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If an effect allows you to cast a spell without paying its mana cost, you can't choose to cast it and pay unless another rule or effect allows you to cast that spell for a cost. Similarly, you can't waive a cost reduction unless that effect says you may.

Adamant effects check what mana was actually spent to cast a spell. If an effect allows you to spend mana “as though it were mana” of any color or type, that allows you to spend mana you couldn't otherwise spend, but it doesn't change what mana you spent to cast the spell.

If you copy a spell that has an adamant ability, no mana was spent to cast the copy at all, so that ability won't apply.

You can't cast Outmuscle unless you choose both a creature you control and a creature you don't control as targets.

Because state-based actions don't destroy a creature with lethal damage marked on it until a player would receive priority, gaining indestructible from Outmuscle will save your creature if it's dealt lethal damage in the fight.

If an instant or sorcery spell has an adamant ability, you perform the spell's instructions in order. You don't perform the adamant instruction before the spell resolves or before any other effects printed above it.

If either target is an illegal target as Outmuscle tries to resolve, neither creature will deal or be dealt damage.

If the creature you control is an illegal target as Outmuscle tries to resolve, no creature gets a +1/+1 counter. If that creature is a legal target but the creature you don't control isn't, it still gets a +1/+1 counter and gains indestructible if appropriate.