Rabbi Alexander said [Shmini Atzeret can be compared] to a king who experienced good fortune [and hosted a feast to celebrate]. All seven days of the feast the king’s son was busy with the guests. Once the seven days of feasting were over, the king said to his son, “My son, I know that all seven days of the feast you’ve been busy with the guests. Now you and I will enjoy one day together. I won’t impose much on you, just [prepare] one chicken and and one litra of meat [about 12 oz./350g].” In such a manner, Israel was busy with the korbanot of the nations of the world throughout the seven days of feasting. About this Rabbi Pinchas said: All these seventy bulls that Israel would offer as korbanot were on behalf of the seventy nations of the world so that the world wouldn’t be harmed because of them. What is the reason? “They respond to my love by accusing me, and I am all prayer” (Tehillim 109:4)—that’s how secure we are in [the power of] our tefillah! Since the seven days of the holiday have elapsed, the Holy One says to Israel, “My sons, I know that all seven days of the holiday you were busy with the korbanot of the nations of the world, so now I and you will celebrate together. I won’t impose much, just one bull and one ram. When Israel heard this, they began to praise the Holy One, saying, “This is the day that Hashem has made—let us delight and rejoice in it.” (Tehillim 118:24). Rabbi Abun said that we do not know the reason for our rejoicing, whether it is on account of the date or time with the Holy One, and Shlomo came and explained, “We delight and rejoice in it” (Shir ha-Shirim 1:4). “In it” [refers to] Your Torah, [also] “in it” refers to Your redemption.

The first day shall be a sacred occasion: you shall not work at your occupations; seven days you shall bring offerings by fire to יהוה. On the eighth day you shall observe a sacred occasion and bring an offering by fire to יהוה; it is a solemn gathering: you shall not work at your occupations.

On the eighth day you shall hold a solemn gathering; you shall not work at your occupations. You shall present a burnt offering, an offering by fire of pleasing odor to יהוה; one bull, one ram, seven yearling lambs, without blemish; the meal offerings and libations for the bull, the ram, and the lambs, in the quantities prescribed; and one goat for a sin offering—in addition to the regular burnt offering, its meal offering and libation.

The Gemara notes: It was taught in a baraita in accordance with the opinion of Rav Naḥman: The eighth day

is a Festival in and of itself with regard to the matter of: Peh, zayin, reish; kuf, shin, beit. This is an acronym for: A lottery [payis] in and of itself, i.e., a new lottery is performed on that day to determine which priests will sacrifice the offerings that day, and the order established on Sukkot does not continue; the blessing of time [zeman], i.e., Who has given us life, sustained us, and brought us to this time, in and of itself, as it is recited just as it is recited at the start of each Festival; a Festival [regel ] in and of itself, and there is no mitzva to sit in the sukka (see Tosafot); an offering [korban] in and of itself, as the number of offerings sacrificed on the Eighth Day is not a continuation of the number sacrificed on Sukkot but is part of a new calculation; a song [shira] in and of itself, since the psalms recited by the Levites as the offerings are sacrificed on the Eighth Day are not a continuation of those recited on Sukkot; a blessing [berakha] in and of itself, as the addition to the third blessing of Grace after Meals and to the Amida prayer (see Tosafot) is phrased in a manner different from that of the addition recited on Sukkot.

(8) After eating unleavened bread six days, you shall hold a solemn gathering [atzeret] for your God יהוה on the seventh day: you shall do no work.

(9) You shall count off seven weeks; start to count the seven weeks when the sickle is first put to the standing grain. (10) Then you shall observe the Feast of Weeks for your God יהוה, offering your freewill contribution according as your God יהוה has blessed you.

It is customary not to get married between Pesach and Shavuot, until Lag BaOmer (the 33rd day), because during that time, the students of Rabbi Akiva died. However, to do "erusin" and "kiddushin" (engagement and betrothal) is OK. And even for "nisuin" (marriage), if someone did so, we do not punish him. Rema: however, from Lag Ba'Omer onwards, all this is permitted (Abudraham, Beit Yosef & Minhagim).

The Gemara concludes: And the halakha is that we reside in the sukka on the eighth day, with regard to which there is uncertainty that it might be the seventh day, but we do not recite the blessing. §

(14) After a person finishes eating on the eighth day [of Sukkot], he should take down his utensils and remove them. When a person has no place to put the utensils, [the following rules apply]: If it is a small [sukkah], he should bring his candelabra into it; if it is a large sukkah, he should bring in his pots, plates, and the like, to bring to mind that it is no longer acceptable and that its mitzvah is completed. Since the day is a holiday, one may not reduce its space and nullify it.

