From Bulbapedia, the community-driven Pokémon encyclopedia.
The Odd Egg (Japanese: なぞのタマゴ Odd Egg) is a specific gift Egg the player receives in Pokémon Crystal. The Pokémon it hatches into has a higher-than-usual chance of being Shiny.
The Egg can hatch into Pichu, Cleffa, Igglybuff, Tyrogue, Smoochum, Elekid, or Magby; regardless of what it will hatch into, it will take 20 Egg cycles, and the hatching Pokémon will know Dizzy Punch.
In the Japanese version of Pokémon Crystal, the player must have an Egg Ticket in order to obtain the Odd Egg. The Egg Ticket was distributed as an event via the Mobile System GB and could be picked up from the Day-Care Man at the Pokémon Day Care on Route 34 if the player had received the event. The player could exchange the Egg Ticket for the Odd Egg by talking to the Pokémon Center Nurse in the Trade Corner at the Pokémon Communication Center in Goldenrod City. Because the Mobile System GB is inaccessible in all Virtual Console releases of Pokémon Crystal, players of the Japanese Virtual Console version of the game cannot receive an Odd Egg.
In the international versions of Pokémon Crystal, the player could receive the Odd Egg from the Day-Care Man at the Pokémon Day Care on Route 34. This change was likely due to the Mobile System GB not being a part of the international version of Pokémon Crystal.
Pokémon hatched from the Odd Egg can have one of two different sets of possible IVs (Attack/Defense/Speed/Special): 0/0/0/0 (not Shiny) or 2/10/10/10 (Shiny). Due to gender being determined by IVs in the Generation II games, this means that all hatched Pokémon will always be female (except Tyrogue, which is always male). Additionally, Shiny Pokémon hatched from the Odd Egg will always have Grass-type Hidden Power with power 49, while non-Shiny Pokémon will have Fighting-type Hidden Power with power 31.
All Pokémon hatched from the Odd Egg hatch with 125 total experience; while this is the minimum experience for a level 5 Pokémon in the Medium Fast experience group experience group (most possible hatched Pokémon), this results in 25 excess experience for Pokémon in the Fast experience group (Igglybuff and Cleffa), which only need 100 experience to be at level 5.
In the Japanese version of Pokémon Crystal, there is a 50% chance the hatched Pokémon will be Shiny (IVs 2/10/10/10) and a 50% chance it will not (IVs 0/0/0/0).
In the international versions of Pokémon Crystal, there is a 14% chance the hatched Pokémon will be Shiny (IVs 2/10/10/10) and an 86% chance it will not (IVs 0/0/0/0), but the chance of any particular Pokémon species hatching from the Egg and the chance of any given species being Shiny are not uniform.
Poké Transporter cannot transfer these Pokémon to Pokémon Bank if they know the move Dizzy Punch. This apparent oversight is presumably due to the fact that the Odd Egg is not obtainable in the Japanese Virtual Console version of Pokémon Crystal, making it impossible for these Pokémon to know Dizzy Punch in the Japanese Virtual Console games.
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