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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.

When using improvise to cast a spell with {X} in its cost, first choose the value for X. That choice, plus any cost increases or decreases, will determine the spell's total cost. Then you can tap artifacts you control to help pay that cost. For example, if you activate the last ability of Archway of Innovation and then cast Wrath of the Skies (a spell with mana cost {X}{W}{W}), choosing 3 as the value of X, the total cost is {3}{W}{W}. If you tap two artifacts, you'll have to pay {1}{W}{W}.

If an artifact you control has a mana ability with {T} in the cost, activating that ability while casting a spell with improvise will result in the artifact being tapped before you pay the spell's costs. You won't be able to tap it again for improvise. Similarly, if you sacrifice an artifact to activate a mana ability while casting a spell with improvise, that artifact won't be on the battlefield when you pay the spell's costs, so you won't be able to tap it for improvise.

You must already control an Island as Archway of Innovation enters the battlefield for it to enter untapped. If it enters the battlefield at the same time as an Island when you control no other Islands, it will enter tapped.

Multiple instances of improvise are redundant.

Because improvise isn't an alternative cost, it can be used in conjunction with alternative costs.

When calculating a spell's total cost, include any alternative costs, additional costs, or anything else that increases or reduces the cost to cast the spell. Improvise applies after the total cost is calculated. Improvise doesn't change a spell's mana cost or mana value.