ABC logic puzzle with only six clue letters!

  A
C.....
C.....
C.....
 .....
 .....
   CA

Solution:

The C in the fourth column must be in the fourth row, forcing the C in the last column to be in the last row, also forcing an A below it, which forces an A and a B in the last two rows of the first column. The C in the third column must then be in the third row, forcing the C in the first row to be in the first column and therefore the C in the second row to be in the second column with an A above it. This means that the third column must have B, A, C in its first three rows and similarly the third row must have C, A, B in its last three columns. There are only three squares left but there are two Bs to be placed so they go in the squares that are not orthogonal to each other.