D. Multiple-Digit Squaring (Rules to Follow)

SSQ is applicable to any number. But SSQ become a little bit complicated when the digits of a number increases. The two-digit SSQ method is the easiest and the very fundamental of the principles behind SSQ. But when the digits of numbers increases, such as in squaring 54,675², it will involve many 'sub-products' and many 'partial sums' and the process of multiplying and adding the set of digits will also become very complex.

But don't worry. As long as you already have the idea of squaring two-digit numbers and learn to adjust and become familiar in squaring three-digit numbers using SSQ, squaring multiple-digit numbers will be just another way of squaring number in a more challenging way. 


Ultimate Rules in SSQ

Now that we are dealing to do the squaring of numbers in four, five or six-digit numbers, it is important that you must know and follow these 'golden rules':


First Golden Rule:

If you're going to square a number, make sure to count the digits. 

 "The count of digits doubles as you square a number."

35² = 1,225

(Take note, a from two-digit number it becomes a four-digit number)

But in certain cases, this rule seems not been followed...

12² = 144

But in SSQ, it is a 'strict rule' that you must follow this golden rule. 12² = 144 will then, become...

12² = 01'44


Second Golden Rule:

The number of sub-products increases as the digit of numbers increases. But the number of sub-products depends on the count of digits. 
"The sub-products are always less than one to the count of digits of the given number".
 
You might noticed that in two-digit SSQ, there is only one sub-product (SP1), while doing the three-digit SSQ, there appears a second sub-product. It is easy to follow the pattern - that in a "four-digit.SSQ", you will then need three sub-products (SP1, SP2 and SP3), to complete the process. So in five digits, there will be four SP's. Six... five, and so on.

Third Golden Rule:
 
As a initial rule, always remember that the count of partial sums increases as the counts of digits of the given number increases. But to be specific, keep it in your mind that:
"The partial-sums are always one less than to the count of sub-products in a multiple-digit number SSQ"

In a way, the partial-sum, when squaring a number having more than two digits, is simply a temporary result of adding the sub-product to the PSL .


Take note, in a two-digit SSQ, there is only one sub-product and "no partial-sum" at all

 (Example 1)

 23²       = 04’09 ← PSL
 +2x3x2 =   1'2   ← SP1  (2 of 12 is aligned to second 0 of 04'09)

................05’29 ← T-Sum 

As a simple way of explaining things, we may consider that a 'two-digit' SSQ is somehow, not part of the multiple-digit SSQ at all - in a sense that it doesn't include any partial sum. In a two-digit SSQ, the sub-product (SP) is directly added to the partial squares line (PSL) to get the 'total sum' (T-Sum).  

Look below. Take note that there are now  two sub-products (SP1 and SP2) and the T-Sum in the above (example1), become the 'partial-sum' (in yellow shades).   

(Example 2)
23..²      = 04’09.. ← PSL 1
+2x3x2 =   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
238²   = 05’29'64   ← PSL 2  (the new PSL)
23x16 = .   36'8  .   ← SP 2  (the new SP) (8 of 36'8 aligned to 6 of 05'29'64)
.............05'66'44    ← T-Sum
  

Fourth Golden Rule:
 "The partial sum will become part of the new PSL along with the next index square"  

In example 2, the 05'29, which is the actual square of 23, become part of the new PSL. We can say that that 05'29 is the "index square" of 23, if we are dealing with multiple-digit SSQ such the example above. 

Fifth Golden Rule:
 "The total sum in a multiple-digit number SSQ will reflect the true square value of the given number".