The Shulchan Aruch OC (580:2) codifies the list of sad days some have the custom to fast for, which includes at the end:

בט' בו נחלקו בית שמאי ובית הלל - The ninth of Adar is when Beis Shammai and Beis Hillel disagreed.

What does this refer to? (Is there one specific day when they disagreed on everything, the first day they argued, etc.?)



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