For how many ordered pairs (A,B) where A and B are positive integers is $AAA_7 + BBB_7 = 444_7$?
Converting everything to base 10:
49A + 7A + A = 57A
57A + 57B = 228.
This means that A+B=4. Since A and B are positive integers:
(1,3)
(2,2)
(3,1)