The speed the Indians calculate at the mental calculation world championships is ridiculous. I mean this in a positive way. Unbelievable!

Here is an example of the current world record holder Shashank Jain taking 10 square roots just a little over 1 minute:

Nodas explained in a separate post

that they do this using the duplex method and if you take some time to learn this, you will be very fast in squaring numbers and taking square roots.

I took some time to play around with this and it really speeds things up.

First, what is a duplex?

If a, b, c, d, e, f, etc. are digits, then

d(a) = a^2

d(ab) = 2ab

d(abc) = 2ac + b^2

d(abcd) = 2ad + 2bc

d(abcde) = 2ae + 2bd + c^2

etc.

In other words, from the outside working inwards, take the outermost digits, multiply them, double them, then take the next adjoining digits, multiply then, double them, then the next digits. etc.

If you are left with one single middle digit, square it.

Then add all these numbers.

If we want to square a 2 digit number: ab then the square is:

d(a) | d(ab) | d(b)

If we want to square a 3 digit number: abc then the square is:

d(a) | d(ab) | d(ac) | d(bc) | d( c )

If we want to square a 4 digit number: abcd then the square is:

d(a) | d(ab) | d(abc) | d(abcd) | d(bcd) | d(cd) | d(d)

Let’s start simple:

11^2 =

a=1

b=1

d(1) | d(11) | d(1) =

1 | 2 | 1 =

121

17^2 =

a=1

b=7

d(1) | d(17) | d(7) =

1 | 14 | 49 =

carry the 1 and the 4:

1+1 | 4+4 | 9 =

289

78^2 =

a=7

b=8

d(7) | d(78) | d(8)

49 | 2*56 | 64 =

49 | 112 | 64 =

carry 11 into the hundreds and 6 into the tens:

60 | 8 | 4 =

6084

alternatively:

78^2 = (80-2)^2

a=8

b=-2

d(8) | d(8, -2) | d(-2)

64 | 2*8*-2 | 4 =

64 | -32 | 4 =

you want to end up with positive digits, so I turn 32 into -40+8 and carry the -40:

64 | -40 +8 | 4 =

carry the -40 into the thousands:

60 | 8 | 4 =

6084

Let’s square 345:

345^2 =

d(3) | d(34) | d(345) | d(45) | d(5) =

9 | 2 * 12 | 2 * 3 * 5 + 4^2 | 2 * 4 * 5 | 5^2 =

9 | 24 | 30 + 16 | 40 | 25 =

9 | 24 | 46 | 40 | 25 = (now handle the carries)

11 | 8 | 10 | 0 | 25 =

11 | 9 | 0 | 2 | 5 = (no carries left, we are done)

119025

A four digit number:

1234^2 =

d(1) | d(12) | d(123) | d(1234) | d(234)| d(34) | d(4) =

1 | 4 | 10 | 20 | 25 | 24 | 16 =

1522756

We can also do this two digits at a time:

12 34^2:

a=12

b=34

12^2 | 2 * 12 * 34 | 34^2 =

144 | 816 | 1156 =

keep in mind that when working with 2 digits at a time, we work with hundreds between two ‘|’'s. So in 1156, we carry 11 and not 115:

144 + 8 | 16 + 11 | 56 =

152 | 27 | 56

1522756

Or with negative numbers:

788^2:

for 788 use 800 - 12, so:

a = 8 and b = -12:

d(8) | d(8 , -12) | d(-12)

64 | 2*-12*8 | 144 =

64 | -192 | 144 =

192 = 200-8, so -192 = -200+8.

64 | -200 + 8| 144 =

Do the carry negatively:

62 | 08 | 144 =

62 | 09 | 44 =

620,944

See how easy this is?