Big Three

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The Big Three is a collective term which refers to the gods Zeus, Poseidon and Hades. After World War II, the Oracle predicted that a demigod child of the big three would destroy or save Olympus if or when they turned 16 so they swore on the River Styx not to have any more children with mortals however both Zeus and Poseidon broke this promise (with Thalia and Percy, respectively). There is a fierce rivalry between the three. Technically, Nico and Bianca were already born before the oath, so Hades remains the only one who has not broken the oath.

The Big Three
The Big Three are the three most powerful gods in all of Olympus: Zeus, God of the Sky, Hades, God of the Underworld, and Poseidon, God of the Seas. The Big Three are equally matched in power for the most part, with Poseidon commanding the seas which cover most of the Earth, and Hades being the God of Underworld, representing every person who has ever died and now is a ghost. Despite Poseidon and Hades' large power, Zeus commands the sky which covers the entire Earth, but has few inhabitants, and even fewer are sentient, while Poseidon and Hades have many more creatures in their domains both non-sentient (fish) and sentient (the dead and mermaids).

The land of Earth (Gaia) is split between the gods and is no one's specific domain, though Zeus is initially believed to control the earth. However that is debatable since Demeter is Goddess of the Earth also. Also the other gods are not to be underestimated because the Big Three only get their raw power from how high they are among the Olympian hierarchy.

Children Of the Gods
The Big Three are the most powerful Gods in Olympus, therefore their demigod children are the most powerful Half-Bloods. The Half-Bloods were causing big trouble during World War II, with the sons of Zeus and Poseidon on one side and Hades' sons on the other. The final straw came when the Oracle predicted the "Great Prophecy". The Prophecy said that a child of the Big Three would decide the fate of all Olympus on their 16th birthday.

When they heard it, Zeus, Hades and Poseidon agreed not to have any more kids from then on, swearing on the River Styx.

Despite them all agreeing not to have any more demigod children (with the exception of Hades who already had two children before the oath), Zeus "fell off the wagon" and had Thalia, and five years later, Poseidon had Percy. Hades having Nico and Bianca is not against the rules since they were born before the oath was made, but were frozen in time in the Lotus Hotel and Casino. It's also hinted that Poseidon might have more kids in the fifth book, The Last Olympian.

Percy had been living with his mom until he was 12 and arrived at Camp Half-Blood. During World Word II Zeus "quaintly" tried to kill Bianca and Nico by obliterating the whole building the were in with a lightning bolt, killing their mother and Hades' lover, Maria di Angelo. After that Nico and Bianca mysteriously disappeared until they were found in "The Titan's Curse". All the children of the Big Three are hated by the other two gods; Percy was hated by Zeus and Hades in The Lightning Thief, though they have grown to dislike him rather then hate him. Hades hates Thalia and tried to kill her, but in the end Zeus spared her and turned her into a pine tree.

At the end of The Last Olympian, Percy relieves the Big Three of their oath.

Children of the Big Three
As said before, all of the Big Three agreed not to have children, but Poseidon and Zeus did. Hades already had two kids when they swore the oath, so technically he never broke it. The worst he did was never sending Nico and Bianca to Camp Half-Blood.

Nico and Bianca
Hades already had Nico and Bianca when the Big Three made their agreement not to have any more kids, for fear of the Great Prophecy. Zeus gave Hades a week to send them to Camp Half-Blood, but Hades never did as he knew that Zeus would destroy them the minute they were out of Hades' territory. It was a week after the deadline and Hades was still trying to convince their mother Maria to let him take them to the Underworld so he could protect them. Maria repeatedly said no, and the last time Hades saw her, she was going upstairs to get her purse. Zeus then struck the building with a lightning bolt, completely destroying it; Hades sensed the attack and had enough time to put a shield around Nico and Bianca but Maria was killed. It was then that he told one of the Furies, Alecto, to take Nico and Bianca to the Lotus Hotel and Casino, where they would be hidden until he sent for them.

Thalia
Thalia was the first demigod born after the Oath, but she ended up becoming the tree that guards camp Half-Blood for almost seven years. She was revived in book two, The Sea of Monsters, when her tree (the source of the magical border around Camp Half-Blood) was poisoned by Luke. Percy, Clarisse, Grover, and Tyson retrieved the Golden Fleece from the sea of monsters.

However the Fleece worked too well and Thalia was revived from the tree. She later joined Artemis' hunters, meaning that she would not age and thus couldn't be the Half-blood in the Great Prophecy.

Percy
Percy was the last Half-blood of the Big Three to be born after the Oath. He is the son of Poseidon and Sally Jackson, and originally did not know anything about his godly heritage. He is probably the most powerful demigod in Camp Half-Blood. Percy's presence was revealed at the worst possible time, as Zeus's Master Bolt, his symbol of power, and Hades's helm of darkness had been stolen. The appearance of a Son of Poseidon made Hades and Zeus suspicious, as Gods can't go into each others' territory, but their children could.

Percy found both of the items and returned them, successfully preventing a war. Since then, Zeus and Hades have stopped hating him and started to merely dislike him. He is more powerful than all other Half-Bloods. It is also implied that the Big Three had already had children that no one knew about at the end of The Last Olympian.