the sumerians often carved the stone steal to identify the owner of an object especially cuneiform writing
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Lord Voldemort wanted to steal the philosopher's stone so he could return to power. He was sharing his body with Professor Quirrell who was attempting to help him steal the stone.
Harry thought that Professor Severus Snape was trying to steal the stone for Lord Voldemort. It was actually Professor Quirrell.
The Arkenstone
No. Professor Snape knew that Professor Quirrell was trying to steal the Philosopher's Stone and was trying to protect it.
It was called the Philosopher's Stone. In some editions this is known as the Sorcerer's Stone.
Hack and steal Lucian's, perhaps?
He didn't. Only someone who wanted the stone, but didn't have the intention of using it, could get the stone through the mirror of Erised.
Yes. Shortly afterwards, Professor Quirrell attempted to steal the stone from the vault. Due to his failed attempt Lord Voldemort decided to share his body to keep a closer eye on him while they worked to steal it from Hogwarts.
The Philosopher's Stone was destroyed to prevent Lord Voldemort, or anybody else trying to steal it.
Severus Snape was going to the third floor trapdoor to make sure Professor Quirrell was not able to go down and steal the philosopher's stone. Although, Harry and Ron thought that Snape himself was trying to steal the stone that night.
Yes. Throughout the book we are led to believe that Severus Snape is the person trying to steal the stone. We then discover that Professor Quirrell is actually trying to steal it. We are also held in suspense during the last chapter when we're not sure if Harry can save the Stone or if Voldemort will get it and kill him.