Gemara

(Mishnah): Techeles is not Me'akev the Lavan (strings that are the like the garment, which is usually white), and Lavan is not Me'akev the Techeles;

Suggestion: Our Mishnah is unlike Rebbi!

(Beraisa - Rebbi): "U'Re'isim Oso" - Techeles and Lavan are Me'akev each other;

Chachamim say, they are not Me'akev each other.

Rejection (Rava): The Mishnah teaches about remnants. If the Techeles strings snapped and the Lavan remains, it is Kosher. If the Lavan snapped and the Techeles remains, it is Kosher;

(Bnei R. Chiya): Remnants of Techeiles are Kosher. Remnants of Ezov branches are Kosher (for Parah Adumah or Metzora).

Question: How much must remain?

Answer (Bar Hameduri): Kedei Anivah (enough to make a loop) must remain.

The Chachamim who argue with Rebbi hold like R. Yochanan ben Nuri;

(Beraisa - R. Yochanan ben Nuri): If one does not have Techeles, he puts Lavan.

39a (Rav): If a string totally snapped, it is Pasul.

39b: Rabah bar bar Chanah expounds that the strings must be Gedil (wound) or Pesil (straight);

Rav requires Pesil in every case.

Rishonim

Rambam (Hilchos Tzitzis 1:4): Techeles and Lavan are not Me'akev each other. If one has no Techeles, he makes Lavan strings. If one made Techeles and Lavan strings, and the Lavan snapped until the corner and only Techeles remains, it is Kosher. Even though one is not Me'akev the other, they are one Mitzvah.

Rambam (18): Similarly, if the Tzitzis were diminished, even if only Kedei Anivah remains, it is Kosher. If even one string was totally cut, it is Pasul.

Rosh (Hilchos Tzitzis 7): Rava taught that the Mishnah teaches about remnants. If Techeles or Lavan remains, it is Kosher. Kedei Anivah must remain. R. Tam derives that if both the Techeles and Lavan snapped, it is Pasul even if Kedei Anivah remains. Therefore, nowadays that we put four strings, two for Lavan and two for Techeles, and we fold and double them to eight (ends), if two of the eight snapped and Kedei Anivah remains, it is Kosher. If three snapped it is Pasul, for perhaps only one intact string remains.

Rosh (ibid.): Some say that the answer of remnants is according to Rebbi. According to Chachamim, the Mishnah simply teaches that Techeiles is not Me'akev Lavan at all. The Gemara struggled to make the Mishnah like Rebbi, and said that it discusses remnants. Only according to Rebbi, who says that they are one Mitzvah, we say that due to (some) whole strings, it is as if all of them are whole. According to Chachamim, they are two Mitzvos. If one of them was cut and the other is intact, it does not help the other. The Halachah does not follow Rebbi, for R. Yochanan ben Nuri disagrees, and the Halachah follows Rebbi against his colleague, but not against his Rebbi.

Rebuttal (Rosh): It is unreasonable that Bnei R. Chiya's law is only according to Rebbi. Rather, it is according to everyone. Some explain 'if Techeles snapped and Lavan remains' is when the Lavan snapped me'Ikaro (at the source), and only a tiny amount remains. Rebbi holds that they are one Mitzvah; the whole strings help for the other kind. Chachamim hold that they are not Me'akev each other. We need two strings of each. If one string totally snapped from where it is attached at the corner it is Pasul, for only three strings remain. If all four snapped and Kedei Anivah remains, it is Kosher. Benei R. Chiya's law was taught after Rava's law, but Rava's law is not totally similar to R. Chiya's law according to Rabanan, who require Kedei Anivah in all four strings. Rava discusses when some are intact and some are snapped. Rava says correspondingly according to Rebbi; if the Techeles totally snapped and the Lavan is intact, or vice-versa (it is Kosher). We bring Benei R. Chiya merely to show that the required remnant is unlike the initial required Shi'ur, but it is different for each Tana. Rav taught that if a string totally snapped, it is Pasul. It need not totally snap; if less than Kedei Anivah remains, it is as if it totally snapped, for a Shi'ur of Kashrus does not remain. This Perush is primary. Benei R. Chiya discuss 'remnants of Techeles', which connotes all four. In several places, the Gemara calls all the Tzitzis 'Techeles'. This is like remnants of Ezov, which Benei R. Chiya also taught, which is when all three stalks were cut. Therefore, even if all eight ends were cut and Kedei Anivah remains, it is Kosher. If even one string has less than Kedei Anivah, it is Pasul.

Nimukei Yosef (Hilchos Tzitzis 12b DH me'Ikaro): Some say that Kedei Anivah suffices only on two strings. We always require two strings to be whole on each corner, i.e. 12 fingers on each end of the string. This suffices even if he initially made them longer than this.

