You can't just match it up by completing a task, you'll have to be extremely lucky as the game generates a random number everyday, if this number matches up with your ID number only then will you be able to visit Mirage Island.
You do not need any items. All you need is a very hard luck. The ID number of the first Pokémon in your party should match with the floating point number of the mirage island. If it matches, talk to the man in Pacifidlog. You can now travel to mirage island.
Just go to the guy in pacifidlog town everyday and eventually you will be able to go and it is eastward from where you are his house * Every day, Mirage Island is given a random number. In order for you to be able to go there, the number has to match a number from a Pokemon in your party. It's super rare to get a match, and most trainers will never gain access to the island. Just adding to the previous answer (:
You don't need a specific species of Pokemon. Every day, a random number between 0 and 65535 is generated by the game. If this number matches the personality value of any of the Pokémon in the party, Mirage Island will appear in the game. To check if the numbers match, there is a man in Pacifidlog Town who tells you.
You need to keep trying to put different Pokemon in your party, but Pokemon with different ID numbers. There is a random ID number for each game to let you see mirage island, and if the ID number is one of the Pokemon in your party, then the man will see mirage island, and you can swim towards it.
Mirage island is extremely hard to get to. Each day, Mirage island gets assigned a random number. To get there, one of your party pokemons must have the same number as mirage island, making the chances very small. However, you can get there, and if you're as lucky to get there, you can catch a bunch of Wynauts there
Its really difficult see heres how you do it, in lilycove city in the dept store you can enter the lottery the numbers must match an ID number of a Pokemon in your party ALL digits you will win a masterball then go to pacifidlog town and go to the house farthest bottom right i believe you will find a old man looking to the right if you speak to him he will mention he can see mirage island quickly leave then use surf and go east and there it is mirage island.
Every day mirage island is given a number (with i think 7 didgits) and if you have a Pokemon in your party with the same ID number then mirage island will b visible, so as you can tell its incredibley unlikely, i have been myself theres not a lot there besides wynaut and a berry (only 1 of these in the whole game) but plenty of people don't ever find this island
You don't need any item, but mirage island is assigned a random # daily that coordinates with the # of one pokemon in the game, if you have that pokemon in your party with the corresponding # of mirage island, you can see it and travel to it.
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