Why on Chanukah do we play with a dreidel and on Purim a gragger?
The miracle of Chanukah was above the laws of nature. The Jewish people were the minority and the Greeks were the majority; we were the weak and they were the strong. Nevertheless, thanks to heavenly intervention, the miracle took place and the Jews were the victors.
On Purim, the miracle was clothed entirely within the laws of nature. The Jewish people gathered in prayer and fasting. Esther pleaded their case before the king. Out of love for his Queen, he killed Haman her arch enemy.
Since the miracle of Chanukah came down from above, we spin the dreidel with the handle on top. The miracle of Purim was through an awakening from below consequently we turn the gragger with the handle below.
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"The students asked Rabbi Shimon Bar Yochai, 'Why were the Jews destined to be annihilated in the days of Achashveirosh?' He said to them, 'What is your opinion?' They replied, 'Because they derived pleasure (ate) from the meal of the wicked person.' "
(Gemara, Megillah 12a)
According to Biblical Law, death is not the punishment for eating non-kosher?
The prophet Yirmeyahu prophesied that seventy years after the destruction of the first Beit Hamikdash, Hashem would bring back the Jewish people from exile. According to Achashveirosh's calculation, the seventy year period had already passed, and since there was no redemption, Hashem had obviously, G-d forbid, forsaken His people. To celebrate his victory, he used the vessels of the Beit Hamikdash and arranged lavish festivities to which he invited everyone, including the Jewish people (Gemara, Megillah 11b).
The food and drink served to the Jews was strictly kosher, even according to the most stringent standards. However, instead of attending the festivities reluctantly, with a sense of grief, "nehenu" -- they had a sense of "pleasure" because they were accepted to Persian high society. Such behavior angered Hashem, to the extent that He considered annihilating the Jewish people, G-d forbid. Had the Gemara meant that their iniquity was the eating of the food, it would have written "she'achlu" -- "Because they ate from his meal."
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Why do we eat "krepchen" (meat covered with dough) on Purim?
On Yom Tov it is forbidden to do any work (except food preparation) and it is a mitzvah to experience the utmost joy. Thus it is customary to eat meat during Yom Tov, because it brings one into a joyous state of mind.
On Purim, one must be joyous and refrain from doing work, but the prohibition is not the same as on other yamim tovim. Therefore, we eat meat in order to bring us into a joyous mood, but because it is not a complete Yom Tov and it is permissible to work if needed, we cover the meat with dough. (This also explains why krepchen are eaten on Erev Yom Kippur and Hosha'ana Rabbah.)
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"If one reads the Megillah backwards, he has not fulfilled his obligation."
(Gemara, Megillah 17a)
Why would one do such a strange thing?
The purpose of reading the Megillah is to bring to our attention that in all generations there are Hamans who plot our destruction. Miraculously, Hashem comes to our salvation thanks to our adherence to Torah and mitzvot.
"Reading the Megillah backwards," means thinking that the Purim narrative is a historical event with no contemporary relevance. One who takes such an attitude is missing the entire significance of Purim and the Megillah.