1 { \S\raise2mu ) } }}
\def \R #1{ \phantom{\Q0} \llap{ \bf #1 \kern 7mu } }
\def \x #1{{\SSS\kern1mu\raise2mu \times \S\kern5mu 3 \kern1mu\SS\raise7mu #1}}
\def \p { \kern8mu{ \S \raise1mu + }\kern8mu}
\def \D { \kern6em }
\def \E #1{ \D \llap{ #1 \kern4mu { \S\raise1mu = } \kern6mu } }
$
$ \kern-12mu \E{ \bf ? } 514 $
Essential $\bf 1 \over 10\raise-4mu{\small\,3}$ of the story
$ \E{ \Q{\T digit} } $
how many enclosed regions, typically loops,
in the drawing of the single $\T digit$
$ \E{ \Q{\T digits} } $
decimal
value of the ternary
number formed
$ \D \raise-6mu\strut $
when each $\T digit$ of $\T digits$ is replaced by $\Q{\T digit}$
$ \E{ \Q{814576} } \Q 8 \x 5 \p\Q 1 \x 4 \p\Q 4 \x 3 \p\Q 5 \x 2 \p\Q 7 \x 1 \p\Q 6 \x 0 $
$ \E{} \R2 \x5 \p\R0 \x4 \p\R1 \x3 \p\R0 \x2 \p\R0 \x1 \p\R1 \x0 $
$ \E{} 201001\raise-4mu{\small\,3} $
$ \E{} 514\raise-4mu{\scriptsize\,10} $
$\endgroup$
Middle $\bf 1 \over 10\raise-4mu{\small\,3}$ of the story
Notice the following retrieval from the
community evidence locker.
It originated from using various
radices
for the numbers being puzzled
and displaying the only two radices with suggestive patterns.
left : right binary ternary
135759 : 1 1 1
151364 : 4 1.. 11
255075 : 9 1..1 1..
279422 : 36 1..1.. 11..
292620 : 91 1.11.11 1.1.1
348777 : 135 1....111 12...
398067 : 147 1..1..11 1211.
417894 : 265 1....1..1 1..211
459431 : 279 1...1.111 1.11..
478926 : 307 1..11..11 1.21.1
609941 : 363 1.11.1.11 11111.
689245 : 435 11.11..11 121.1.
Watch what happens, starting with the dots, when ...
... the middle two columns are ignored
and left-column digits without enclosures are dotted out.
left right ternary
.....9 1
....64 11
...0.. 1..
..94.. 11..
.9.6.0 1.1.1
.48... 12...
.9806. 1211.
4..894 1..211
4.94.. 1.11..
4.89.6 1.21.1
60994. 11111.
689.4. 121.1.
No wonder the ternaries had so few $\small\tt 2$s,
a peculiarity that had been noted early without direct result.
Only the digit $8$ has two loops
and thus transforms into $\small\tt 2$,
while each of four digits$-0 , 4 , 6 , 9-$has
just one loop (or enclosed region)
and transforms into $\small\tt 1$.
And no wonder that a few ternary numbers were exactly as wide
as the left-column numbers and that none were wider,
which was not noted along the way.
Final $\bf 1 \over 10\raise-4mu{\small\,3}$ of the story
How would anyone notice the above relationship?
By following up on clues, such as those just mentioned,
and allowing for dumb luck coincidence.
The records room at Puzzling HQ has a dossier on The Case of the
Really, really,
really hard sequence,
which was reported just 11 days after the present puzzle
and hinges on an eerily comparable ternary modus operandi.
Sure enough, that case was
cracked
by the poser of the present puzzle, in barely 86 minutes and 48 seconds.
To see a new puzzle that uses a similar device must have been precious!
And then to write a solution that flaunts
the very key to their own unsolved puzzle.
The present puzzle's poser's pathological ternary obsession
did not go unnoticed, nor uncontracted,
by the detective assigned to this case.
Sooner or later, well, much after the hint ...
... $0 \kern2mu {:} \kern1mu 1$
revealed that two wildly different numbers, $135759$ and $0$,
can produce the same degenerate result, $1$,
it was time to consider that individual digits of the puzzling
numbers might be nullifying each other
or even being ignored.
The binary and ternary patterns above
seemed like the simplest leads to follow when embarking on this.
(Not) one of the (first) possibilities to touch on,
if only to shake up thought patterns,
was the graphical way of looking at digits in that other case.
And it happened to work. Tickle me pink.
