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I'm rank 41 and have some observations.
>Saving gold balls from previous fish MAY increase chance of a shiny Karp
>Doing training in favour of eating MAY increase chance of skelly Karp (tried levelling up normal Karp with training exclusively)
Had my first shiny at Rank 29 and first skelly at Rank 41 and the above was all I did differently.
Could just be coincidence but I figured I'd put it out there.