Who knows sixty-one?
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61 esronim (an isaron = the minimum amount of flour from which challah is to be separated; between about 3-5 lb., depending on whom you ask) is the largest amount of meal-offerings (menachos) brought for the public sacrifices on any day. This would happen if the first day of Sukkos coincides with Shabbos. The breakdown is as follows:
For the daily (tamid) offering:
For the additional (musaf) offering for Shabbos:
For the additional (musaf) offering for Yom Tov:
13 bulls, 3 isaron for each, total 39
2 rams, 2 isaron for each, total 4
14 lambs, 1 isaron for each, total 14
Aggregate total: 61.
The significance of this figure is that R' Yehudah gives it as the reason behind the halachah that an individual may not bring a minchah offering larger than 60 isaron: "it is sufficient that the individual's contribution be one less than that of the community."
Sixty-one are the kosher eggs required to "nullify" one non-kosher egg.
One reason suggested (that's easy to understand) is that people will assume sixty eggs have 60x the volume of one egg, when in fact some additional variation is likely -- maybe there were a few smaller eggs in the batch of 60. So go with 61 eggs, which is most likely to cover it.
(Challenge exercise: explain this with statistics and probability. Does anyone have the probability distribution function for chicken-egg volumes?)