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Craft abilities are activated abilities with costs that have a mana component as well as an additional "materials" component.

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Craft abilities are written "Craft with [materials] [mana]", which means "[Mana], Exile this permanent, Exile [materials] from among permanents you control and/or cards in your graveyard: Return this card to the battlefield transformed under its owner's control. Activate only as a sorcery."

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If a card that isn't a transforming double-faced card becomes a copy of a card with craft, it'll stay in exile if you activate the craft ability. It won't return to the battlefield.

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If a resolving spell or ability instructs a specific creature to explore but that creature has left the battlefield, the creature still explores. If you reveal a nonland card this way, you won't put a +1/+1 counter on anything, but you may put the revealed card into your graveyard. Effects that trigger "whenever a creature explores" trigger as appropriate.

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If an ability instructs a creature to explore, its controller reveals the top card of their library. If it's a land card, they'll put it into their hand. Otherwise, they'll put a +1/+1 counter on that creature, then choose to either leave that card on top of their library or put it into their graveyard.

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If no card is revealed, most likely because that player's library is empty, the exploring creature receives a +1/+1 counter.

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If the materials required include multiple objects, you may exile some of them from among permanents you control and the rest from among cards in your graveyard. You don't have to choose all permanents or all cards from your graveyard.

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In some unusual cases, noncreature permanents may explore. For example, if the creature card returned by Defossilize is somehow not a creature once it's on the battlefield, it can still explore. You'll take all the same actions, and you may end up putting a +1/+1 counter on the permanent. (Note that some effects target a creature, and those effects would still require a legal target to have it explore.)

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Lodestone Needle's enters-the-battlefield ability can target an artifact or creature that's already tapped. If so, you'll just put two stun counters on that artifact or creature.

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Once an ability that causes a creature to explore begins to resolve, no player may take any other actions until it's done. Notably, opponents can't try to remove the exploring creature after you reveal a nonland card but before it receives a counter.

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Some spells or abilities might cause a creature to explore multiple times in a row. If you reveal a nonland card when a creature explores and leave it on top of your library, then the creature explores again immediately afterwards, you'll reveal the same card again.

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The back faces of some cards with craft refer to cards "used to craft" it. This refers to the cards exiled as part of the cost of the craft ability of the front face. Those cards are considered to be "used to craft" that permanent as long as they remain exiled and the permanent remains on the battlefield, even if the permanent's controller changes or some of its characteristics change (because of a copy effect, for example.)

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You may exile tokens you control as part of the materials required. However, because they aren't cards and won't stay in exile, any abilities that refer to what you "used to craft" the back faces won't refer to anything.