אורח חיים קצ״דOrach Chayim 194

Se’if 1א׳

שְׁלֹשָׁה שֶׁאָכְלוּ כְּאֶחָד, וְשָׁכְחוּ וּבֵרַךְ כָּל אֶחָד לְעַצְמוֹ, בָּטֵל מֵהֶם הַזִּמּוּן וְאֵין יְכוֹלִים לַחֲזֹר וּלְזַמֵּן לְמַפְרֵעַ; וְכֵן אִם בֵּרְכוּ שְׁנַיִם מֵהֶם; אֲבָל אִם שָׁכַח אֶחָד מֵהֶם וּבֵרַךְ, הַשְּׁנַיִם יְכוֹלִים לְזַמֵּן עִם הַשְּׁלִישִׁי, אַף עַל פִּי שֶׁכְּבָר בֵּרַךְ יָכוֹל לוֹמַר: בָּרוּךְ שֶׁאָכַלְנוּ מִשֶּׁלּוֹ, וְהֵם יוֹצְאִים יְדֵי חוֹבַת זִמּוּן וְהוּא אֵינוֹ יוֹצֵא יְדֵי זִמּוּן; שֶׁאֵין זִמּוּן לְמַפְרֵעַ. הַגָּה: וְאִם הָאֶחָד זִמֵּן עִם אֲחֵרִים, אַף שְׁנַיִם הַנִּשְׁאָרִים אֵינָן יְכוֹלִים לְזַמֵּן (בֵּית יוֹסֵף בְּשֵׁם רַשְׁבָּ"א).

Three people that ate together, but forgot and benched by themselves, missed the mitzvah and they are not allowed to return and say the Zimun out of order. And also if two of them benched together but one forgot, the other two can have a Zimun with the third, even though he already benched, he is able to say "Baruch She'achalnu Mishelo," and they [the other two] fulfill their mitzvah of Zimun through this, but he [the third one] doesn't fulfill his Mitzvah, because he did it out of order. Rema: and if the third one did the Zimun with others, the first two cannot do the Zimun with him (Beit Yosef in the name of Rashba).

Se’if 2ב׳

שְׁלֹשָׁה שֶׁאָכְלוּ וְיָצָא אֶחָד מֵהֶם לַשּׁוּק, קוֹרְאִים אוֹתוֹ וּמוֹדִיעִים לוֹ שֶׁרוֹצִים לְזַמֵּן, כְּדֵי שֶׁיְּכַוֵּן וְיִצְטָרֵף עִמָּהֶם וְיַעֲנֶה עִמָּהֶם בִּרְכַּת זִמּוּן, וְיוֹצְאִים יְדֵי חוֹבָתָן אַף עַל פִּי שֶׁאֵינוֹ בָּא וְיוֹשֵׁב עִמָּהֶם; וְהָנֵי מִילֵי בִּשְׁלֹשָׁה, אֲבָל בַּעֲשָׂרָה כֵּיוָן שֶׁצְּרִיכִים לְהַזְכִּיר אֶת הַשֵּׁם אֵינָם מִצְטָרְפִין עַד שֵׁיָּבֹא וְיֵשֵׁב עִמָּהֶם. וְע"ל סִימָן ר' עַד הֵיכָן בִּרְכַּת הַזִּמּוּן.

If three people ate together and one of them went out to the market [outside], they call out to him and inform him that they want to make the Zimun [the quorum for Grace After Meals], so that he will focus his intention, join them, and answer the Zimun blessing with them. They fulfill their obligation even though he does not come and sit with them. This applies specifically to three [people]. However, with ten, since they need to mention the Divine Name, they do not join together [to form the quorum] unless he comes and sits with them. [See later in Siman 200 regarding how far the blessing of the Zimun extends.]

Se’if 3ג׳

שְׁלֹשָׁה שֶׁאָכְלוּ כְּאֶחָד וְאֵין אֶחָד מֵהֶם יוֹדֵעַ כָּל בִּרְכַּת הַמָּזוֹן, אֶלָּא אֶחָד יוֹדֵעַ בְּרָכָה רִאשׁוֹנָה וְאֶחָד הַשְּׁנִיָּה, וְאֶחָד הַשְּׁלִישִׁית, חַיָּבִים בְּזִמּוּן וְכָל אֶחָד יְבָרֵךְ הַבְּרָכָה שֶׁיּוֹדֵעַ, וְאַף עַל פִּי שֶׁאֵין בָּהֶם מִי שֶׁיּוֹדֵעַ בְּרָכָה רְבִיעִית אֵין בְּכָךְ כְּלוּם; אֲבָל לַחֲצָאִין אֵין לְבָרֵךְ, אִם הָאֶחָד אֵינוֹ יוֹדֵעַ כִּי אִם חֲצִי הַבְּרָכָה, שֶׁאֵין בְּרָכָה אַחַת מִתְחַלֶּקֶת לִשְׁתַּיִם.

If three people ate together as one, and not a single one of them knows the entirety of the Grace After Meals (Birkat HaMazon), but rather one knows the first blessing, one knows the second, and one knows the third—they are still obligated in the Zimun [the formal invitation]. Each one shall recite the blessing that he knows, and even if there is no one among them who knows the fourth blessing, it is of no consequence. However, they may not recite it by halves if one only knows half of a blessing, for a single blessing cannot be divided between two people.