Head of Mimir
There was a different story on how Odin gained knowledge from Mimir, but this version took place in a different circumstance that had nothing to do with the Well of Mimir.
After the war against the Vanir, the Aesir and Vanir exchanged hostages as a means of securing peace. Mimir was one of the hostages for the Vanir, as well as Hoenir (Vili), the brother of Odin. The Aesir received Njörd and Freyr, Njord's son, as hostages. The Aesir also received Kvasir, the wisest god of the Vanir. See Mead of Poetry to read about Kvasir's fate.
Hoenir was a handsome and noble looking As, but he was not too bright. Every decision Hoenir made seemed highly thought out, but only because Mimir was there to advise him. However, when Mimir was absent, Hoenir either gave strange advice during the meeting or said that he wanted to wait for Mimir's return.
The Vanir grew increasingly suspicious of Hoenir's intelligence. When their suspicions were confirmed that Hoenir was not really intelligent, the Vanir felt cheated. The Vanir were angry enough to cut off Mimir's head and send it back to the Aesir. The Vanir left Hoenir unharmed since he was Odin's brother.
Odin preserved Mimir's decapitated head with herbs so that the head would not decay. It seemed that for any knowledge Odin wished to gain, all he had to do was talk to the bodiless head. Odin often received counsel from Mimir's talking head.
See also the War of the Aesir and the Vanir, for more details about Mimir.
Related Information
Sources
Voluspa ("Sibyl's Prophecy") and Sigrdrifumal ("Lay of Sigrdifa") from the Poetic Edda.
Ynglinga Saga written by Snorri Sturluson.
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By Jimmy Joe