You can calculate them step by step in your head by adding to the previous sqaure the sum of the number you are squaring and the previous number you squared.
1²=1
2²= 1²+ (2+1)= 4
3²= 2²+ (3+2)= 9
4²= 3²+ (4+3)= 16
and so on.
Got em all with about 15 seconds left and that felt great!!
28^2 and 29^2 were my hardest two ... b/c I didn't have any clue what those two were off the top of my head... so I just went about doing 28^2 = (30-2)^2 = 900 - 2(-2 x 30) + 4 {which is just a quick way of multiplying those two binomials together} ... and this gave ---> 900 - 120 + 4 = 784 !!!!
Similarly, 29^2 = (30 - 1)^2 = 900 - 2(30 x 1) + 1 = 900 - 60 + 1 = 841 !!!
I'm totally gonna nerdify myself now. I used to lay in bed and do math problems in my head as a way to fall asleep (I never counted sheep). Doing the squares was one thing I did. I also calculated how many possible combinations there were on the combo locks at my school and how long it would take to try them all at various different times (1 second each, 10 seconds each, etc). I was a lonely child. :P
There is a good trick with perfect squares in ascending order.
You just take the difference of the last two squares and add two to it. Then add this to the last square. For example:
20^2 = 400. This is 39 greater than 19^2 (361). Add two to 39 to get 41. Then add this to 400 to get 21^2.
Saves getting the calculator out as it becomes simple addition.