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Square Any Two-Digit Number In Your Head .:. kentbrewster.com

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?

Comments from before Disqus:

Tim .:. 2010-07-15 22:20:45
(x + y)^2 is the easiest way I've found to do squares of double digits (Other than those ending in 5 which has an even easier method, (x*(x + 1)) and concatenate 5 * 5 = 25 )
MrBacon .:. 2009-12-05 13:00:07
Jenny, let x and y be any natural number and z their product. The powers of x and y multiplied are equal to z squared. (if xy=z, then xy(xy)=z(xy) or x^2y^2=z^2). For example, 9*4=36, as in (3^2)(2^2)=6^2, or 3*2=6.
Jenny .:. 2009-11-29 10:20:22
if a square number is multiplied by a square number the product is always a square number. How do I convince someone this is true?
Michael Hernandez .:. 2009-11-17 19:52:49
I am not going to tell you exactly how I got three
Michael Hernandez .:. 2009-11-17 19:45:09
This is an easier way to square a number AB² 36²
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
thande .:. 2009-09-21 10:06:15
this is so ??????????????????????????????
Kent Brewster .:. 2009-08-11 17:35:32
Welcome, Google friends. Sorry, but this isn't Albert Clay's method for multiplying any two numbers together in your head. Now that I know it's out there, however ... I'm working on it. :)
Khalifah .:. 2008-07-23 10:44:02
Got one more way, though probably not the easiest but works well for even numbers
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
Ron .:. 2008-04-14 00:10:43
I just figured this out and started looking to see if it was on the internet any were. I think the way I figured out to do numbers ending in 1 is much easier than either of the 2 here. I will do 71 since it is the two examples shown already. 71*71=5041 So to the trick 7*7=49 7+7=14 and 1. So 49+ the 1 from the 14 = 50 drop the 4 in 504 and 1 = 5041. It does get to be a harder as the numbers go up, but for most people up to 121*121=14641 should be pretty easy. 12*12=144, 12+12=24 and 1. 144+ the 2 from 24= 146, drop in the 4 =1464, and 1 =14641.
Raghu Srinivasan .:. 2008-03-08 08:23:37
Hi Kent,

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
Eddie .:. 2008-02-06 14:45:14
Wow thats awsome , many thanks for that , wish id been taught that methjod at school , life would have been so much easier.
Im now an engineer and wonder just how many qiock methods would have made my struggle so much easier to qualify
Rumey Jiffrey .:. 2007-12-11 00:18:07
There is an easier way eg.
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
Benny .:. 2007-11-26 09:15:09
Hi Ken,

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..
Kent Brewster .:. 2007-11-26 09:15:09
Thanks! The dirty little secret here is that I didn't actually learn how this worked until my daughter went to fifth grade. :)

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