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This page tells you how to get shiny pokémon. A shiny pokémon is a rare pokémon that has a slightly different color pattern from the usual pattern for its type. Being shiny only affects the pokémon's appearance, and doesn't have any effect on its stats, moves, breeding results, or anything at all other than how it looks.
Aside from the Red Gyarados in the Lake of Rage, finding a random shiny pokémon in the wild depends entirely on luck, and is extremely rare. Out of all of the random pokémon that you encounter, only 0.012207% will be shiny. In other words, statistically, only one out of every 8192 random pokémon is shiny. There is nothing that you can do to make it more likely to find a shiny pokémon in the wild.
Usually, when you get an egg after putting two Pokémon into the Day Care, it has the same probability of being shiny as listed above. However, the Director of Game Freak, Junichi Masuda, revealed that if you breed pokémon who each come from a different country's version of the game (for example, breeding a pokémon from a North American cartridge with a pokémon from a Japanese cartridge), the resulting egg is more likely to have a shiny pokémon inside. Specifically, the egg is four times as likely to have a shiny pokémon inside than it normally would be. Because Masuda is the one who revealed this information, many fans now call this the "Masuda Method."
Statistically, one in 2048 eggs created using the Masuda Method will be shiny, or in other words, 0.048828125% of eggs created with the Masuda Method will be shiny. This is still pretty rare, but it is definitely better than one out of every 8192 eggs being shiny.
As mentioned above, being shiny only affects a pokémon's appearance, so using a shiny pokémon to create an egg won't increase your chance of getting a shiny egg.
Shiny Pokémon will be a different color from the usual color for that type of pokémon. However, sometimes the color difference is pretty subtle. The way to tell for sure is to look at the summary screen for your pokémon and see if there is a red star in the corner of its picture. The red star means that that pokémon is a shiny.
Also, when a shiny pokémon goes into battle, it will make a sparkling sound after it makes its usual pokémon sound, followed by a circle of stars appearing on it for a moment.