Seven-headed beast is a popular expression that means you are facing some difficulty that may not have a simple resolution. It's something that's almost impossible to solve.
When a situation arises that becomes a big problem and a quick solution is not envisioned, it is said that the situation is an animal with seven heads. For example: Learning Finnish was becoming a beast.
The expression is also used as a way to reassure someone who is afraid of facing an apparently complicated situation or to minimize the seriousness of a problem. Example: Getting a driver's license is not a seven-headed beast.
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The expression "seven-headed beast" is believed to have its origins in the almost invincible Hydra of Lerna, a character in Greek mythology. According to legend, Hidra of Lerna was a gigantic serpent that had many heads (seven or nine) and inhabited the swampy region of Lerna, in ancient Greece. She was an extremely dangerous animal and difficult to be extinguished because when one of the heads was cut off another one was reborn in its place. It fell to the Greek hero Heracles (Hercules to the Romans) the mission to defeat the monster. To prevent the renewal of heads, Hercules incinerated each cut head. It was in this way that he managed to defeat the terrible beast.