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19. A says "Atleast one of us is knave"
    
    if A (knave) B(knave) ---> A said truth but not possible as A is knave
    if A (knave) B(Knight) ---> A said truth but not possible as A is knave
    if A (Knight) B(knave) ---> A said truth so A is Knight and B can be knave
    if A (knight) B(knight) ---> A said truth since A is Knight but not possible as both are Knights

    So A is Knight and B is Knave

20. A says Both of us are Knights
    B says A is Knave
    
if A(Knave) B(Knave) ----- then Both are knaves Then B said truth but not possible as B is knave
if A(Knave) B(Knight) -----  A is knave and said false that both are knights B is knight said truth that A is Knave
if A(Knight) B(Knave) ----- As A is knight A said truth => B is knight and says A is Knave Not possible
 if A(Knight) B(Knight) ---- same as above

   So A is Knave and B is Knight

21. A says I'm Knave or B is knight (since their is or if one statement is true then A said truth)
      
       A         B
     Knave     Knave       A is knave and it said truth not possible    
     Knave     Knight      same as above
     Knight    Knave       If A is knight then I'm knave is false so B is to be Knight
     Knight    Knight       same as above

   So A is Knight and B is Knight

22.  A says I'm knight
     B says I'm knight
     
     if A says true Then  A is Knight
     if A says false Then A is knave

    
     if B says true Then  B is Knight
     if B says false Then B is knave
     
     So A  can Be either Knave or Knight  and B can Be either Knave or Knight

23. A says we both are Knaves

       A         B
     Knave     Knave       A is knave and it said truth not possible    
     Knave     Knight      A is Knave said false that both are Knaves if B is Knight
     Knight    Knave       If A is knight then it said false that both are knave not possible
     Knight    Knight       same as above
 
   so A is Knave B is Knight

Please Let me know if I'm Wrong

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Put A's assumed value and get B from the given stmts and if both A's and B's values matches with the assumed value then that combination is the correct ans.

19.

   
A                B B
Knight        Knight Knave
Knight        Knave Knave
Knave        Knight If A is Knave then it means both are Knight 
Knave        Knave If A is Knave then it means both are Knight 

20.

   
A                B B
Knight        Knight Knave
Knight        Knave Knight
Knave        Knight Knight
Knave        Knave Knave

21.

   
A                B B
Knight        Knight Knight
Knight        Knave Knight
Knave        Knight Knave
Knave        Knave Knave

22.

   
A                B B
Knight        Knight Knight
Knight        Knave Knight
Knave        Knight Knight
Knave        Knave Knave

23.

   
A                B B
Knight        Knight Knave(we had assumed A as Knight but first stmt says both are knave so this one clearly is  not the correct solution.)
Knight        Knave Knave(we had assumed A as Knight but first stmt says both are knave so this one clearly is  not the correct solution.)
Knave        Knight Knight
Knave        Knave Knight (Knave said truth is not possible so, this one also in correct).
edited

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