Do I have to tap Monolith for mana before I cast Transmute Artifact, or can I wait until it resolves and tap it before sacrificing it? I'm guessing this doesn't work but just wanted to make sure.
Neither. You can tap it before or after casting it, but you can't tap it once Transmute Artifact begins to resolve.
(I'm assuming you're not trying to use the mana to cast Transmute Artifact.)
When you add something to the stack, you automatically pass priority unless you say you're keeping it. Since you passed, if your opponent passes, that thing resolves. If you say you retain priority, you can tap the Monolith but then your opponent gets a chance to respond.
If your opponent responds, though, you'll also get a chance to respond - at that point you could tap Monolith.
When Transmute Artifact resolves, you may choose to sacrifice Grim Monolith to the effect.
“It's possible. But it involves... {checks archives} Nature's Revolt, Opalescence, two Unstable Shapeshifters (one of which started as a Doppelganger), a Tide, an animated land, a creature with Fading, a Silver Wyvern, some way to get a creature into play in response to stuff, some way to get a land into play in response to stuff (a different land from the animated land), and one heck of a Rube Goldberg timing diagram.”
-David DeLaney
So you have to tap it either before you cast it, or retain priority if you want to float the mana before it gets sacrificed on resolution. If Transmute gets countered, you have a tapped monolith and 3 mana.
EDIT: HOWEVER... I just realized you can use the 3 mana + 1 additional mana (that you needed for Wurmcoil anyway) to simply untap Monolith. Not as much of a blowout as I feared.
Remember also that a) Transmute doesn't have any targets, so you make all choices on resolution, b) the sacrifice isn't optional. If they, say, destroy your Monolith in response to Transmute you have to sacrifice a different artifact if you have one. At least Transmute lets you pay additional mana during resolution if your plan changes.
“It's possible. But it involves... {checks archives} Nature's Revolt, Opalescence, two Unstable Shapeshifters (one of which started as a Doppelganger), a Tide, an animated land, a creature with Fading, a Silver Wyvern, some way to get a creature into play in response to stuff, some way to get a land into play in response to stuff (a different land from the animated land), and one heck of a Rube Goldberg timing diagram.”
-David DeLaney
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