I play Yugioh at school and the main person I play against has a Blue-Eyes deck. He uses Maiden with Eyes of Blue and says I can't destroy it. But I read somewhere that you can banish it.
Is this correct?
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Let's read Maiden with Eyes of Blue's effect, emphasis mine:
When a card or effect is activated that targets this card (Quick Effect): You can Special Summon 1 "Blue-Eyes White Dragon" from your hand, Deck, or GY. When this card is targeted for an attack: You can negate the attack, and if you do, change the battle position of this card, then you can Special Summon 1 "Blue-Eyes White Dragon" from your hand, Deck, or GY. You can only use 1 "Maiden with eyes of Blue" effect per turn, and only once that turn.
We can see that nowhere on it's card text reads that it cannot be destroyed. Thus, otherwise there is another card your opponent has, the Maiden is not indestructible by herself.
When you target her with any effect, she special summon a Blue-eyes, but she does not negate that effect.
When you attack her, she negates the first attack, and special summons a Blue-eyes. However, if you attack her again she will take the hit.
In short, you can target and destroy the Maiden by various means, as nowhere on it's text is indicated otherwise. Just, be ready to handle the Blue-Eyes White Dragon that comes with you doing that.