The Wikipedia page suggest that in the epic, Gilgamesh is a demigod of superhuman strength.
How do we know he was a demigod? What makes someone a demigod?
Did he possess some special powers that make him a superhuman?
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In general, English "demigod" refers to a person who is partially divine and partially human. Most typically, this refers to someone who has one divine and one human parent (like Hercules, or the Pandavas), making them 50% divine. But the case of Gilgamesh is slightly different. Rather than being half-divine, Gilgamesh is in fact two-thirds divine.
When the gods created Gilgamesh they gave him a perfect body. Shamash the glorious sun endowed him with beauty, Adad the god of the storm endowed him with courage, the great gods made his beauty perfect, surpassing all others, terrifying like a great wild bull. Two thirds they made him god and one third man.
From a mathematical perspective, you can only be two-thirds something if 1.) you have a very finely-crafted family tree extending infinitely far back [i.e. successive binary approximations of 2/3]; or 2.) you have loops in your family tree [which requires time travel, probably]. (This is because 2/3 is not a binary fraction).
So how is it that Gilgamesh was two-thirds divine if born from a divine mother (Ninsun) and a mortal father (Lugalbanda)? I don't know, but there you have it. I can't imagine mathematical accuracy would have been a priority of the Sumerians when they crafted the Epic.
The definition for demigod differs from culture to culture. But the definition of demigod, which can be used here in the case of Gilgamesh would be "half-god".
Gilgamesh was a Sumerian king who wished to become immortal. Endowed with superhuman strength, courage, and power, he appeared in numerous legends and myths, including the Epic of Gilgamesh.
From Myth Encyclopedia article on Gilgamesh (emphasis mine):
According to legendary accounts, Gilgamesh was the son of the goddess Ninsun and of either Lugalbanda, a king of Uruk, or of a high priest of the district of Kullab.
The fact that his mother was a goddess and his father, a mortal, suggests than Gilgamesh was a demigod. And his superhuman abilities could have been due the fact that one of his parents was a god. In the Epic, his strength and feats were described as superhuman.
I read this story and thought about it for many years, and always try to figure out how Gilgamesh with 2/3 God. With his genealogy being a mortal man and mother a goddess. And after much. The only way I could figure out that he was 2/3 God, is that his father was a demigod, either the son of shamash or the son of Adad and his mother the goddess was full blood God and was the daughter of either shamash or Adad. And in this way one of Gilgamesh parents his mother would be 100% God and his father would be 50% God and in Gilgamesh that would make him roughly two-thirds.
And the last and maybe not the most known or common way of making him 2/3 God, it is through genetics like how people who want kids today go pick their partners and the DNA is manipulated to create boys and girls or specific eye colors and things like that. And do these things happen many thousands of years ago if you read the Lost Books of enki and you read the way they described the process in creating humans that we see today, I would explain but it's a in-depth and in detail process and it would take mini paragraphs. But pretty much they were mixing certain Siemens an incubating women's eggs in clay vessels and then putting the eggs and goddesses to carry them. But with this process they explain it is very possible to create a precise genetic specimen. So the first answer is one that may be very well correct and if it isn't that it is this way as explained in this answer.