Multiplication Trick

Doug Reiss from Memphis, Tennessee wrote to me that he found the trick in a book by Shakuntala Devi called, Figuring: The Joy of Numbers published in 1977. So GAMES must have gotten it from there.

From GAMES:
We don't know how this multiplication trick works, but it's never failed us yet. Take any two numbers: say, 116 and 3,011. Halve the first number again and again, discarding any fractional remainder, until you reach the number 1. Thus: 116, 58, 29, 14, 7, 3, 1. Double the second as many times as you halved the first. Thus: 3,011; 6,022; 12,044; 24,088; 48, 176; 96,352; 192,704. Write these series alongside each other, and cross out every even number in the halves column and its partner in the doubles column. Thus, as shown in the following columns, the even numbers in the halves column (116, 58, and 14) are crossed out along with their companions in the doubles column (3,011; 6,022; and 24,088), regardless of whether these are even or odd.

Add the numbers that remain in the doubles column only. The resulting sum will be equal to the product of the two numbers you started with.
Thus: 12,044 + 48,176 + 96,352 + 192,704 = 349,276 = 116 X 3,011.

Explanation of the Trick

In February 2015, Shekhar Dutta from the Sarwan Memorial School in West Bengal, India sent me an explanation of the trick. He showed that it can be explained by using base two! I will give a short summary of his idea and then attach his letter below.

Example 1 -- the problem above:
116 x 3011 = 349,276

The number 116 can be written in terms of powers of two:

Now use the distributive property:

Example 2 -- 13 x 7 = 91

Here is the trick:

Here is the explanation using the powers of two:

Example 3 -- 15 x 8 = 120

Here is the trick:

Note that there are no even numbers in the first column, so nothing was crossed out.

Here is the explanation using the powers of two:

Example 4 -- 8 x 15 = 120 Here is the trick:

Here is the explanation using the powers of two:

Examples 3 and 4 were the same problem but they look different in the trick;
in example 3, there were no even numbers in the first column; but in example 4, all the
numbers in the first column were even except the last one (which was the answer).

Thanks again to Shekhar for providing the explanation!

Here is Shekhar's letter (some plus signs were omitted):

I am a student from India. I am 15 years old. My English may be a
little poor. I have visited your website 'Mr. P's Math Page'. From
there 'Puzzles and Games', 'Math Trick', 'Number and Computational
Tricks', 'Multiplication Trick'. In this sequence I have reached the
trick of multiplying two numbers.

It said that we have to half one number till it reaches 1, and double
the second number. After canceling the even number(s) in the first
part and the corresponding numbers in the second part, the sum of the
leftover numbers in the second part will be the product of the two
numbers. You said that you don't know an explanation for this trick.
Here I am trying to give an explanation.
(I have not taken any help from anyone to find this explanation. I
would like to get a reply of this mail as soon as possible.)

You said if the first number is odd after halving we will discard the
fraction. It is like subtracting 1 and dividing by 2.
13/2=6.5, after discarding fraction, =6
Also 13-1=12, 12/2=6
After subtracting 1 (if necessary) and dividing by 2 as we make 1 we
make a binary representation of the number.


13-1=12, 12/2=6  (1)
6/2=3   (0)
3-1=2, 2/2=1   (1)
1   (1)
Or 13   (1)
     6     (0)
     3     (1)
     1     (1)
Therefore 13(base 2) =1101
This can also be written as  
2^3 * 1  +  2^2 * 1 +  2^1 * 0 +  2^0 * 1 = 13
Or  8 +  4 +  1 = 13

Therefore multiplying a number by 13 = 
Multiplying the number by 8 +
Multiplying the number by 4 +  
Multiplying the number by 1

13x = 8x +  4x +  x
*Canceling the even number(s) in the first 
part means canceling the
0(s) in the number's binary form.
13     1
6       0
3       1
1       1
this become after canceling
13     1

3       1
1       1
This digits can be represented as
13   1   1

3     1   4
1     1   8
The last column sums to the number taken
*This can be expanded as
13   1   1     x
6     0   0     2x
3     1   4     4x
1     1   8     8x
After canceling the even digits
13   1   1     x

3     1   4     4x
1     1   8     8x
The last column sums to 13x = 13 * x
*As it is seen, by this method the first 
number is braked as multiples of 2. The 
second number is multiplied with this 
multiples of 2 and then the products 
are added.

Examples
15 * 8
15   1   1     8 * 1 = 8
7     1   2     8 * 2 = 16
3     1   4     8 * 4 = 32
1     1   8     8 * 8 = 64
No even numbers to cancel
15 * 8 = 8  + 16 + 32 +  64 = 120

Or 8 * 15
8   0   0     15 * 1 = 15
4   0   0     15 * 2 = 30
2   0   0     15 * 4 = 60
1   1   8     15 * 8 = 120
After canceling the even numbers



1   1   8     15 * 8 = 120
8 * 15 = 120
I hope this explanation will help.
Waiting for reply.