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All of the Princes are related.
It's very possible that the original Prince could be the father of the Sands of Time
Prince, while the New Prince from the 2008 reboot could be a distant cousin.
- Alternately, the new prince could be the first of the line of action-guy Persian princes. He may be a random tomb robber now, but he has just become very friendly with a real live princess. If they get married, he will become a prince, and his male descendants will be princes also.
If Sand Monsters are near the Dagger of Time, they are immortal.
In the Sands of Time when a Sand Monster is killed, it revives unless it's sand is absorbed. When the Prince loses the dagger, the monsters stay dead when killed instead of being revived. In Warrior Within the Prince doesn't have the Dagger and the monsters die when you kill them. Clearly, this means being near the Dagger gives them a bizzare immortallity.
- The sword that the Prince found at the end of Sands has some sort of nasty effect on Sand monsters: they die in one hit, whether he has the dagger or not(which makes it very difficult to collect sand when you have it). In Warrior Within, they aren't sand zombies, they're the armies of the Empress of Time. Perhaps everything from that island releases Sand when killed.
We've been going backwards through the series
The 2008 Prince is the first one, he marries Elika and they have a child, the child is either the Sands of Time Prince or his father. Then The SOT Prince has a son, the Classic Prince.
The Ubisoft games, at least. The 2008 game takes place thousands of years ago. Ahriman and Ormazd are two of "Those who came before", and Elika and the Prince have quite a lot of TWCB blood in their veins, and the much later Sands of Time Prince is of the same bloodline. The Empress of Time is either one of TWCB, or a powerful weapon invented by them. Same goes for the Dahaka. The hourglass, medallion and the dagger are all TWCB artifacts and the human bad guys are all Templars.
- But if this theory was true, is the dagger a Piece of Eden or some random artifact?
- The Sands could be a powerful bioweapon, but it would be nigh-impossible for even the Templars to wipe the Indian invasion of Babylon from history. Remember, the world's history, in AC, is basically the same except for the bits the Templars altered.
Malik from Forgotten Sands is The Mourning King
Both of them look pretty similar, and wear similar clothes/armour. All we see of Malik at the end of Forgotten Sands is him collapsing. He went into some sort of coma, and when he woke up, didn't tell his family that he was still alive out of shame and embarrassment at his possession. He fled to another kingdom, eventually became its ruler, and this led to the events of the 2008 Prince of Persia game.
The trips to the fountain of life in the first game are really an early representation of the prince fighting the very onset of the darkness that would take the form of the dark prince two games later
The game seems to make a point of giving you an easy to find fountain when you have Farah near the end of the first game, and she doesn't seem to notice you were gone. Not to mention that each hole leading to the fountain always disappear as if they never existed. Its really just a glimpse into the prince's soul and partaking of the water is a means of countering the corruption the sands bring onto him.
The builders for the locales were slaves...
- Forced to build, they took the instructions literally, creating the Benevolent / Malevolent Architecture.
King: "I want columns that reach the [two-storey] ceilings!"
King: "What is this?! The columns are just hanging from the ceiling!"
Spokes-slave: "You said you wanted the columns to reach the ceiling. You never said the columns had to reach the floor."
King: "... If I compensated you for your labour, I'd dock your wages."
Spokes-slave: "Hey, you get what you pay for."