Poskim

Shulchan Aruch (OC 12:1): If all the strings on a corner snapped, and there remained Kedei Anivah enough to loop around all the snapped strings, it is Kosher. If Kedei Anivah does not remain even in one string, for all snapped, it is Pasul. Therefore, since each string (of the four) is folded (through the corner) into two, if two ends snapped, it is Pasul, for perhaps they are from one string.

Beis Yosef (DH v'Im): The Rosh rejected the opinion that the answer of remnants is only according to Rebbi, so the Tur did not bring it. The Rosh gave one explanation of Benei R. Chiya. If Kedei Anivah remains in every string, it is Kosher. If less than this remains even in one, it is Pasul, like Rav taught. Even though Rashi explained 'me'Ikaro' to be the end attached to the Talis, the Rosh says that it is anything less than Kedei Anivah.

Beis Yosef (DH u'Mah): According to the Ri, if only one end snapped it is Kosher, even if less than Kedei Anivah remains, since the other end remains Kosher. Kedei Anivah on the other end suffices. If it snapped above the Gedil, where it is attached to the corner, even if both ends are intact, it is as if it is not here at all, for it is not attached to the Talis.

Gra (DH Im): According to Rebbi, Techeles and Lavan are Me'akev, but the number of strings is not Me'akev. According to Chachamim, Techeles and Lavan are not Me'akev, but four strings are Me'akev. All agree that if all snapped and Kedei Anivah remains, it is Kosher.

Shulchan Aruch (ibid.): The custom is to be meticulous when making Tzitzis to make a Siman on four ends, so that they will always be on the same side of the knot. If two strings on one side snapped, it is Kosher, for surely they are from different strings, and the other end of each string has more than Kedei Anivah. R. Tam is Machshir only if two strings are whole, i.e. all four of their ends are 12 Godelim (thumb-breadths) long. Then we are Machshir if the other two strings snapped and Kedei Anivah remains. If three strings snapped, even if Kedei Anivah remains, it is Pasul. Therefore, when three ends snapped, if one was not meticulous when making the Tzitzis that the four ends on one side of the knot be evident, we are concerned lest each end is from a different string, and only one string is whole. Therefore, it is Pasul due to Safek.

Beis Yosef (DH veha'Tosfos): The Mordechai (939) says that a case occurred in which three ends snapped, and R. Tam disqualified it.

Beis Yosef (DH v'Rabbeinu): R. Yerucham says that if two ends snapped, even if Kedei Anivah remains, it is Pasul. Perhaps this is according to the opinion that Tosfos and the Rosh rejected. However, why does R. Yerucham bring it?!

Magen Avraham (5): If one was meticulous, all the more so it is (surely) Pasul when three strings snapped (on one side), like the Rema (below) says! Some say that the Mechaber mentioned 'he was not meticulous' for the continuation, in which he is Machshir when two ends snapped on one side only if Kedei Anivah remains. If he was meticulous, we are Machshir even if Kedei Anivah does not remain. One can make a different mark for (both ends of) each string, and avoid all Safek.

Machatzis ha'Shekel: If one did like the Magen Avraham suggests, it can be Kosher even if three ends snapped (and Kedei Anivah remains). However, it is difficult to say that the Mechaber alludes to this.

Kaf ha'Chayim (7): Our custom is to tie a knot at the end of each of the four strings on one side, and to always keep them on one side.

(Note: In standard editions of the Shulchan Aruch and Mishnah Berurah, the next comment of the Rema is at the end of this Sa'if, but clearly it applies here, and Shulchan Aruch ha'Shalem puts it here.)

Rema: All the more so, if one was meticulous that the four ends on one side be recognizable, and three ends snapped on one side, it is Pasul, for then surely three strings snapped. Also if the three snapped ends are not all on one side it is Pasul, for perhaps they are from three strings.

Shulchan Aruch (ibid.): If only two strings snapped, we are Machshir if Kedei Anivah remains. The Halachah follows the first opinion. However, when possible, is good to be concerned for R. Tam's opinion.

Beis Yosef (DH ul'Inyan): Since it is not clear how to explain the Rambam (see the Beis Yosef in the Halachah for Menachos 39), we follow the first opinion, which the Ri and Rosh agreed to. When possible, it is good to be stringent like R. Tam.

Beis Yosef (DH veha'Mordechai): Some are stringent (like R. Tam), and some are lenient (like the other Perush, i.e. the Ri). Semag holds like R. Tam. The Rosh holds like the Ri. The Tur says that even though Teshuvas ha'Rosh (2:12) is like R. Tam, and in (9:23) he brought both opinions, what he wrote in his Pesakim is primary.

Mishnah Berurah (7): The first opinion is Machshir even if all four ends on one side were cut, and the four ends on the other side have only Kedei Anivah.

Rema: The custom is like R. Tam.