Who knows sixty?
Please cite/link your sources, if possible. At some point in the next few days, I will:
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Sixty describes a verse in Song of Songs that Artscroll interprets two different ways.
Artscroll siddur page 293:
Behold! The couch of Shlomo! Sixty mighty ones round about it ...
Same Hebrew verse, same exact Artscroll siddur, eight pages later:
Behold the resting place of Him to Whom peace belongs, with sixty myriads of Israel's mighty encircling it.
Sixty is the rule of thumb for the point at which most normal foodstuffs are diluted, by volume, beyond tasteability. (As Rabbi Moshe Dovid Tendler has pointed out, the notion of homoeopathic dilutions really doesn't click with this.)
The practice of some Sephardic Jews is to prefer to use a non-Jewish taste tester instead, but Ashkenazic practice relies on this rule of thumb, more or less exclusively*.
It's not per se that sixty-to-one is a halachically-magical number; I'd asked a medical-ethics posek if 1.6% is ever a threshold used in Halacha in the various discussions of odds and risk in Halacha; he said no, that's just about food.
Flavorings, including some food concentrates, are famously not nullified by 1:60; thus, if a non-kosher coffee flavoring is usually added 1:100 into plain coffee, that coffee remains non-kosher. However, the resulting flavored coffee is now a normal foodstuff, and would in turn be nullified if mixed 1:60 with plain coffee.
The Achronim debate, in the case of a non-kosher flavoring added 1:60+, whether the foodstuff is Biblically or Rabbinically prohibited. The logic to say it's Biblically prohibited is straightforward: it tastes non-kosher, it's non-kosher; 60 was just a rule of thumb. Those who argue it was Biblically permitted say (e.g. Chochmas Adam) that the Torah prohibition is only on "normally tasteable" proportionate tastes, not "trace" tastes. Which is awfully close to saying that 60 is a magic number ... (Well at least it's a unitary perfect number ...)
(* Yes there are exceptions, but that's too complicated for here.)
60 are the number of letters in Birkas Kohanim. These are compared to valiant warriors who guard and protect the Jewish People. (Tanchuma, Naso 9, et al, interpreting Song of Songs 3:7)
In Bavel, they asked for rain (tal umatar) 60 days after Tekufas Tishrei (Taanis 10a)
Hineh mitaso sheliShlomo, shishim giborim saviv lah Migiborei Yisrael. Shir Hashirim 3:7
This can refer to: 60 myriads that left Egypt (Rashi); The 24 priestly shifts + 24 Levite shifts + 12 divisions of Israel during the time of the Beis Hamikdash who conquered their (bad) inclinations (Metzudas David); The 60 letters that the Kohanim bless Yisrael with (Unkolos); etc.
60 is the gematria of גאון (Gaon) - the title given to one who has mastered all of the 60 mesechtos of Shas by heart. (The three Bavas are counted as one mesechta, and Sanhedrin and Makkos are counted as one - רדב"ז; רלט-שלג, or by counting Ma'aseros and Ma'ser Sheni as one, see here)
Bechoros 8a says that a pig's pregnancy lasts 60 days.