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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

For each library, the search is complete only when you explicitly say it is. For example, you can look through one player's library, set that library down, look at another player's library, choose a nonland card in the first library, then choose a nonland card in the second library. Don't reveal any cards from those libraries to any other player until you exile them.

When resolving Jace's third ability, you search each player's library (including yours) and exile the nonland cards before casting any of them.

Jace's first ability creates a delayed triggered ability that triggers whenever a creature an opponent controls attacks. It doesn't matter which player or planeswalker that creature is attacking.

You cast the cards by putting them on the stack one at a time, choosing modes, targets, and so on. The last card you cast will be the first one to resolve.

If a card has {X} in its mana cost, you must choose 0 as its value.

When casting a card this way, ignore timing restrictions based on the card's type. Other timing restrictions, such as “Cast [this card] only during combat,” must be followed.

If you cast a card “without paying its mana cost,” you can't pay alternative costs such as overload costs. You can pay additional costs such as kicker costs. If the card has mandatory additional costs, you must pay those.

Piles can be empty. If one of the piles is empty, you choose to put all the revealed cards in your hand or on the bottom of your library.

You pick one of your opponents when Jace's second ability resolves. The ability doesn't target that opponent. All players may see the revealed cards and offer opinions. You (not your opponent) choose which pile is put into your hand and which pile is put on the bottom of your library.

If you can't cast a card, perhaps because there are no legal targets available, or if you choose not to cast one, it will remain exiled. Jace's ability won't allow you to cast it later.