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[ThunderCloud]

Spoiler for well...

My point is C's number is irrelevant with the given circumstances (that A and B are both 3, which limits it to 2,3).

More importantly, I do believe A and B determine their numbers based strictly on their responses if C always say "I don't know".

Spoiler for Here is (cryptic) thought chart to show it

This is all the thought process of A. He can deduce his number solely based on what he thinks B thinks he thinks..etc (without a "C thinks...").

 

C's 3.
A know's he's 2,3.
    (A)CASE 2 B:3 A-2,3
                         4 A-1,2            
             (B)CASE 4 A:1 B-4,5
                                  2 B-3,4                     
                      (A)CASE 1 B: 4 A-1,2
                                            5 A-1! ~ {A would have voiced this}
                                           (A would know A's 1)    _Round 1
                      (B would know B's 4 [!5])
             (A would know A's 2 [!1])                           _Round 2
    (B would know B's 3 [!4])    
(A knows A's 3 [!2])                                                _Round 3

 

(B) CASE x A:  <--> Suppose B considered he was x. Then B knows A's thinking would be:

[!x] <--> because he cannot be x (via proof by contradiction for the hypothetical posed directly above [equally spaced])

 

Spoiler for

You are right -- for A, C's responses are indeed redundant. But not so for B; in fact, if A and B were the only players in the game (knowing that their sum is either 5 or 6), B would be unable to deduce his number.