Start at olivine city and pretend your going to Cianwood city. The small island is whirl islands. First you need whirlpool
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go to whirl islands
Whirl Islands only exist in Gold, Silver, Crystal, HeartGold and SoulSilver.
In heartgold, Lugia is obtainable at the Whirl Islands but u need to obtain the silver feather from a guy in Pewter City. Once u have that silver feather, u can go deep into the Whirl Islands and find Lugia.
First you must have a badge from the Mahogany Town gym leader (he uses ice types). Next you must find the HM Whirlpool. Equip Whirlpool to a Pokemon and go to one of the Whirl Islands and use the move Whirlpool on a whirlpool.
A person in Pewter City will give you the Silver Feather, then go to Whirl Islands. When you get to the end of the cave you will see Lugia.