Heuristic

I will take a rest from tables for a while and demonstrate how Heuristic can be used to solve
all multiplication problems without resorting to remembering tables.
Let’s work out the following problems :
6 x 7 , 9 x 8 , 7 x 4 , 11 x 3 , 12 x 5
6 x 7 = 1111111   1111111    1111111

1111111   1111111    1111111

We create 10(s). We need a group of 3 for every group of 7.
There are 2 ways :

• break down groups of 7(s) to create groups of 3(s)
• borrow groups of (3) then subtract the amount borrowed after the fact

Variation of method 1

combine the 6 groups into 3 groups
1111111     1111111     1111111     11111111111111     11111111111111
1111111     1111111     1111111     11111111111111 ( 1 group from 2 smaller groups )
Break the 3 groups into groups of 10(s)
1111111111    1111     1111111111     1111     1111111111     1111
Combining the left-behinds and arranging the groups we have
1111111111    1111111111     1111111111     111111111111
Finally break the recombined group into 10
1111111111    1111111111    1111111111     1111111111     11
Answer = 42
9 x 8 = 8 + 8 +8 + 8 +8 + 8 +8 + 8 +8
= 8 + 8 +8 + 8 +8 + 8 + 8 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
= 10 + 10 + 10 + 10 + 10 + 10 + 10 + 2
= 72
9 x 8 = 8 x 9 = 9 + 9 + 9 + 9 +9 + 9 +9 + 9
( I chose the bigger number to be the multiplicand )
borrowing 8 (1s) we get 10 + 10 + 10 + 10 + 10 + 10 +10 + 10 = 80
return the 8(1s) or 8 we get 80 - 8 = 70 + 10 - 8 = 70 + 2 = 72
Using the variant of the above method
9 x 8 = 8 + 8 +8 + 8 +8 + 8 +8 + 8 +8
= [ 8 + 8 ] [ 8 + 8 ] [ 8 + 8 ] [ 8 + 8 ] + 8
= [10 + 6 ] [10 + 6 ] [10 + 6 ] [10 + 6 ] + 8
= [ 10 + 10 + 10 + 10 ] [ 6 + 6] [6 + 6 ] + 8
= [ 10 + 10 + 10 + 10 ] [ 10 + 2] [10 + 2 ] + 8
= [ 10 + 10 + 10 + 10 + 10 + 10] [ 2 + 2 + 8 ]
= [ 10 + 10 + 10 + 10 + 10 + 10 + 10] [ 2 ]
= 72
As you can conclude, it is possible to find the answer to any multiplication by using the Heuristic methods presented above. However, they do take a bit of time to finish. Hence, in the next section I will introduce a new method - its less wordy and easier to manage.

Before I do that, I have to introduce 2 groups of numbers.
3 x 3 or 111+ 111+ 111 = 9
3 x 2 or 111+ 111 = 6

Let’s calculate 9 x 8:
we will split 9 into 3 + 3 + 3
and 8 into 3 + 3 + 2
Next let’s draw a rectangle which is 3 columns by 3 rows

Another way , Heuristic
12 x 5 = 5 x 12

Since 11 + 11 + 11 + 11 + 11 = 1111111111
We have 6 groups of 10 = 60
In Mathematical parlance we write this heuristic as
5 x 12 = 5 x 10 + 5 x 2
= 50 + 10 = 60