From Nookipedia, the Animal Crossing wiki
Wisp (ゆうたろう, Yūtarou) is a character in the Animal Crossing series. He is absent from Doubutsu no Mori, Animal Crossing: Wild World, and was initially absent from Animal Crossing: New Leaf until the Welcome amiibo update. In the earlier installments, he is a spirit, while in Animal Crossing: City Folk, he is a genie. He appears at night, usually between 12 and 4 a.m. (8 p.m. to midnight in Animal Crossing: City Folk).
In Animal Crossing, Wisp's appearance is governed by a player's play history, and if certain conditions are met he will appear in a random acre between 12:00 AM and 4:00 A.M. Wisp can appear concurrently alongside other special visitors such as Katrina, and must be sought out by the player when in town; Copper will not mention anything about Wisp. To determine whether or not Wisp will appear the game keeps track of how often the player loads their game within a given time period. While the exact algorithm used to determine Wisp's appearance is unknown, the following logic provides an approximate representation:
The game checks the value of the 'Wisp counter'. If this value is 3 or greater, Wisp will appear. The value of the counter starts at 0, and increases by 1 the first time the game is loaded and saved on a given day. The value is reset if: the player time travels backwards any length of time, the player goes more than 7 days without loading their game, the month changes, or the player finds Wisp and captures all five spirits.
Therefore, it is possible to see Wisp at maximum every three days. This also means that the player is in direct control of Wisp's appearance and it is not based on specific dates, nor is it random, and it is not affected by extraneous factors such as town status, or the amount of weeds in a player's town.
Wisp will not be initially visible when the player enters the acre in which he is hiding. He makes himself known by calling out to the player, and will only appear when the player reaches the spot he is hiding at. Wisp will ask the player to catch five spirits found around town with their net. After the player catches the five spirits, Wisp will offer them one of three rewards as thanks:
Animal Crossing: City Folk
In Animal Crossing: City Folk, on certain nights the player can find an Empty lamp somewhere in the town. After the player finds the lamp and returns it to Wisp, the genie will ask the player to rub the lamp in their attic, where he will offer to grant them one wish. The player does not have to rub the lamp straight away, and can save the wish for later use. As in his original appearance, Wisp will offer three choices:
Animal Crossing: New Leaf
In Animal Crossing: New Leaf, a magic lamp will appear randomly while walking around town. If the player places the lamp in their house, Wisp will appear. Unlike Animal Crossing and Animal Crossing: City Folk, Wisp helps with amiibo. Wisp can only be used once a day. While Wisp did not appear in Animal Crossing: New Leaf at the game's launch, he was added as a major character in the Welcome amiibo update. When the player rubs the lamp, Wisp will appear and the player can use an amiibo card. When the amiibo is scanned, the player can choose for the villager to move to their town or to get something, usually furniture.
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