Final answer:
King Eurystheus did not count Hercules' task of cleaning the Augean Stables as one of the Twelve Labors because Hercules either received payment or because he used a river to clean the stables, which was seen as circumventing the expected manual labor.
Step-by-step explanation:
The reason King Eurystheus did not count Hercules' task at the Augean Stables is rooted in the narrative of the Twelve Labors of Hercules, a series of feats that Hercules was required to perform as penance for his actions.
According to certain versions of the myth, Eurystheus decided not to count the cleaning of the Augean Stables as one of the labors because Hercules had either been paid for the work or because he completed the task by rerouting a river to wash out the stables, rather than doing the cleaning by hand, which Eurystheus deemed as not meeting the labor's requirements.