Why 3 days?
I can't understand it by reading the story, but maybe someone else can?
Why that number of days.
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Sefer Manos HaLevi says that these 3 days are related to 3 sins regarding which Esther expected to be held responsible: eating prohibited food, her relation with Ahashverosh, and some level of abetment in the death of Hathach.
P.s. This article made me raise my eyebrows with science referring to three days of fasting with regeneration of the immune system. But maybe there is still some skepticism about it and that's for another question.
more than 3 days without water is life threatening.
so it is the maximum fast
If you're ever stuck out in the wilderness, remember what survival experts call 'the Rule of Threes'.
You can live 3 minutes without air, though we don't recommend trying. In a harsh environment — it's snowing, say — you have 3 hours to survive without shelter. After 3 days, you need water or you'll perish. You can make it 3 weeks without food, though we promise you that won't be fun.
Talmud (b. Megilah 15A) indicates that Esther suspended the 3-day Feast of Passover (13-15 Nisan). The First Targum Esther corroborates this oral tradition of the Jews.
First Targum Esther 4:17:
ונעם מרדכי ועבר על חדוות חגא דפסחא צומא גזר ויתיב על קיטמא ועבד ככל דפקידת עילוהי אסתר
So Mordecai wailed, and passed the joy of the Feast of Passover [as] the decreed fast. He thus sat upon ash, and made all do likewise which Esther had commanded him.
The significance of the 3-day fast was alignment to the Exodus narrative, when the Jews were saved from evil Pharaoh. That is, like Pharaoh and his army, the evil Haman had perished on the same day that the Egyptian Army perished in the Red Sea. In this regard, the Jewish Bible scholar Adele Berlin (2001) makes the following astute observation in her commentary on Esther.
In summary, the narrative of Pharaoh's desire to obliterate the Jews and the narrative of Haman's desire to destroy the Jews are chronologically parallel.
To commemorate the festival of Purim, the dates of Purim were moved one month earlier from Nisan to the month of Adar so as not to interfere with the celebration of the Feast of Passover (and yet still preserve the memory of the salvation of the Jews not from Pharaoh but from Haman).
Berlin, Adele (2001). The JPS Bible Commentary: Esther. Philadephia: Jewish Publication Society, 27.