Difficulty understanding Gemara's logic about number of necessary sukkah walls

Question

Some background information

At Sukkah 6b, the gemara presents a dispute between R' Shimon and the Rabanan about how many walls are necessary for a sukkah to have. The Rabanan hold that a sukkah needs two full walls (at least 7 tefachim long) plus one wall of at least a tefach. R' Shimon holds that a sukkah needs three full walls plus one wall of at least a tefach. The gemara attempts to explain the nature of their dispute.

The first possible explanation is that they dispute whether the כתיב (how a word is written in the Torah) or the קרי (how the word is pronounced when reading from the Torah) is considered more important. They consider three times the word sukkot is written in the Torah. Twice it is chaser (the vav that makes the word plural is not included in the כתיב), and once it is malei (the vav is written).

Since the Rabanan hold that the כתיב is more important than the קרי, they count the instances of sukkot that are chaser (spelled סכת) as 1 and they count the instance that is malei (spelled סכות) as 2. So all-together they get 4 (סכת + סכת + סכות) and they subtract one because they hold that you do not make a derasha on the first instance of a word. They are left with three, and they make the derasha that that means 3 walls for the sukkah and they say that there is a halacha l'Moshe miSinai that one of the walls only needs to be one tefach long.

R' Shimon, on the other hand, holds that the קרי is more important, so he counts each instance of the word sukkot as 2 for a total of 6 (סכות + סכות + סכות). He subtracts one instance of the word sukkot because he doesn't make a derasha on that instance and so he's left with a total of 4 (סכות + סכות), meaning 4 walls. And he holds (like the Rabanan) that there is a halacha l'Moshe miSinai that one of the walls only needs to be one tefach.

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My Question

After all of that background information, here is my question. After the above discussion, the gemara gives a series of alternative explanations about the nature of the dispute between R' Shimon and the Rabanan. The first alternative explanation leaves me baffled, though.

It says that alternatively, we can even say that both the Rabanan and R' Shimon hold that the קרי is more important. So they both count a total of 6. Subtract one instance of סכות because we don't make a derasha on the first instance of the word, and we're left with 4. Here's the part I don't understand:

מר סבר סככה בעיא קרא ומר סבר סככה לא בעיא קרא

Based on Rashi, I understand that to mean that the Rabanan hold that we would not have been able to know that a sukkah needs skhakh without one of these instances, while R' Shimon holds that we don't need to use one of the instances to show that a sukkah needs skhakh since it wouldn't be called a sukkah if it didn't need skhakh.

So for R' Shimon, the case is the same as above; he has a total of 4 walls, and there is a halacha l'Moshe miSinai that one of them only needs to be 1 tefach. But I don't understand how the Rabanan come out to 3 walls based on this understanding. It seems to me that the Rabanan require two of the uses of sukkot for other things, and they are only left with one. That should be two walls, not three.

So what about the Rabanan's position according to this second explanation of the dispute am I misunderstanding?

Answer 1

Art Scroll 6b2 goes into more detail.

All agree that the pronounced form has primacy so that of the 6 (3*2) we are left with four (2*2) allusions. The rabbanan then say that one of the four allusions is required for schach leaving three allusions. (note 23). The reason is that as once the two of the actual references are available for learning an allusion, we can use each of the four allusions for teaching something. This is the reason for the machlokes on what we use the fourth allusion to teach.