Explanation #1: Rashi explains that cleaning one's mouth (with bread) is more effective than rinsing it with liquid. They do not mention rinsing with liquid. Beis Hillel says that one rinses with liquid and does not require cleaning (with bread). Cleaning is not enough, rather one must rinse with liquid which is the main cleaning here, and he does not need to clean.

Question #1: There is a difficulty according to his explanation. Beis Hillel should not have said, "And he does not require." Rather, their argument should be "he cleans and does not rinse" versus "he rinses and does not clean!"

Question #2: Additionally, why does the Gemara try to conclude that they do not argue? Why not say that Beis Shamai holds that cleaning is better, and Beis Hillel holds that rinsing alone is enough just as cleaning would be enough?

Explanation #2: It therefore appears that it is obvious to the editor of the Gemara that rinsing is more effective than cleaning. This is the explanation of the Gemara. Beis Shamai holds that one cleans and does not rinse. This means that cleaning alone is enough, even though it is less than rinsing. They do not require rinsing which is more effective.

Explanation #2 (cont.): Beis Hillel holds that rinsing alone is required. They seemingly should have said that cleaning is not good enough. However, because Beis Shamai wrote "And they do not require (rinsing)," Beis Hillel is also quoted as saying "they do not require (cleaning)."

Implied Question: The Gemara now should have asked if so, this is one of the leniencies of Beis Shamai and stringencies of Beis Hillel! (Why didn't the Gemara ask this question?)

Answer: Rather, the Gemara has other questions it wanted to ask (instead).

Implied Question: The Gemara could not have answered that Beis Shamai requires both, and while they said cleaning, they meant that rinsing is also required. It also could not have said that Beis Hillel holds either rinsing or cleaning is enough. (Why not?)

Answer: If so, Beis Hillel should have said cleaning, as this is a greater inclusion than rinsing which is more effective.

Explanation: Both are required (as opposed to one being good enough) as explained by Rashi. This is also the explanation of Rabeinu Chananel.

Explanation: This does not refer to the regular meals that people are accustomed to having, one in the morning and one at night. Rather, one is even permitted to eat (cheese) immediately after clearing away the table and saying Birkas ha'Mazon, as the Rabbanan did not differentiate (in their decree from having a meal in a long time or having another meal right away).

Opinion: In Berachos (53b), the Gemara derives from the Pasuk, "And you will sanctify yourselves - this refers to Netilas Yadayim." It also derives "And you will be holy - this refers to Mayim Acharonim." This is only an Asmachta (not an actual derivation).

Proof: This is apparent from the fact that Mayim Acharonim was instituted due to salt of Sedom, as stated later (in our Gemara).

Opinion: Since we do not commonly have salt from Sedom, we are not accustomed to wash Mayim Acharonim.

Proof: Another proof (to the fact that these Pesukim are Asmachtos) is from the Gemara's statement (ibid.) that "for holy - refers to the oil used to anoint their hands" to take away any residue from the meal. This is clearly an Asmachta, as we never have found anywhere that it is an obligation or even a Mitzvah to rub oil on one's hands after the meal.

Implied Question: The Gemara there says that just as one who is sweaty is invalid for Birkas Kohanim, so too hands that are full of residue are unfit for Berachos. (This sounds like it is discussing an actual Halachah!)

Answer: This means that since for them (they had salt from Sedom) it was obligatory to wash Mayim Acharonim after their meal they are therefore considered sweaty (akin to a Kohen who is sweaty for Birkas Kohanim). They therefore must ensure that they wash before saying Birkas ha'Mazon.

Question: Why does it make a difference that Netilas Yadayim is a Mitzvah and Mayim Acharonim is an obligation? Why say two different terms if we know that they are both obligatory, as stated in the Beraisa, and are not optional?

Answer #1: Rabeinu Tam says that the difference between them is in a Milchamas ha'Reshus (permitted battle). This is as the Mishnah states in Eiruvin (17a) that they are exempt from washing hands. Rav Chiya says, this is only regarding Netilas Yadayim. However, they are not exempt from Mayim Acharonim. This is because Rav Yehudah the son of Rav Chiya says, why did they say that Mayim Acharonim is obligatory? This is because salt from Sedom etc.

Answer #2: The Bahag explains that Netilas Yadayim is called a Mitzvah due to the fact that it was instituted due to the eating of Terumah. It therefore requires a blessing. However, Mayim Acharonim that are for the good of the person due to salt from Sedom do not require a blessing (and are called an obligation).

Opinion: Based on this logic, Rabeinu Yehudah writes in Tosfos (perhaps Tosfos Rabeinu Yehudah) in Bameh Madlikin (second chapter of Meseches Shabbos) that one does not make a blessing on reciting Kriyas Shema before going to sleep by saying "Asher Kidishanu b'Mitzvosav v'Tzivanu Likros Kriyas Shema."

Question: Why don't Mayim Emtzayim, for example washing between cheese and meat, which are for the purpose of a Mitzvah require a blessing?

Answer: This is only a way to be able to eat, similar to doing Nikur on forbidden fat and salting meat (we do not say a blessing on either process).

Opinion #1: Rabeinu Shmuel explains that the case is where both are dishes containing meat or both are dishes containing milk. However, between a dish of cheese and a dish of meat it is obligatory. However, the Gemara does not discuss eating a cheese dish after a meat dish, as even if one would wash his hands it would be forbidden to eat a cheese dish until the next meal. This is as stated earlier that a person who eats meat cannot then eat cheese.

Question #1: This does not appear correct according to Rabeinu Tam, as between a dish and cheese implies that the dish is eaten first. This is as stated earlier, "How long should one wait between eating meat and eating cheese? No time at all. Didn't Rav Chisda say that if one eats meat he cannot then eat cheese? Rather, he meant how long does one have to wait between eating cheese and then eating meat?" (This shows the order is significant!)

Question #2: Additionally, if between one dish and another means that both are meat dishes or both are milk dishes, why does he have to wash at all?

Opinion #2: Rabeinu Tam explains that between dish and dish means between a dish cooked with meat and a dish cooked with cheese. Since the meat and cheese are not actually present in these dishes, and there is only the taste that they gave to these cooked dishes, they were not stringent to require washing hands between the dishes. It is optional. However, between a dish cooked with meat and actual cheese one is obligated to wash his hands.

Observation: This explanation can even be correct according to those who forbid eating cheese after meat in the same meal even if one washes his hands and cleans out his mouth.