Who knows thirty-one?
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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 are the Canaanite kings whom the Jewish People defeated and killed. (Joshua 12:7-23)
31 are the pages in Maseches Megillah (starts on 2, goes to 32)
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)
Thirty-one is no. (Gematria of "lo")