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.
Look at the difference between one square and the next. for the next square, add two to the difference and add that to the previous square. 4 - 1 = 3. 3 + 2 = 5. 5 + 4 = 9 = 3 times 3.
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 !!!!
Yep, I had to use that method for the upper teens, lower twenties, and upper twenties. Only have 1-13, 20, 25, and 30 memorized. Definitely would not have had time had I tried to multiply the ones I didn't have memorized.
A method I had was look at what first digit will equal, look at what last digit will equal, guess numbers in the middle. Doesn't really work when you get to 4 digit answers though
And if you combine knowing that each increment is roughly the size of the previous increment with working out the final digit (which is trivial), then you don't really need to do the addition precisely, just ballpark it, then guess the closest number with the right final digit. Worst case you need two guesses.
You can go really fast on this by just knowing what each digit the answer will end in. 1=1, 2=4, 3=9, 4=6, 5=5, and so on. Then when you get to the higher numbers just jump up about 30 or 40 and put in the appropriate end digit. You can smash through this quiz in about a minute without too much trouble. :)
1²=1
2²= 1²+ (2+1)= 4
3²= 2²+ (3+2)= 9
4²= 3²+ (4+3)= 16
and so on.
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 !!!
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.
(x - 1)^2 - x^2 = 2x + 1
As I look at the list of answers, I see a pattern and I could now do this in half the time.
Example:
0x0=0 (it may not really be a square, but bear with me)
1x1 = 1 (previous answer + 1)
2x2 = 4 (previous answer + 3)
3x3 = 9 (previous answer + 5)
4x4 = 16 (previous answer + 7)
5x5 = 25 (previous answer + 9)
6x6 = 36 (previous answer + 11)
7x7 = 49 (previous answer + 13)
8x8 = 64 (previous answer + 15)
9x9 = 81 (previous answer + 17)
10 x 10 = 100 (previous answer + 19)
And so on. I'm NOT a math-minded person but love patterns and that is just too cool! :P