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Why are there three matzot on the Seder plate?

Probably most people know the explanation that they represent the three types of Jews: Cohen, Levi and Israel. AskMoses gives as a source for this explanation "Rav Shrira Gaon and Maaseh Rokeach 16:58, cited in the Rebbe’s Haggadah, and explained in Migdal Ohr by Rabbi Ezra Schochet, vol. 7." – I haven’t checked, but I’ve heard the explanation many many times*.

Another explanation that I’ve heard a few times is that they represent the three patriarchs, Abraham Isaac and Jacob. Wikipedia attributes this to the MaHaRal (Judah Loew ben Bezalel, 16th century, Prague), but I haven’t found any real source for this*.

This year at the Seder I heard for the first time the the "practical" reason: We need two whole loaves to make the blessing on bread. Before the point where we make the blessing on bread (matzot) in the Seder, we break one of the matzot. So we need to start with three. Here is at least something of a source for this explanation:

Of the three matzot on the table, why is it that the middlte matza is the one which is broken? The halakhic answer is: On Pesach three matzot are required. One for lechem o’ni (poor man’s bread) and two for lechem mishna (the two loaves required on each Shabbat and Yom Tov). Since it is customary to lay one’s ten fingers on the lechem mishna [as a reminder that ten mitzvot are fulfilled in the process of producing and eating bread] these matzot must be placed on the top and bottom of the pile.
(R. Nachman Cohen, The Historical Haggada, page 14)

While looking for sources for the above explanation, I came across a reason that was new to me. The three matzot are supposed to be a rather confusing allusion to flour. When the angels came to visit, Abraham told Sara to make break from three "se’a" of flour (Genesis 18:6). And of course she was baking matzot as they came on Passover. Why was it Passover? — stay tuned for a separate post on this topic!

* please write a comment if you know more!