My friend and I were playing Magic the Gathering when he tried to summon Ghor Clan Rampager. He used all his mana to cast him, but in response I played Twiddle and tapped one of his lands. What would happen to his creature?
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That's impossible. There is no chance to use spend mana between the point where you start casting the spell and the point where you pay for it.
"In response to" means "after it has been cast but before it resolves". When Ghor Clan Rampager is cast, the lands have already been tapped and the mana has already been produced.
(Or did you mean in response to activating the land's ability? One cannot do anything in response to activating a mana source as it does not use the stack.)
You would have to cast Twiddle before he casts Ghor Clan Rampager (which means before he even mentions he's going to cast it), but then he could just tap the land for mana in response to you casting Twiddle, so it would have no effect.
The answer by @ikegami correctly describes the specific situation in this question. For the more general question (as stated in the headline), the answer is that if a player finds he is unable to pay any costs (e.g. because he miscalculated the mana available), the game will be restored to the point before the spell casting was started, see comprehensive rule 601.2.
Tapping mana doesn't go on the stack, so therefore you cannot react to it and tap his land before he creates mana off it. The casting cost of his spell is determined at the time of casting, and before putting the spell on the stack (casting) he has to pay the cost.
He has to tap his lands (for mana), something you cannot react to, then cast his spell. But the mana is already in his pool.
What probably happened (I see it a lot). He put his creature on the battlefield and he wanted to tap his lands, but you wanted to react to it. It should be the other way around, first tap for mana, then cast the card.
If somehow you got priority before he taps his lands (for example as a reaction on another spell), you can tap his lands with Twiddle. But even then, when you target his land with your spell, he can react to it and tap it himself (for mana).