The crown exchange is a system in which collected crowns can be exchanged for special fish. Each formula requires a specific amount of 3 different crowns and there are 2 or 3 options for each exchange. For example, to exchange for fish A, you'd need x amount of crowns b, c, and d. Or there are one or two additional formulas which would be x amount of crowns e, f, g or x amount of crowns j, k, l.
Like the Gem Exchange, most of these exchanges require fish(es) that are only obtained by purchasing fishbucks or by other special means. In the Buckfree section in the chart, the number after Yes is which option in the game is the Buckfree option. There is always a chance that you can obtain any of the fishbuck fish needed for free! These options include the shell game, activities, and rewards for completing certain stamp collections.
In order to get a fish crown, you must grow a fish until it is full grown. That's when it receives a crown (you will see it on the fish's head in your tank as well as in the description). Once it's grown you can sell the fish and you will receive the crown.
List of Fish[]
EDIT TIP: Place new formulas at the top of the list (sorted by "new formula first" in the game)