The methods outlined below are intended as aids to mental arithmetic in multiplication utilizing fundamental knowledge of square root.

If you wanted to find the product of two EVEN or two ODD numbers you simply find the square of the mean of the two numbers and subtract the square of half the difference between the two numbers.

For instance, to find the product of 17 and 23 we subtract the square of half the difference (3² = 9) from the squared mean (20² = 400). The result is 400 - 9 = 391.

Remember, only EVEN numbers by EVEN numbers and ODD numbers by ODD numbers.

Odd Numbers

To find the product of 23 and 31 we get [(23+31)/2)=27²)=729] - [(27-23)=4²)=16] = 713.

But the product of 23 and 41 gives [(23+41)/2=32²=1,024] - [32-23=9²=81] = 943.

Even Numbers

To find the product of 22 and 32 we get [(22+32)/2=27²]= 729 - 25[(27-22)=5²] = 704.

But the product of 22 and 42 gives 1,024 - 100 = 924.

An interesting twist is that in special cases we can have an all squares result. For instance :-

The product of 11 and 99 is 3,025 (55²) - 1,936 (44²) = 1,089 (33²).

To get the product of 22 and 88 we have 3,025 (55²) - 1,089 (33²) = 1,936 (44²).

To get the product of 44 and 176 we have 12,100 (110²) - 4,356 (66²) = 7,744 (88²).