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1. Shemini Atzeres differs from the days of Sukos in six ways.
2. The full Hallel is recited on all eight days of Sukos.
3. There is an obligation to rejoice through the eating of meat of Korbanos during all of Sukos, except for the first night.
4. At the end of the seventh day of Sukos, one should do something to indicate that he is moving back to his house.
5. The Gemara describes how the water libations were performed.
A BIT MORE
1. This is known by the acronym PZR KShV, referring to Payis (a lottery was performed for the bull offered, as opposed to the days of Sukos where all Mishmaros offered bulls in order), Zman (the Berachah of "Shehecheyanu"), Regel (it has its own name), Korban (it has its own group of Korbanos), Shirah (it has its own Shir for the day), Berachah (it has its own blessing).
2. Additionally, Shalmei Simchah (Shelamim offerings used to fulfill the Mitzvah of Simchas Yom Tov) may be offered all eight days of Sukos.
3. This happiness is attained from eating the meat of Korbanos slaughtered throughout Sukos. The verse implies that while this applies on Shemini Atzeres, it does not apply to the first night of Sukos, since the meat must be from a Korban slaughtered on Sukos (and Korbanos cannot be offered at night).
4. The Mishnah suggests that he should move his special vessels back into his house. The Gemara gives additional suggestions (for example, he should remove four by four Tefachim of Sechach).
5. The Tzedukim did not agree that the water libations should be performed. One year they managed to have a Kohen of their own perform the Avodah, who attempted to pour the water on his legs instead of in the appropriate hole in the Mizbe'ach. The entire congregation present realized what he was doing and pelted him with their Esrogim.
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