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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.

Tamiyo’s ability affects sacrifices, but not any other ways permanents can leave the battlefield. It won’t stop a creature from dying due to lethal damage or having 0 toughness, and it won’t stop a permanent from being put into its owner’s graveyard due to the “legend rule.” None of these are sacrifices; they’re the result of game rules.

If a spell or ability an opponent controls states that something happens unless you sacrifice a permanent (as Mogis, God of Slaughter does) or discard a card (as Painful Quandary does), you can’t choose to sacrifice or discard. On the other hand, if a spell or ability an opponent controls instructs you to sacrifice a permanent unless you perform an action (as Killing Wave does) or discard a card unless you perform an action, you can choose whether or not to perform the action. If you don’t perform the action, nothing happens, since the spell or ability can’t cause you to sacrifice any permanents or discard cards.

If a spell or ability your opponent controls reduces your maximum hand size, Tamiyo’s first ability won’t stop you from discarding cards when the game rules cause you to discard during your cleanup step.

As a spell or ability an opponent controls resolves, if it would force you to sacrifice a permanent or discard a card, you just don’t. That part of the effect does nothing. If that spell or ability gives you the option to sacrifice a permanent or to discard a card, you can’t take that option.

If Tamiyo has 3 loyalty, you can’t activate her last ability to return her to your hand. She won’t be in your graveyard while you’re choosing targets for the ability.