Shabbat 15 - Shammai and Hillel argued no more than three times
Although there are numerous disagreements between the schools (students) of Shammai and those of Hillel, Shammai and Hillel themselves disagreed only in three instances. What are they?
When making dough, one has to separate a portion for a priest (Kohen) and give it to him. Nowadays, when the priests are no longer ritually pure, one separates and destroys a symbolic amount. How much dough is considered significant enough for separating the Kohen's portion, called challah? Shammai says that it is one kav (the volume of twenty-four eggs), while Hillel maintains that it is twice that amount, or two kavim. Both base their measurements on the amount of manna that the Jews would have for their meal in the desert, except that Shammai says that one meal forms this basis, and Hillel - that daily amount of manna, collected in the morning and then used for the two meals of the day, should be used in the calculation.
A mikveh is a pool of rainwater used for ritual purification. If some water is poured into it from a vessel, before it collects the required volume of rainwater, it becomes disqualified. How much poured water would disqualify it? Hillel says that it is a hin (faithfully using the word "hin"that his teaches used), or about a gallon (because one can wash himself in that), and Shammai is much more lenient and says that is it nine kavin, about four gallons (because this is enough for a shower). The other Sages are stricter than both and say that it is three lugs (about one quart).
Their third disagreements was learned in the tractate Niddah here .
Art: Bernardino Luini - The Gathering Of The Manna