Polukranos, World Eater

two generic mana one green mana Legendary Creature — HydraPower and toughness5/5

X generic mana one green mana : Monstrosity X. (If this creature isn't monstrous, put X +1/+1 counters on it and it becomes monstrous.)

When Polukranos becomes monstrous, it deals X damage divided as you choose among any number of target creatures your opponents control. Each of those creatures deals damage equal to its power to Polukranos.

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Polukranos, World Eater rulings

2013-09-15
The value of X in Polukranos's last ability is equal to the value chosen for X when its activated ability was activated.
2013-09-15
You choose the division of damage as you put the ability on the stack, not as it resolves. Each target must be assigned at least 1 damage. In multiplayer games, you may choose creatures controlled by different opponents.
2013-09-15
If some, but not all, of the ability's targets become illegal, you can't change the division of damage. Damage that would've been dealt to illegal targets simply isn't dealt.
2013-09-15
As Polukranos's triggered ability resolves, Polukranos deals damage first, then the target creatures do. Although no creature will die until after the ability finishes resolving, the order could matter if Polukranos has wither or infect.
2013-09-15
Once a creature becomes monstrous, it can't become monstrous again. If the creature is already monstrous when the monstrosity ability resolves, nothing happens.
2013-09-15
Monstrous isn't an ability that a creature has. It's just something true about that creature. If the creature stops being a creature or loses its abilities, it will continue to be monstrous.
2013-09-15
An ability that triggers when a creature becomes monstrous won't trigger if that creature isn't on the battlefield when its monstrosity ability resolves.

Polukranos, World Eater keyword rules

Monstrosity Rule 701.31

“Monstrosity N” means “If this permanent isn’t monstrous, put N +1/+1 counters on it and it becomes monstrous.”

Monstrous is a designation that has no rules meaning other than to act as a marker that the monstrosity action and other spells and abilities can identify. Only permanents can be or become monstrous. Once a permanent becomes monstrous, it stays monstrous until it leaves the battlefield. Monstrous is neither an ability nor part of the permanent’s copiable values.

If a permanent’s ability instructs a player to “monstrosity X,” other abilities of that permanent may also refer to X. The value of X in those abilities is equal to the value of X as that permanent became monstrous.