Who knows six?
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Six are the vowels in the Babylonian system of nikkud.
Six are the books of the Mishna Berura.
Six are the ways Israel was rewarded for Avraham's service to the angels:
Because Avraham called for the angels to be brought water, G-d repaid by providing water in the Midbar, in the Yishuv (settlement) and in the Messianic Age (e.g. "Spring water will flow out of Jerusalem (Zech. 14:8));
Because Avraham offered the angels a chance to wash their feet, Israel was repaid in the Midbar, Yishuv and the Messianic Age, with "water" to cleanse our sins;
Because Avraham invited the angels to recline beneath a tree, G-d spread out a cloud for shelter in the Midbar, succahs in Israel, and a tabernacle in the Messianic Age;
Because Avraham fetched a morsel of bread, Israel was repaid with manah in the Midbar, crops in the Yishuv, and the promise of abundant grain in the Messianic Era;
Because Avraham ran to the cattle, Israel was repaid with meat in the Midbar, abundant livestck in the Yishuv, and each man will have a heifer and two sheep in the Messianic Era;
And because Avraham stood over the angels, G-d attended our needs in the Midbar,leading us with the pillar of fire, listened to our prayers in the Yishuv, and will give us the Messiah.
Source Midrash Rabbah, Genesis, 48:10.
6 are the number of steps taken by the bearers of the Ark between each sacrificing of an ox and fatling in the bringing of the Ark to the city of David.
Shmuel II 6:13
וַיְהִי, כִּי צָעֲדוּ נֹשְׂאֵי אֲרוֹן-יְהוָה--שִׁשָּׁה צְעָדִים: וַיִּזְבַּח שׁוֹר, וּמְרִיא.
Six are the words in the verse גדלו לה' אתי which we say when we take out the Sefer Torah, which correspond to the six steps taken by the bearers of the Ark between each sacrificing of an ox and fatling in the bringing of the Ark to the city of David (Tosefos Megillah 32a s.v. גוללו)
The Midrash in Bamidbar Rabbah 12:17 says that there are six specific mitzvos that a Jewish king is commanded about every day:
Those who went from Bavel to Y'huda with Z'rubavel (Ezra 2) traveled with pack animals in the ratio of six (more precisely, 6.13) people per animal (verses 64–67: 49,897 people to 8136 animals).
Branches coming out the side of the menorah in the mikdash.
Hours the Rambam says to wait between meat and milk.
Wings on an angel.
Years till average child is bar daas.
Steps one takes after shmoneh esrei till one can walk away, three backwards and three forwards.
Total count of all the lechem mishna from shabbos.
All the matzahs from both seder nights.
Six are the things that preceded the creation of the world:
R. Menachem and R. Yehoshua b. Levi said in the name of R. Levi: A builder requires six things: water, earth, timber, stones, canes, and iron. And even if yon say, He is wealthy and does not need canes, yet he surely requires a measuring rod, as it is written, And a measuring reed in his hand (Ezekiel 40:3).
While a person may "technically" be exempt from providing food or sustenance to their children past the age of 6, Beis Din would never allow this to happen and they would force the father to provide the child with food and sustenance until at least the child's Bar/Bat Mitzvah. (SA, EH ibid).
Regarding the Shoham stones worn on the Cohen Gadol's shoulders,
שִׁשָּׁה֙ מִשְּׁמֹתָ֔ם עַ֖ל הָאֶ֣בֶן הָאֶחָ֑ת וְאֶת־שְׁמ֞וֹת הַשִּׁשָּׁ֧ה הַנּוֹתָרִ֛ים עַל־הָאֶ֥בֶן הַשֵּׁנִ֖ית כְּתוֹלְדֹתָֽם׃
six of their names on the one stone, and the names of the six that remain on the other stone, according to their birth.
For the "original" menorah:
וְשִׁשָּׁ֣ה קָנִ֔ים יֹצְאִ֖ים מִצִּדֶּ֑יהָ שְׁלֹשָׁ֣ה ׀ קְנֵ֣י מְנֹרָ֗ה מִצִּדָּהּ֙ הָאֶחָ֔ד וּשְׁלֹשָׁה֙ קְנֵ֣י מְנֹרָ֔ה מִצִּדָּ֖הּ הַשֵּׁנִֽי׃
And there shall be six branches going out of the sides thereof: three branches of the candlestick out of the one side thereof, and three branches of the candle-stick out of the other side thereof;