There is an almost universal custom to place three pieces of matzah on the Seder table. In the list of instructions at the beginning of the Haggadah the three matzot are always mentioned. Judaica shops around the world sell plates with three sections for each of the matzot. So it might come as a surprise that it is not at all clear why one should have three pieces of matzah at the Seder.
How many matzot would we expect to be on the table? The Torah tells us that the Children of Israel in the desert ate manna, a special delicacy that fell from the sky. Each day, people received only the quantity needed for that day. As the manna did not fall on Shabbat, a double portion of manna was given on Friday. In recognition of the double portion we start each Shabbat and festival meal with two loaves of bread. Therefore, we would expect to find two slices of matzah at the Seder, not three.
There are other considerations. The Talmud discusses whether one should recite the blessings over whole loaves of bread or over slices. The conclusion is that one should always use whole loaves. Rav Papa, a Babylonian scholar, adds that on Passover one should use a whole loaf together with a slice. The reason is that the matzah is called lechem oni, bread of the poor. As poor people are more likely eat bits of bread rather than whole loaves, on Passover one should uniquely recite the blessing over a broken loaf. We should therefore expect to find on the Seder table one and a half pieces of matzah -- the half symbolising the bread of poverty. Nowhere does the Talmud suggest that we should use three matzot.
To find the source of the three matzot we need to take a journey through history and visit Jewish communities around the world. The custom of using three matzot is not mentioned by any of the early rabbinic authorities in North Africa or Spain. Maimonides's code simply states that one should recite the blessing over one and a half pieces of matzah, following the ruling of the Talmud.
However, from very early times, all Ashkenazi rabbis mention the custom of the three matzot - even though none of them know the reason why! For example, Rashi's school taught that the first matzah should be used for the blessing over eating bread, the second for the special blessing for eating matzah and the third should be used for making the Hillel sandwich: "Since three of them come for the purpose of mitzvah, he should do a mitzvah with the three of them … but other people leave the third one whole, and it's not correct to do that" (Machzor Vitri p271). In other words, they know that there is a third matzah on the plate and strive to find something to do with it.
Why leave an unused matzah on the table?
Why are there three matzot on the Seder table? The only source that gives an independent reason for the custom is an obscure manuscript of a book by Rabbi Eliezer of Worms, one of the leaders of the Chasidei Ashkenaz movement in the 13th century. He writes: "and why do we leave a third matzah on the table unused? In order to recite the Grace after Meals over it. For the rabbis have said: anyone who does not leave a whole loaf of bread on his table will see no blessing." The third matzah is there specifically so that there is an unused matzah on the table at the end of the meal.
And why leave an unused matzah on the table? To understand this we need to travel to Provence. Rabbi Jonathan of Lunel (13th century) writes: "It is the way of the world that after finishing eating, one breaks bits of bread for the sake of the poor from the bread which is left on the table. If he is a mean person, he only puts on the table the amount he needs for his family and no more... But if he has leftovers on the table, then his eye isn't narrow and he gives it to the poor."
So the matzah is lechem oni, the bread of the poor. We symbolically mark this in two ways at the Seder. We recite the blessing over a half slice to remind us that there are those who do not have enough to eat and are left eating scraps. And we place a third, unnecessary, matzah on the table so that there are leftovers at the end of the meal for those who do not have.
