Commentary for Berakhot 7:4

אינן רשאין לחלק – Since an obligation of invitation [to a group recitation of ברכת המזון] falls upon them.

Introduction Our mishnah teaches that if people are eating a meal together they may not separate to recite Birkat Hamazon if this will cause them to recite a lesser zimun , or no zimun at all.

וכן ארבעה וכן חמשה – The “three” [of that group of four or five] (cannot separate) to recite ברכת המזון for themselves and the individual should separate himself off from them, for he also has an established obligation for the appeal to participate in the group recitation [of ברכת המזון], it says Psalms 68:27), “In assemblies bless God, [the LORD, O you are from the fountain of Israel].” And the determined Halakhic practice is that from three individuals until ten individuals, but not including ten, the person reciting the blessing says, “Let us thank Him whose food we have eaten/נברך שאכלנו משלו,” and everyone responds, “ברוך שאכלנו משלו ובטובו חיינו/Blessed is [He] whose food we have eaten and through whose goodness we live.” And with ten or more than ten people, the leader says, “נברך לאלהינו שאכלנו משלו/Blessed is our God whose food we have eaten.” And everyone answers: “ברוך לאלהינו שאכלנו משלו ובטובו חיינו/Blessed is our God whose food we have eaten and through whose Goodness we live.”

Three persons who have eaten together may not separate [to recite Birkat Hamazon]. Similarly four and similarly five. A group of three, four or five people eating together cannot separate into two groups to recite Birkat Hamazon because that will leave some people without the ability to say any zimun.

ששה נחלקין – in order to have a group recitation (of three individuals for ברכת המזון) and another equivalent group over there, until one reaches the number “ten.” But with ten individuals, we do not divide the group since it would obligate them to a have a group recitation including the mention of God’s name until there would be twenty and only then could they divide into two [separate] groups if they desired.

Six may separate, up until ten. A group of six to ten, but not including ten, can separate because their zimun will not change, provided that each group still has at least three.

And ten may not separate until there are twenty. Once the group reaches ten it can’t separate because if it does, not everyone will be able to say God’s name as part of the zimun. Since the zimun for ten is greater than the zimun for three, the group of more than ten people cannot split up until there are ten left for each group, meaning there are twenty.