Unusual products

We all know that 9 × 7 is equal to 7 × 9. However, the following pairs of products appear unusual.

Example 1:

Example 2:

Usually, ‘reversed products’ are not the same:

Example 3:

What distinguishes the identical products from the non-identical products?  Do the factors offer any clues?

Here is another look at Example 1:

Note that 2 × 3 = 6 × 1, or in other words the product of the Units is the same as the product of the Tens.  When we do long multiplication our first ‘sub-product ‘(or should that be sub-total) is  U × U, followed by   U × T, then T × U and finally T × T. Therefore, when we ‘reflect’ 13 and 62 all of the ‘sub-products’ stay the same, although the middle pair U × T and T × U are in a different order.

For Secondary Students:

We can use algebra to give the general pattern.  Let A, B, C and D represent any of the digits from 0 to 9.  Please note that AB  and CD are not products but concatenations (or strings). That is, in polynomial notation, AB would be written as 10 A + 1 B and likewise CD would be 10 C + 1 D.

Extending the Pattern

Now that we have some understanding of the pattern, it is natural to ask if it can be extended to three digit numbers.    Have a go!