Rosh, or Reverse Chinuch Method

Some prefer 7 Techeiles Chulyos instead of 6. This method allows for it by doing blue-white-blue in between knots 2-3 and 3-4.

From VilnaGaon.org:

“A Discussion of the Viewpoint of the Rosh on Tying Tzitzit With Tekhelet

In the commentary of the Rosh on the Torah in the book Hadar Z’kainim, at the end of Parshat Shelach it is written:
And I heard that the windings for each corner be 3 sets of 9 windings and the last set of 12 windings. The three sets of 9 windings and the 1 set of twelve windings = 39 which has the same numeric value as the words G-d is one. Furthermore I have heard that the sum of all the windings and sets of windings called Chulyot and knots and strings = 248, like the limbs of a man, and check and find if you are proficient in numbers: 4 * 39 = 156, and 5 knots and 4 Chulyot and 8 strings behold this is equal to 23***; if so if we multiply 4 * 23 we arrive at the sum 248.

*** Now this needs further investigation for we need another 6 to reach the number 23; now it is possible that we can complete the number in the following manner: 5 double knots which behold is 10 + 8 strings + 4 chulyot + the hole in the garment to which they are tied = 23 or perhaps one can reach the number 23 by counting the G’dil (the upper third of the strings of the Tzitzit) instead of counting the hole.

Now in Hilchot Ketanot, Hilchot Tzitzit of the Rosh, printed in the back of tractate Menachot in point # 15 it is written:
“Tekhelet wrapped around the majority of the length of the strings is kosher. And even if he just wrapped one Chulya {singular of Chulyot} it is kosher. And the beautiful fulfillment of the mitzvah of Tekhelet is a third Gedil and two thirds Anaf {that is that one third of the length of the strings are tied and wrapped together and two thirds of the length is loose as separate strings}. Now, what is the measurement for a Chulya {set of windings}? It was taught that Rebbe says, so that one can wind {around the other strings} and repeat this a second time and a third time. It was taught in a Baraita one who wishes to do less should not do less than 7 and one that wishes to add should not do more than 13. One should not do less than 7 in correspondence to the 7 heavens. and one should not add more than 13 in contrast to 7 heavens and the 6 spaces {or atmospheres} between them. It was taught in a Baraita, one begins with the white and ends with white, one goes up in holiness and does not lower it. Now it seems that specifically when there is Tekhelet do we have to be scrupulous about the number of Chulyot. For the Tekhelet is recognized by the Chulyot and resembles the heaven {or sky} and the white resembles the spaces {or atmospheres} between each heaven. However in our times that we lack Tekhelet we do not have to be scrupulous about the number of Chulyot {as defined by Rebbe earlier, these are sets of three windings} and for this reason we do not have to knot  the strings to differentiate between each of the Chulyot, for such or knotting is to recognize the number of Chulyot. And as far as making 5 knots, we have not found a support for this; for in the days of the sages they would make at the very least 7 knots. As it was stated, “one who wishes to do less should not do less than 7” and upon each Chulya {set}, a knot. But if we will explain that the obligation to tie that was stated,  refers to one set of white together with one set of Tekhelet, then we will find a basis for the 5 knots. For at the beginning one ties a knot close to the corner of the garment and then wraps a Chulya {set} of white and a Chulya of Tekhelet and then ties another knot, behold there are now 2 knots. Then another white and another Tekhelet which is then tied with a knot, behold there are now three. And still another white and another Tekhelet, which is then tied with a knot, behold this is a total of 4. Then one ends with white alone and ties a knot and behold there are 5. Now Rashi commented that in regards to what they said that the mitzva of Tzitzit is equivalent to all the commandments, the reason being is that the Gematria {numerical value of the letters} of the word Tzitzit equals 600 which together with 8 strings and 5 knots comes to the value of 613 {the total number of Torah commandments}.
Now on first glance there is a contradiction between the words of the Rosh in Hadar Z’kainim {which I quoted at the beginning} and his words in Hilchot Ketanot.
Now in my humble opinion one can explain this in one of two ways.
If the words of the Rosh on the Torah were stated after his words in Hilchot Ketanot, one could explain that the Rosh discovered that indeed there is a strong basis for the words of our sages that one should tie Tzitzit with 5 knots (also in the time that Tekhelet exists}. For behold the Zohar and so too the Targum, Yonatan Ben Uziel \ Targum Yerushalmi on the end of Parshat Shelach, obligate us to tie 5 knots. And so too, did Midrash Tanchuma write in accordance with Rashi’s view that the Gematria of the word Tzitzit together with 8 strings and 5 knots equals 613.
After the Rosh saw that there was room to establish the custom of 5 knots, he decided that it is fitting to combine the custom with the simple wording of the Talmud. That is to say that the first and last Chulya {set} be white and that there should be a total of 7 Chulyot {sets} of Tekhelet, corresponding to the 7 heavens and 6 Chulyot {sets} of white corresponding to the empty spaces {or atmospheres}.
And it is possible to tie in the following way in order to fulfill all the required numbers:
1] A double knot close to the corner of the garment, a Chulya of white, a Chulya of Tekhelet and then another Chulya of white.
2] A double knot, a Chulya of Tekhelet, a Chulya of white, and a Chulya of Tekhelet.
3] A double knot, a Chulya of Tekhelet, a Chulya of white, and a Chulya of Tekhelet.
4] A double knot, a Chulya of Tekhelet, a Chulya of white, and a Chulya of Tekhelet followed with a final Chulya of white.
5] A double knot
Or else, even if we say that the words of the Rosh in Hilchot Ketanot come after his commentary to the Torah, it is possible to say that the Rosh himself, was not convinced that there is an obligation to tie with 5 knots. But his rabbis were convinced. So the Rosh, though he did not find a strong proof to support the words of his rabbis also did not find a reason to oppose the words of his rabbis.”

As a point of reference, Rav Shabtai Sabato from Yeshivat Mitzpeh Yericho High School does Chinuch tie but does Blue/White/Blue between the third and fourth double knots to give seven blue chulyot and six white.

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