

Squaring two digit numbers in your head - jgrahamc
http://www.jgc.org/blog/2010/03/squaring-two-digit-numbers-in-your-head.html

======
ZeroGravitas
Works even better for squares ending in 5 because you'll round up and down to
multiples of ten which means the number ends in 00 and you'll always be adding
25.

So:

    
    
      35 squared is 4x3 x100 = 12 00 + 25 = 1225
    
      45 squared is 5x4 x100 = 20 00 + 25 = 2025
    
      55 squared is 6x5 x100 = 30 00 + 25 = 3025
    

When I heard about this trick it was referred to as vedic, hindu or indian
math, but I don't know if that's actually historically/geographically accurate
or not. There's some Japanese nintendo DS games that aim to train you in it, I
think it was a bit of a craze over there.

~~~
hackermom
The two game series are called Brain Age, and Big Brain Academy. Both are
great fun - and helpful.

~~~
jerf
While those are fun, IIRC (and I've spent some time with both) neither of them
do large multiplication, and I'm quite confident neither of them try to
_teach_ this technique to you; they just present math problems with no
technique. (Memorization is appropriate for anything up to 12x12 or 15x15
anyhow, and if they exceed that it's not by much.)

~~~
ZeroGravitas
Yeah, that's a different thing, the games I was thinking of were:

 _Pa to Tokeru! India Suugaku Drill DS_ [http://www.yesasia.com/global/pa-to-
tokeru-india-suugaku-dri...](http://www.yesasia.com/global/pa-to-tokeru-india-
suugaku-drill-ds-japan-version/1005152732-0-0-0-en/info.html)

 _Indo Shiki Keisan Drill DS_ <http://www.play-
asia.com/paOS-13-71-9g-49-en-70-2fox.html>

I think the latter might be the same as the Korean title, _Indian Math Brain
DS_.

For those that don't want to import stuff that they probably can't read
(though I find you can usually get by) there's also _Make 10: a journey of
numbers_ which is a weird maths based adventure game, kind of Brain Age meets
WarioWare:

[http://www.nintendo.co.uk/NOE/en_GB/games/nds/make_10_a_jour...](http://www.nintendo.co.uk/NOE/en_GB/games/nds/make_10_a_journey_of_numbers_9256.html)

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barrkel
In case this observation isn't immediately obvious:

    
    
        Observe that 27^2 = 30 x 24 + 3^2
    

It's derived from this:

    
    
        (x + c)(x - c) = x^2 + cx - cx - c^2
                       = x^2 - c^2
        => x^2 = (x + c)(x - c) + c^2
    

(I note that the blog explains this later, but I had to stop when I read the
observation and work it out for myself on paper - when reading the text, it
seemed like this step was missing.)

~~~
bkz
IMHO, a clearer version:

    
    
                     x^2 = x^2
       x^2 + (r^2 - r^2) = x^2
       (x^2 - r^2) + r^2 = x^2

(x + r)(x - r) + r^2 = x^2

If we continue you'll also notice that:

    
    
          (x + r)(x - r) = x^2 - r^2 
    

Which is useful if you need to calculate the opposite where two numbers are
mirrored around a suitable even number, i.e. 28 * 32 -> (30 + 2)(30 - 2) =
30^2 - 2^2

~~~
barrkel
I don't think that's clearer. Factoring quadratic equations is taught in
schools, and that's where you learn how to multiply out expressions of this
form:

    
    
        (ax + b)(cx + d) = ax(cx + d) + b(cx + d)
                         = acx^2 + adx + bcx + bd
    

A special case of these factorizations is:

    
    
        (ax + c)(ax - c) = ax(ax - c) + c(ax - c)
                         = a^2x^2 - acx + acx - c^2
                         = a^2x^2 - c^2
    

The reason it's a special case is because there's cancellation of the terms.
So starting out with this is useful, as it's likely to be something people
have met in school - but people needn't remember the factorization (which
isn't strictly speaking needed much - we have an equation for solving this),
just how to multiply the terms (which is used more often). From it, we can
derive all that's needed for the trick, without any odd moves.

With your own version, you have to introduce a term which adds up to zero -
(r^2 - r^2) - for no obvious reason, and then you need to already know how to
factor x^2 - r^2 to (x + r)(x - r) in order to understand the last step.

So I think your version is less easy to understand because it's not clear why
the first step is what it is, and it relies on more recollection of school
arithmetic.

(The only reason I expound at length on this is because I think what makes
something obvious or easy to understand is a deep topic, linked to what's
assumed to be known beforehand and the size of inferential jumps between
steps, and is important for programming, writing, and indeed all
communication.)

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nileshtrivedi
A good shortcut I use is for calculating cumulative interest / growth.

x% growth followed by y% growth is equal to a growth of (x+y).xy % overall.
Eg. 6% followed by 8% growth is 14.48% overall growth.

You can easily extend this to account for negative growth rates, more decimal
places and so on. Very useful in tracking business metrics, portfolio
valuation and so on.

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tvchurch
I remember reading how Bethe taught Feynman how to square numbers around 50 in
his head.

1) Start at 50^2 = 2500. 2) Pick a number, determine how far away it is from
50, call it N (N can be negative). 3) Multiply N by 100, add that number to
2500. 4) Square N, add that number to your current total.

For example, 52^2. 1) Start at 2500. 2) N = 2. 3) N _100 = 200. 2500 + 200 =
2700. 4) N^2 = 4. 2700 + 4 = 2704.

Example when N is negative: 47^2. 1) Start at 2500. 2) N = -3. 3) N_100 =
-300. 2500 + (-300) = 2200. 4) N^2 = 9. 2200 + 9 = 2209.

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jgrahamc
Another interesting thing from the book is square root estimation. For example
what's the square root of 73?

Well you know it's between 8 and 9 because 8^2 is 64 and 9^2 is 81, so it's 8
point something. The difference between 64 and 73 is 9. His estimate would be
8 plus 9 divided by 2 times 8. Or how much the low estimate is off by, divided
by twice the guess.

In this case that's 8 9/16 = 8.5625. The actual square root of 73 is 8.544.

