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Tying strings together so they meet the minimum required measurement? 
Prior to tying the string to the corner: Prior to tying the Tzitzis strings to the corner, it is valid to tie the ends of two strings to each other for the sake of achieving a single long string. Thus, if one has a string that is too short, and will not reach the minimum length of 24 cm after it is tied, he can tie its end to the end of another string and then use it to tie to the corner if it now meets the minimum length of 24 cm. It is permitted to do so even initially. [There is no minimum amount of strings that can be tied to each other, and hence even if all the strings of all four corners of the Tzitzis are made up of multiple strings that are tied to eachother, it is nevertheless Kosher.]
After the Tzitzis were tied to the corner: All the above is regarding tying the strings together prior to tying them onto the corner, however once the strings have been tied onto the corner, then tying a loose string to a string of the Tzitzis that does not meet the minimum length, is not always valid. This depends on the current Kashrus status of the Tzitzis. If after tying the Tzitzis to the corner the strings tore in a way that invalidates the Tzitzis, then it does not help to tie a loose string onto the torn string to help it reach its minimum Kashrus length, and if one does so the Tzitzis remains invalid. For example, if after tying the Tzitzis to the Tallis, two strings tore from opposite sides and neither have a 4 cm length remaining, in which case the Tzitzis is invalid, it does not help to now tie a loose string onto the torn string, and even if one does so the Tzitzis remain invalid. Rather, one must un-tie the entire Tzitzis from the corner and then attach the loose string to the torn string and then retie it to the corner. [If however the Tzitzis remained Kosher despite the tear, such as two strings tore and one of the strings retains a 4 cm length, then tying a loose string to the torn string is valid to extend its length. This benefits the Tzitzis being that it considers it to now retain its full initial length, and prevents future invalidation if a third string tears.]
Prior to tying the Tzitzis strings to the corner one my tie strings together to create the minimum required length of the string. Once the Tzitzis are tied to the corner, then tying a loose string onto a torn string in order to extend its length is only valid if the Tzitzis still remained Kosher after the tear. If however the tear rendered it invalid, then it does not help to extend the length of the torn string.
 15/3; M”A 15/1
 The reason: As a knot is considered a complete attachment and makes the two strings actually considered as one single string. [ibid]
 15/4; Taz 12/3; M”B 12/7; Elya Raba; Peri Megadim; Derech Hachaim; Siddur Yavetz; Ashel Avraham Butchach; Aruch Hashulchan 12/12; Kaf Hachaim 12/6
Other opinions: Some Poskim rule it does not help to tie a string to a Tzitzis string that is already attached to the corner, and hence for all intensive purposes of invalidation, it is considered that the strings length has not been extended. [Beis Yitzchak 3; Kiryah Nemana 45; Maharsham 8/9; Chazon Ish 3/15] Other Poskim rule it always helps to tie another string to lengthen the Tzitzis even if the Tzitzis is now deemed invalid due to the tear. [Beir Heiytiv 12/3 in understanding of Taz ibid] Some rule one may be lenient like this opinion if the Tzitzis is still Kosher according to the opinion of Rabbeinu Tam, meaning that three Tzitzis tore to below the 24 cm mark. [Eretz Tzvi ½]
 The reason: As since the Tzitzis is currently invalid the performance of an action does not help to fix it unless one does an action to the actual original tying of the Tzitzis to the corner, which is to completely untie the Tzitzis and then attach a loose string to the torn strings and then retie it to the corner in a Kosher way. If however one does do an action to the original tying of the Tzitzis to the corner, then the attachment of a loose string to the torn string is worthless, as the Torah states “Gedilim Taaseh” which teaches us that the Tzitzis must be originally tied in a Kosher status and is invalid if it made Kosher only after it was tied invalidly. See 10/10 for further details on this issue. [Admur ibid]
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