Here's something I wrote for Slashdot a while back:
To square any two-digit number X, decide what number N it will take to raise or lower X to P, the nearest multiple of 10. Add the opposite of N to X to get Q, multiply P times Q, and add N^2 to the result. For instance:
67 * 67 = (67 + 3) * (67 - 3) + (3 * 3) = 70 * 64 + 9 = 4489
The hard part is 70 * 64, but if you teach yourself to ignore the zero at the end of the 70 and multiply from left to right, it sounds like this:
"Seven sixes make forty-two, times ten makes four hundred and twenty. Seven fours make twenty-eight, plus four hundred and twenty makes four hundred and forty-eight, times ten makes forty-four hundred and eighty, plus three squared--that's nine--makes forty-four hundred and eighty-nine."
Presto, you've figured out the answer in less time than it takes to say it. Note: don't be discouraged if you forget what number you needed to add at the end, or what number you were originally squaring; they're going to drop out of your short-term memory storage until you practice enough.
Here's an easier one, 52 * 52:
52 * 52 = (52 + 2) * (52 - 2) + (2 * 2) = 54 * 50 + 4 = 2704
Numbers ending in 4 and 6 are the hardest, because you've got to add 16.
86 * 86 = (86 + 4) * (86 - 4) + (4 * 4) = 90 * 82 + 16 = 7396
Numbers ending in 1 are easy:
71 * 71 = (71 + 1) * (71 - 1) + (1 * 1) = 72 * 70 + 1 = 5041
... and numbers ending in 5 are trivial:
45 * 45 = (45 + 5) * (45 - 5) + (5 * 5) = 50 * 40 + 25 = 2025
This is math that the average fourth-grader can handle ... can you?
Multiply A by two A*2 3*2 equals 6
Multiply answer of A*2 with B Answer * B 6*6 equals 36
Then Add That answer from the table Answer + ? 36 + 3 equals 39
Add that answer with the A0+? 39+30² equals 39+90 equals 129
two digits of the square of A0 6*6 equals 36
square last digit and the take the last 1296
digit of the answer and put it on the
final answer
For example to get the square of 72 you break 72 into halves till you get to the smallest that you can easily manage 36, 18, 9
Then you do 9*9 = 81 and multiply this by 64 or by 8 and then 8 again. If you had used 18, you would have done 18*18 = 324 and multiply that by 16 or by 8 and then 2
If you used 36, then 36*36=1296 and multiply that by 4. As you can see, for every division of the original number (72) you have to multiply the end result with a multiple of 2, 4, 16 ... depending on how many divisions you make
And here's a trick to square numbers that end in 5 - it works for any number of digits but gets difficult after the 3 digits.
The square of a number A5 (where the value of A is 1 in 15 and 4 in 45) is A*(A+1) concat 25
So 45 squared is 4*5 concat 25 = 2025
95 squared is 9025
115 squared is 13225
You can even cheat with simple multiples of 10 and wow people with instant calculations of 995*995 = 99 00 25 ie 990,025
Im now an engineer and wonder just how many qiock methods would have made my struggle so much easier to qualify
71 * 71 = 70^2 + 71^2 - 70^2 = 70^2 +(70+71)(71-70) = 70^2 + 141 = 5041 or simply put
72^2 = 70^2 + (70+72)(2) etc...
I find this much easier... my dad showed this to me a long time ago
Quite a great deal of tutorial you got here. I was very impressed with your trivial math skills above. For someone who's made it point of duty to help his kids, your box of tricks above would certainly come handy.
Keep up the good work..