~~~
anatoly
_The actual square root of 73 is 8.544._

No it isn't.

</pedantry>

~~~
pgbovine
<pedantry>your XML failed to parse</pedantry>

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Freebytes
One technique I like to use when doing any kind of multiplication is to round
the numbers, multiply, then add the multiplication of the remainders together.
222 * 15 = 200 * 15 + (15 * 22) = 3000 + (15 * 20) + (15 * 2) = 3000 + 30 +
300 = 3330.

I would do this for all multiplication basically, though. I really like the
trick presented here for doing squares. It will make life much easier if I can
just remember the formula.

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manish
I had read in Surely You are joking Mr Feynman, squaring numbers near 50 is
really easy, for example, 49xx2 is (50 - 1)xx2 which is 2500 - 100 + 1 that is
2401.

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liuliu
I have an odd habit to factorize the number first: 24 * 35 = 6 * 5 * 7 * 4 =
840. Oh, the article is talking about addition and squaring...

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tokenadult
It's interesting that the book cited in the submitted article

[http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-
Cal...](http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-
Calculation/dp/0307338401)

(this link is to the United States Amazon site, while the submitted article
link was to amazon.co.uk)

is written mostly by a professional mathematician, but with an introduction by
an author who is a historian by higher education. Michael Shermer writes a
number of interesting books,

<http://www.amazon.com/Michael-Shermer/e/B001H6MCNY/>

of which my favorite is Why People Believe Weird Things: Pseudoscience,
Superstition, and Other Confusions of Our Time.

[http://www.amazon.com/People-Believe-Weird-Things-
Pseudoscie...](http://www.amazon.com/People-Believe-Weird-Things-
Pseudoscience/dp/0805070893/)

~~~
frankus
The other author (also my frosh Calculus prof) did a TED talk a while back:

<http://www.youtube.com/watch?v=M4vqr3_ROIk>

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baby
Woow, that is really effective ! I recently listened to Derren's Brown
audiobook about memory and he teaches a really effective method to remember
list of words. With his technique I can memorize really easily 20 words and
still remember them days after. I never tried more but I'm sure I can do 50
words (I really have to try).

His technique :

1\. okay first imagine that the list of words begins with "telephone, sausage
and monkey"

2\. When you listen to the first word, imagine something grotesque with vivid
colors and unscaled. For a telephone just imagine an immense telephone, let's
say a pink huge telephone.

3\. When you hear the next word, imagine something in relation with the
previous word. Like you need to use smelly sausage to type on the digital
numbers.

4\. Continue to do the same. Next word : "A red monkey is eating saussage"

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proee
Having the ability to do quick calculations in your head is critical for good
engineering.

I've worked as an electrical engineer and knowing a lot of 'envelope-style'
calculations is important when you are trying to trace down problems. Being
able to calculate parasitic inductance, or capacitance is key when say trying
to find out why and opamp is oscillating.

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signa11
another approach might be to treat numbers as letters, squaring them is then
represented algebraically as (a+b) ^ 2 == a^2 + b^2 + 2ab which can then be
easily combined:

79^2 = 49, 126, 81 => 6241 a == 7, a^2 == 49

b == 9, b^2 == 81

2ab == 2 _7_ 9 = 126

combination == (a^2+2ab+b^2)

from 81 take 1

add '8' to 2ab == 134, take 4

add '13' to 49 == 62

answer == 6241

~~~
araneae
I do that for all 2 digit multiplication I do in my head- I just use foil.

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bho
something i do for more general two digit multiplies is to work from left to
right, like he states. take his example, 27 squared:

2x2 = 4 2x7 + 2x7 (cross) = 28, carry the 2 now you have 68 7x7 = 49, carry
the 4 729

it's usually fast enough for me, and i don't have to remember other tricks.

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swah
I do 67^2 as 60 _60 + 7_ 60 + 7 _60 + 7_ 7 ...

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crazydiamond
All this and _much more_ is available in Vedic Mathematics (and a lot is
freely available on the internet).

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hackermom
Very interesting approach. For those who are interested in more "different"
math, google (or wikipedia) for "vedic math" (from Sanskrit _Véda_ ,
"knowledge"); an ancient Indian system of mathematics a bit different from the
typical western teachings, said to better adhere to how the human mind works.
Some amazingly efficient techniques for arithmethics on both paper and in head
are to be found there.

