Which of the following numbers are definitely NOT perfect squares and why? (i) 153 (ii) 257 (iii) 408 (iv) 441

A perfect square can only end in the digits 0, 1, 4, 5, 6, or 9.
Numbers ending in 2, 3, 7, or 8 are never perfect squares.

• (i) 153: The one's digit is 3.
• (ii) 257: The one's digit is 7.
• (iii) 408: The one's digit is 8.
• (iv) 441: The one's digit is 1.

Numbers ending in 3, 7, or 8 are not perfect squares.

• 153 ends in 3, so it is surely not a perfect square.
• 257 ends in 7, so it is surely not a perfect square.
• 408 ends in 8, so it is surely not a perfect square.
• 441 ends in 1, which means it could be a perfect square (in fact, $21^2 = 441$).

The numbers which are surely not perfect squares are $\boxed{153, 257, \text{ and } 408}$.