(1) The order of prayers on the eighth night and day, 2 Seifim: 1. On the night of Shmini Atzeret, we say in the Amidah "and You gave us this eighth day as a festival of Atzeret." Rem"a: We do not refer to the eighth day as a "festival" because there is nowhere [in the Torah] that it is called a festival. Rather, we say "the eighth day of Atzeret." Outside of Israel, they eat in the sukkah at night and during the day because it might be the seventh day. They do not bless on this sitting.

In the diaspora we eat in the sukkah on Shimini Atzeret, at night and during the day, for it is possibly truly the seventh day of Sukkot And we do not recite a blessing on sitting in the sukkah. And we do say a "she'hechiyanu" blessing during kiddush on Shimini Atzeret night since it is a distinct holiday. And, indeed this is the conclusion of the Gemara (Sukkah 47a) that we sit in the sukkah on the eigth day but do not recite a blessing on sitting in the sukkah. Nonetheless, we find in the halakhic writings many different customs: The Mordechai in the name of Ravyah writes that it is forbidden to sleep in the sukkah on Shimini Atzeret because of "bal tosif" [the prohibition against adding to mitzvot]. For when it comes to eating in the sukkah on Shimini Atzeret there is a means of distinguishing that Sukkot is over in the absense of the blessing on sitting in the sukkah. But when it comes to sleeping in the sukkah, what means of distinguishing could there be? (Magen Avraham). And even though this is not fully "bal tosif" as I wrote above in Siman 666 for "bal tosif" at the wrong time for the original mitzvah requires intention to violate the prohibition nonetheless it looks like "bal tosif." Furthermore, in truth one is sitting in the sukkah to do a mitzvah out of doubt concerning the date and so some sort of distinguishing is needed since we truly do now the date of the month's start. And I will explain this in paragraph 5 with God's help.

And for this reason we leave the sukkah after our morning meal on Shimini Atzeret. And there are some who do not eat in the sukkah on Shimini Atzeret at night but do eat in the sukkah on Shimini Atzeret during the day but this is not a valid practice (Tur). And some will make kiddush and eat a little bit in the sukkah and then go home for their meal. And this practice was loudly criticized by many great scholars since the Gemara concludes that the halakhah is that we do sit in the sukkah on Shimini Atzeret and if so we must sit in the sukkah all night and all day and we should also sleep in the sukkah without any distinction with the other days of Sukkot.

However, according to my humble opinion there is a rationale for the custom of the Jewish people an the matter is all true it just requires explanation. For in truth the Gemara does conclude that we should sit in the sukkah, but our rabbis the Tosafists wrote there, and this is their language, "they did not want to enact that the lulav should be taken as well on Shimini Atzeret out of doubt for Shimini Atzeret is yom tov and the lulav and etrog would be muktzah and taking it would be a problematic weekday activity. But regarding the sukkah, sometimes a sukkah is a pleasant place to eat and one might eat in a sukkah even on yom tov that was not Sukkot." And the Tur writes and this is his language, "it is not relevant to question sitting in the Sukkah on Shimini Atzeret as a problematic weekday activity since many times people sit in the shade of a sukkah." So it is clear that the essential permissibility of sitting in the Sukkah on Shimini Atzeret is due to there being no way to discern that one is sitting in the sukkah for mitzvah purposes but were it apparent that one was only sitting in the sukkah because of the command to sit in the sukkah it would not be permissible to do so on Shimini Atzeret.

And according to this, all is well. For certainly in warm climates like Babylonia, and the western and southern lands, where it is warm and mild during Sukkot, certainly they must eat and sleep in the sukkah on Shimini Atzeret just like on Sukkot itself. But in northern lands like ours, where it is mostly cold on Sukkot, if one did not force oneself into the sukkah despite the cold, one would miss the mitzvah entirely. And we force ourselves to eat and sleep in the sukkah on Sukot and indeed, most avoid sleeping in the sukkah because of the cold as I wrote earlier. However, on Shimini Atzeret, where the sages commanded us to sit in the sukkah in a way that makes it clear we have no mitzvah to sit in the sukkah is something that is nearly impossible in our lands. Heaven forfend that we would ignore the enactment of the sages, but we still have to make it clear that we are not sitting in the sukkah out of obligation. This means that one who sleeps in the sukkah during Sukkot should not sleep in the sukkah on Shimini Atzeret and one who does not sleep in the sukkah on Sukkot would now have no distinguishing feature and so must find some other distinguishing feature such as not eating in the sukkah at night or to eat part of his meal outside of the sukkah. These two distinguishing features are not ideal and it would be best to simply leave the sukkah after the daytime meal to show that the meal was not eaten in the sukkah for a mitzvah. And in this way all of the customs are upheld (and so it seems in my humble opinion).