Gemara

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

Suggestion: Our Mishnah is unlike Rebbi!

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.

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.

(Rava): We learn from Bnei R. Chiya that one must tie the strings together after each Chulya (set of windings);

If this were not necessary, if only a small remnant remained, the last knot would become undone, and the whole Gedil (windings) would become undone!

Rejection: Perhaps it is not necessary. Bnei R. Chiya taught that remnants are Kosher only in a case in which the strings happened to be tied!

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

Question (Rava - Beraisa): Shirei (remnants) and remains of Tzitzis of any size are Kesherim.

Suggestion: 'Remnants' are when something remains. 'Remains' are when nothing remains.

Answer (Rav Nachman): No; the Beraisa teaches one matter. Remnants of remains of Tzitzis of any size are Kesherim. It says Shirei (remnants) to teach that a Shi'ur (a minimal required size) must remain, i.e. enough to make a loop.

39b: Rabah bar bar Chanah expounds that the strings must be Gedil 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.

Question (Beis Yosef OC 12 DH veha'Rambam): In Halachah 4, the Rambam connotes that even if all the Lavan totally snapped, it is Kosher, since the Techeles remains! Also, it seems that he rules like Rebbi. Why does he rule like Rebbi against Rebbi's Rebbi, R. Yochanan ben Nuri? Also, if he rules like Rebbi, he should say that Techeles and Lavan are Me'akev each other! Also, why did he illustrate 'Techeles is not Me'akev Lavan' in a case when he has no Techeles, and to illustrate 'Lavan is not Me'akev Techeles', he discusses strings that snapped? Also, how can the Rambam be Machshir when nothing remained of Lavan? The Gemara requires Kedei Anivah!

Answer #1 (Beis Yosef DH v'Nir'eh): The Rambam rules like Chachamim, like the simple reading of the Stam Mishnah. He taught about one who has no Techeles, and surely Lavan is not Me'akev Techeles. This is a Kal va'Chomer, for the primary Mitzvah is Techeles! The same applies to one who has no Lavan. He is Machshir when the Lavan snapped and the Techeles is intact, or vice-versa. Even though the Gemara said so to establish the Mishnah like Rebbi, all agree that the law is true. The Gemara is Machshir when Lavan was cut and Techeles remains, i.e. even if the Techeles was totally cut. The Gemara did not ask how much must remain regarding Benei R. Chiya's law, rather, in a case when all of the Tzitzis snapped. We require Kedei Anivah. If only one Min (Techeles or Lavan) snapped, we do not require any remnant of the other, like Benei R. Chiya. Even one string of Techeles suffices, even if the Lavan totally snapped until the garment. If all of one Min snapped, and also one string of the other Min, it is Pasul, like Rav taught. We are Machshir only when the other Min is intact, but not if one string was totally cut. Likewise, if all the strings were of one Min, if one string totally snapped it is Pasul. This explains Halachah 18. According to this, we can say that Benei R. Chiya are Machshir only when Kedei Anivah remains. The Gemara asked about how much must remain according to them, like the simple reading. Rava brought a proof from Benei R. Chiya, even though they require that Kedei Anivah remains (and Rava does not). This is because they discuss when all the strings snapped. Rava merely proves that the remnant need not be as big as what is required initially. Halachah 4 is Rava's law. Halachah 18 is Benei R. Chiya's law.

Answer #2 (Beis Yosef DH v'Yesh): Alternatively the Rambam explains that Rava discusses when one Min was cut, but Kedei Anivah remains, like Benei R. Chiya. If it was totally cut, it is Pasul, like Rav taught. If so, the Rambam teaches only two laws. 1) If one Min was cut, but Kedei Anivah remains (it is Kosher). Initially (Halachah 4) the Rambam wrote the general law, and in Halachah 18 he explained that Kedei Anivah must remain. This is when the other Min is totally intact; he relied on what he wrote above (Halachah 4). 2) If a string was totally cut, it is Pasul. This is Rav's law. If so, the Rambam holds like R. Tam.

Answer #3 (Beis Yosef DH v'Yesh): Alternatively, the Rambam teaches three laws. 1) If only one Min was cut, even if less than Kedei Anivah remains, it is Kosher, since the other Min is totally intact. Any amount that remains suffices, but something must remain. It was diminished 'Ad (until) the corner', but not totally to the corner. This is why Rava discusses a remnant. If nothing remained, he would have said 'it was cut me'Ikaro.' If Kedei Anivah remained, he would not need to say that the other Min is intact. Even if both were cut, it is Kosher if Kedei Anivah remains. Rava proved from Benei R. Chiya only that the remnant need not be as big as what is required initially (like I said above). 2) When all of the Tzitzis snapped, we require Kedei Anivah. This is Benei R. Chiya's law. 3) If even one string was totally cut, it is Pasul.

Rosh (Hilchos Tzitzis 7): R. Tam derives that if both the Techeles and Lavan snapped, it is Pasul even if Kedei Anivah remains. Nowadays we put four strings, two for Lavan and two for Techeles, and we fold and double them to eight (ends). If two ends snapped and Kedei Anivah remains, it is Kosher. If three strings snapped it is Pasul, for perhaps only one intact string remains. Some explain that 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. This Perush is primary. If even one string has less than Kedei Anivah, it is Pasul.

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'Heicha): If both ends of a string snapped, and Kedei Anivah does not remain on either end, but the combined length is Kedei Anivah, it seems that it is Kosher. Perhaps the Tur's words 'if two ends snapped, it is Kosher if Kedei Anivah remains' discuss this case. However, I did not find anyone who says so. We can say that if neither side has Kedei Anivah by itself, each side is not important, and it is as if it is not here (so they do not join).

Mishnah Berurah (3): It is not clear whether it is Pasul in this case. Eliyahu Rabah and Derech ha'Chayim say that one should be stringent, even if it is a Safek whether the two ends are from the same string or from two strings. There is a Sefek-Sefeka (two doubts) - perhaps they are from two strings, and perhaps the Halachah is that they join. Even so, one should be stringent, for Tzitzis are available. Artzos ha'Chayim is lenient in this case. If one end has enough to loop around one string, surely one may be lenient (for there is a third Safek; perhaps this is Kedei Anivah).

Shulchan Aruch (ibid.): 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. If only two strings snapped, we are Machshir if Kedei Anivah remains.

Beis Yosef (DH u'Svara): When two ends snapped, why does the Tur require (according to R. Tam) that Kedei Anivah remains? If they are from different strings, no string totally snapped. If they are from the same string, Rav disqualifies only if it totally snapped! The Tur must understand that the Rosh said 'less than Kedei Anivah is considered totally snapped' also according to R. Tam, even though the Rosh did not explicitly say so. R. Yerucham says so.

Kaf ha'Chayim (5): Mishbetzos Zahav was unsure about a string that became untwined, and one side was cut less than Kedei Anivah and the other side is more than Kedei Anivah. See what I wrote above in 11:12. (If less than Kedei Anivah of a string remains twined, it is Pasul.)

Shulchan Aruch (ibid.): 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, we follow the first opinion, which the Ri and Rosh agreed to. When possible, it is good to be stringent like R. Tam.

Mishnah Berurah (11): Surely one may (rely on the Ri and) bless on it.

Kaf ha'Chayim (11): Chayei Adam says that if two whole strings do not remain, he may wear it only if he cannot get others, or he would miss Tefilah or similar concerns. He may not bless on it.

Kaf ha'Chayim (12): It is proper to fix a string even if one end snapped. Surely the number of 32 strings (in all) has great secrets that go up to Shamayim.

Rema: The custom is like R. Tam.