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When is the Holiday of Shavuos?
On the 50th day of the counting of the Omer is the holiday of Shavuos, which is called “Atzeres” in the wording of the Sages. The verse did not depend the Holiday of Shavuos at all on the day we received the Torah, which was the 6th of Sivan, or on any specific date of the month, but rather solely on the 50th day of the count. [The holiday is thus not dependent on a set Hebrew date in the calendar but rather on the 50th day of the count of Sefira.]
The date of Shavuos in times of the Temple: In the times that the new month was sanctified based on testimony of sighting of the new moon, the festival of Shavuos would at times fall out on the 5th of Sivan and at times on the 6th of Sivan and at times of the 7th of Sivan. How? When both Iyar and Sivan were sanctified on the 31st day, both Nissan and Iyar were 30 day months. This would cause the 50th day of the count of the Sefira, which is the Holiday of Shavuos, to fall on the 5th of Sivan. When only Iyar or Sivan were sanctified on the 31st day, then one of the months was a 30 day month and the other a 29 day month. This would cause the 50th day of the count of the Sefira, which is the Holiday of Shavuos, to fall on the 6th of Sivan, which is the day that the Torah was given. When both Iyar and Sivan were sanctified on the 30th day, both Nissan and Iyar were 29 day months. This would cause the 50th day of the count of the Sefira, which is the Holiday of Shavuos, to fall on the 7th of Sivan. Now, although based on this there were times that the Holiday of Shavuos fell on the day before or the day after Matan Torah, as Matan Torah took place on the 6th of Sivan, this matter holds no Halachic significance being that the verse did not depend the Holiday of Shavuos at all on the day we received the Torah, or on any specific date of the month, but rather solely on the 50th day of the count.
The date of Shavuos today: In today’s times, in which we no longer sanctify the month based on testimony and every month has a set amount of days based on the tradition of our calendar, the Holiday of Shavuos will always fall on the 6th of Sivan, which is the day of Matan Torah. The reason for this is because the tradition we received for our calendar is that the month of Nissan is always a full month of 30 days and the month of Iyar is always a lacking month of 29 days. Thus, the counting of the 50 days will always fall on the 6th of Sivan in today’s calendar, as the Sefira begins on the 16th of Nissan. There are 15 days of count in Nissan, between the 16th of Nissan and the 30th of Nissan, and there are 29 days of count in Iyar, and 6 days of count in Sivan, for a total of 50 days.
 Admur 494:1; Michaber 494:1; Rivash 97; Peri Chadash 494:1; Chok Yaakov 494:1; Chok Yosef 494:3
 Sefiras HaOmer always begins on the 16th of Nissan which is the day of the Omer offering.
The source: This is learned from the verse [Vayikra 23:16 and 21]: “Count fifty days and then bring an offering….. And you shall call on the essence of this day a day of Holiness”.
 This is also the wording of Michaber 494:1, thus implying like Admur that Shavuos is not dependent on a date.
 Admur 494:1; Chachamim in Shabbos 86b; Kesef Mishneh in Avos Hatuma 5 in opinion of Rambam and Rashba that they rule like Chachamim; M”A 494:1 that our custom is like the Rambam; Aruch Hashulchan 494:2
Other opinions: Some say the Torah was given on the 7th of Sivan. [Rebbe Yossi in Shabbos 86b; See M”A 494:1 that from Y.D. 196 it is evident we rule like Rebbe Yossi, as we rule a woman is Poletes Zera Soteres for six Onos, as rules Rebbe Yossi] Rebbe Yossi holds that Hashem wanted to give the Torah on the 6th day, although Moshe added one day on his own, and therefore the Torah was given on the 7th day. [Shabbos 87a; See Likkutei Sichos 28:18 footnote 18]
 Admur ibid; Tzaruch Iyun from Admur 600:1 in parentheses who writes that in times of Kiddush Hachodesh they would send emissaries to notify the public of the day of Rosh Chodesh Sivan. This implies that Shavuos is set to the 6th day of Sivan and not the 50th day of count, as otherwise, what necessity is there for them to know which day was the 1st of Sivan, either way they will keep Shavuos on the 50th day from the 16th of Nissan. Vetzaruch Iyun!
 According to our tradition, whenever one month begins on the 30th the next begins on the 31st, and when one begins on the 31st the next begins on the 30th. This is with exception to the months of Kislev and Cheshvan which can at times both be 30 day months and at times both be 29 day months. [See Rambam Hilchos Kiddush Hachodesh 8:5; Tur 428]