Who knows thirty-one?
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31 are the Canaanite kings whom the Jewish People defeated and killed. (Joshua 12:7-23)
31 is the gematria of "ויהי", "and it was ...". Two of the five Megilot (Esther and Ruth) begin with this word; the other three (Song of Songs, Ecclesiastes, Lamentations) don't contain it at all.
Chazal says "ויהי" indicates bad news. An explanation I'd heard was that this word is made of יהי, "will be" in the future, then reversed to the past with the "vav ha-hipuch." If we're taking our future and just forcing it to be just like the past, that's a very bad sign.
(Conversely והיה -- learning from our past and applying it to our future -- is a good sign.)
I think this is from R' Sorotzkin, though I could be mistaken.
31 is the most English dates possible in a Hebrew month. i.e., if the molad was at 1pm on Jan 1, the next molad would be around 1:44am on Jan 31.
(on a related note, 3 is the most Hebrew months in an English month, and 3 is the most English months in a Hebrew month; in the example above, Kislev, Tevet, and Shevat all fall out in Jan, while Jan, Feb, and March all overlap with Shevat)
Yehoshua battled with 31 kings of the land of Canaan (source: Keshushat Levi, Parashas Korach and Shenei Luchot HaBerit, Torah Shebikhtav, Chukat, Torah Ohr 37)
II Malachim 22:1 - the length of Yoshiyahu's reign in Yerushalayaim:
בֶּן־שְׁמֹנֶ֤ה שָׁנָה֙ יֹאשִׁיָּ֣הוּ בְמׇלְכ֔וֹ וּשְׁלֹשִׁ֤ים וְאַחַת֙ שָׁנָ֔ה מָלַ֖ךְ בִּירוּשָׁלָ֑͏ִם וְשֵׁ֣ם אִמּ֔וֹ יְדִידָ֥ה בַת־עֲדָיָ֖ה מִבָּצְקַֽת
Josiah was eight years old when he became king, and he reigned thirty-one years in Jerusalem. His mother’s name was Jedidah daughter of Adaiah of Bozkath.
Also see II Divrei HaYamim 34:1
פָּסְקָה לְהַכְנִיס לוֹ אֶלֶף דִּינָר, הוּא פוֹסֵק כְּנֶגְדָּן חֲמִשָּׁה עָשָׂר מָנֶה. וּכְנֶגֶד הַשּׁוּם, הוּא פוֹסֵק פָּחוֹת חֹמֶשׁ. שׁוּם בְּמָנֶה וְשָׁוֶה מָנֶה, אֵין לוֹ אֶלָּא מָנֶה. שׁוּם בְּמָנֶה, הִיא נוֹתֶנֶת שְׁלֹשִׁים וְאֶחָד סֶלַע וְדִינָר. וּבְאַרְבַּע מֵאוֹת, הִיא נוֹתֶנֶת חֲמֵשׁ מֵאוֹת. מַה שֶּׁחָתָן פּוֹסֵק, הוּא פוֹסֵק פָּחוֹת חֹמֶשׁ
If a woman agreed to bring her husband one thousand denarii he must agree to give her a corresponding sum of fifteen maneh. As a corresponding sum for appraised goods, he agrees to give one-fifth less. [If a husband is requested to enter in his wife's ketubah] “goods assessed at one maneh”, and these are in fact worth a maneh, he only [must agree to] a maneh. [Otherwise, if he is requested to enter in the ketubah:] “goods assessed at a maneh”, his wife must give him thirty-one sela and a denar, and if “at four hundred”, she must give [him goods valued at] five hundred. Whatever a bridegroom agrees to give [his wife in her ketubah] he writes one fifth less [than the appraised value].
31 is the small sum gematria of the miluy of Shadai. Meaning שדי as: שי״ן, דל״ת, יו״ד like is found in Sefer Raziel. This follows the posuk from the Torah found in Shemot 6:3.
וָאֵרָ֗א אֶל־אַבְרָהָ֛ם אֶל־יִצְחָ֥ק וְאֶֽל־יַעֲקֹ֖ב בְּאֵ֣ל שַׁדָּ֑י וּשְׁמִ֣י יְהֹוָ֔ה לֹ֥א נוֹדַ֖עְתִּי לָהֶֽם׃
This is the name of G-d affiliated with procreation which is alluded to via the 6 letters of miluy from this name which total 500 and is the sum or union of the 248 limbs of the husband and the 252 limbs of the wife. As the Rokeach teaches this is also the basis for the obligation to have both a son and a daughter.
He also points out that this is the gematria of “Be fruitful and multiply.” (פרו ורבו)
Rabbi Nehunia ben Kaneh points out in Sefer HaTemunah that the Aleph of 31 (אל) is the sign of the father (the reproductive organ). This is also found in Pardes Rimonim by Rabbi Moshe Cordevero.
Similarly, the Lamed is the sign of the nursing mother (the source of milk for the child) which is called אל דד, the right breast. When nursing, one is to start from the right breast.
And if considering the partzufim (the limbs or parts) which comprise the two letters א and ל of 31, it is seen that they are equal. Namely that the Aleph is compromised of a Yud above, a Yud below and a Vav between (26, יהוה). So too the Lamed is comprised of a Kaf below and a Vav above (26). That G-d is one.