1600 + 669 = 2269 = Not a perfect square
1500 + 669 = 2169 = Not a perfect square
1540 + 669 = 2209 = Square of 47
1690 + 669 = 2359 = Not a perfect square
Thus, by adding 669 to each of the alternatives, we note that 1540 + 669 = 2209, and this is the only square number.
So each side of the square contains
√2209 = 47 balls
∴ Each side of the triangle will contain 55 balls.
Thus the triangle will contain
1 + 2 + 3 + ... + 55 =
= 1540 balls