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Citing the Torah’s epithet for matzah, "lehecm oni", ("Poor Man’s Bread"), the Gemarah in Maseches Pesachim (115b) derives that the matzah of seder night must be broken: "ma darko shel ani beprusa...just as a poor person eats a broken piece of a loaf, so too matzah must be eaten as a broken piece". In addition to the Gemarah’s textual derivation, Chazal perceive numerous symbolic elements in the breaking of the matzah. The Da’as Zekeinim Al HaTorah (Shemos 12:8), for example, suggests that the breaking of the matzah may represent the splitting of the Yam Suf and the Jordan River. Similarly, the Chidah (Haggadas Simchas HaRegel) adds that the splitting of the matzah into two halves may symbolize the teaching of Pirkei D’Rabbi Eliezer (ch. 48), that HaKadosh Baruch Hu "halved" the time of the Egyptian servitude from 430 years to 215 years. These various understandings of Yachatz underscore that both slavery and salvation are within the broken matzah, thereby highlighting the central theme that salvation can instantly emerge from the most abject situations of suffering.
While all agree that the matzah must ultimately be broken, the Rishonim debate precisely when the matzah should be broken. The Rambam (Hilchos Chametz U’Matzah 8:6) implies that one breaks the matzos immediately before reciting the beracha of "Hamotzi lechem min ha’aretz." R’ Avraham Gershon Zaks (Hagadas HaGershuni, Yachatz) suggests that this is consistent with the Rambam’s general understanding of lechem mishneh during the seder. Although lechem mishneh generally requires two whole and intact loaves, the Rambam rules that lechem mishneh of seder night consists of one whole and one half matzah, in keeping with the Gemarah’s teaching of "ma darko shel ani beprusa". Apparently, the Rambam understands the halacha of broken matzah as defining the mitzvah of lechem mishneh. Therefore, explains R’ Zaks, it is perfectly consistent to maintain that the breaking of the matzah should be performed adjacent to the beracha and consumption of the lechem mishneh as well. R’ Zaks adds that this explanation also accounts for the Rambam’s omission of any broken matzah requirement for the afikomen. Because the halacha of broken matzah relates specifically to lechem mishneh, the halacha does not necessarily apply to afikomen, which is not consumed as lechem mishneh.
Most rishonim and achronim disagree with the Rambam, maintaining that the breaking of the matzah must immediately precede Maggid. Several explanations are suggested for this timing. The Beis Yosef (O.C. 473) suggests that the matzah is broken at this point in order to recite "Ha lachma anya," which describes the poor man’s bread. Alternatively, the breaking and hiding of the matzah serves to pique the curiosity of the children, encouraging them to ask questions. The Shulchan Aruch HaRav (ibid. 36) suggests a further possibility. The Gemarah derives a connection between the consumption of matzah and the recital of the Hagaddah, noting that matzah is "lechem sheonin alav devarim harbeh - bread upon which many things are recited." Based on this relationship, one must recite the Hagaddah over matzah which is halachically fit for fulfillment of the mitzvah of achilas matzah. Because achilas matzah requires that the matzah be broken, by extension, the recital of the Haggadah must be performed over a broken piece as well.
The Ran (Pesachim 25b in Rif, s.v. "mihu") cites Rav Hai Gaon, who understands the early breaking of the matzah as a function of the general halachos of lechem mishneh. As the Netziv explains (Shu"t Meishiv Davar I: 21), lechem mishneh generally requires two intact loaves of bread. For purposes of lechem mishneh, however, "intact" is defined in a relative sense: if the mevarech breaks a whole loaf prior to reciting the beracha, the loaf is certainly not considered "intact." If, however, the mevarech receives a loaf which has already been broken, the loaf is halachically considered "intact," since the completeness of the loaf is defined relative to its form at the time of reception. Hence, explains the Netziv, by breaking the matzah a significant amount of time prior to the actual beracha, the mevarech can be considered to have received matzah which was already broken. As such, the broken matzah is halachically considered "whole" for purposes of lechem mishneh, and one thereby fulfills both the need for "poor man’s bread" as well as the need for "wholeness" for lechem mishneh.
One other possibility for the early breaking of the matzah is suggested by R’ Shlomo Zalman Aurbach (Halichos Shlomo, Pesach 9:29). Although halacha dictates that Kiddush must immediately precede the seuda of yom tov, the recitation of Maggid on seder night seems to create a hefsek, separating the Kiddush from the seuda. By breaking the matzah before the recital of Maggid, one links the Maggid to the meal, demonstrates that Maggid is considered mei’tzorchei achila - for the purposes of eating. The Maggid is therefore not considered a hefsek, but rather a bridge between the Kiddush and the meal, despite the possible passage of several hours in between.
Regarding the actual execution of Yachatz, the rishonim debate which of the three matzahs should be broken. The Smag (283) and the Rokeiach (241) rule that the top matzah should be broken, in accordance with the Talmudic dictum of "Ain ma’avirin al hamitzvos - we do not pass over mitzvos." Of note, Rashi (Yoma 33a s.v. "ain"), quoting the Mechilta, maintains that the very source for this principle comes from the mitzvah of matzah. As the Mechilta expounds,"ushemartem es hamatzos - you shall guard the matzos" to read "ushemartem es hamitzvos - you shall guard the mitzvos." Thus, if a mitzvah presents itself, one should not let it pass. While Tosafos (ibid.) understands this derivation to be mid’oraisa, the Radbaz (ShU"T I:559) opines that it is a mere asmachta (see also Divrei Malchiel O.C. 16).
Despite the cogent argument of the Smag and Rokeiach, the accepted halacha follows the Rosh (Pesachim 10:30) and the Mordechai (Pesachim 38) who maintain that the middle matzah is broken. In defense of the Rosh and the Mordechai, the Bach (Orach Chaim 473 s.v. "v’yikach") explains that breaking the middle matzah does not violate the principle of Ain ma’avirin al hamitzvos. Eating the matzah, as opposed to breaking the matzah, constitutes the primary mitzvah of seder night. When it comes to eating the matzah, the beracha of "Hamotzi lechem min ha’aretz" is recited on the top matzah, while the second bracha of "Al achilas matzah" is recited on the broken second matzah, in perfect adherence to the principle of "Ain ma’avirin al hamitzvos." Thus, by breaking the middle matzah, one ensures that he will not "pass over the mitzvos," as this breaking ensures that the first bracha of "Hamotzi" will be appropriately recited on the first matzah.