Inspired Idea

Cleave {3}{U}{U} (You may cast this spell for its cleave cost. If you do, remove the words in square brackets.)
Draw three cards. [Your maximum hand size is reduced by three for the rest of the game.]

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A cleave cost is an alternative cost that's paid instead of the spell's mana cost. Casting a spell for its cleave cost doesn't change the spell's mana value.

If a player has more cards in their hand than their maximum hand size during the cleanup step of that player's turn, that player discards until they have that many cards. A player's maximum hand size isn't checked at any time other than their own cleanup step.

If an effect allows you to “cast a spell without paying its mana cost,” you can't cast that spell for its cleave cost.

If multiple effects modify your hand size, apply them in timestamp order. For example, if you are affected by Null Profusion (an enchantment that sets a player's maximum hand size to two) and then resolve Inspired Idea without paying its cleave cost, your maximum hand size would be zero. However, if Null Profusion entered the battlefield after Inspired Idea resolved, your maximum hand size would be two.

If you cast a spell for its cleave cost, that spell doesn't have any of the text in square brackets while it's on the stack.

If you currently have no maximum hand size, then reducing your maximum hand size by three will have no effect. If the effect causing you to have no maximum hand size ends, Inspired Idea's effect will apply at that time.

You can't cast a spell for both its cleave cost and another alternative cost. For example, if an effect gives an Alchemist's Retrieval in your graveyard a flashback cost of {U}, you can't cast it from your graveyard for its cleave cost.