C. Squaring Three-Digit Numbers

Squaring a three-digit number using SSQ is as simple as squaring a "two-digit" number. It is my advise that you must first read my first two articles (entitled, "Introduction" and "Squaring Two Digit Numbers") to fully appreciate how SSQ works for three-digit numbers.

Question: What is the square of 238?

238² = ?


General Rule:

In  squaring number using SSQ,  no matter how many digits it has (or involved) - you must always 'remember' that it is much easier to deal it by 'doing things' two separate digits at a time.


238 is a three-digit number. By general rule, we must deal (or consider) 238 as three separate digits as 2 - 3 - 8.

Now, focus your attention on the first two separate digits 2 and 3...

23..² = (leave this space blank)
23..²      = 04’09.. ← PSL  
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
................05’29..   ←Partial Sum (remember, this is not yet the final answer)

You might notice that the above computation is the same as to the example given on Squaring Two-Digit Numbers.


Groupee

What we actually done is the same process or procedure as we learned in squaring two-digit number. Now, in this case, there is a remaining digit 8, in the given value 238 that we must deal with.

At the moment, don't bother yourself about that remaining digit. Let's first  consider the square of 23.

23..² = 05'29.. 

In reality, 05'29 is the 'actual square' of 23. But in SSQ, as fun way of memorizing words, let's call 05'29 as the square of the 'groupee' 23.

Groupee, is a group of digits considered as one unit. 2 and 3 as separate digits in the first procedure that we had done (Two-Digit SSQ)., will now be a groupee known as 23.

Its corresponding 'square' will then become part of the new PSL (Partial Square Line) along with a new 'index  square' of the remaining digit 8.

238² = 05'29'64

Now, as a general rule of SSQ, we must get the sub-product (new SP), using the DouLAL Multiplication Pattern

Activity 1: "DouL" means Double the Last digit...

8 x 2 = 16  

Activity 2: "AL" means Multiply to All in Its Left

You may notice that the 'all to the left' of 8 is 23 in the given value 238, so...

23 x 16 = 368

The last procedure will then be, add the new sub-product (SP) to the new Partial Square Line (PSL) to get the T-Sum.

238²   = 05'29'64   ← new PSL 
23x16 =     36'8  .   ← new SP1 
............05'66'44    ← T-Sum


Practical Technique for Three-Digit SSQ

Step 1: Consider squaring the first two digits using SSQ method..


238² = (leave this space blank)
23..²      = 04’09.. ← PSL  
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
................05’29..   ←Partial Sum 

Step 2:  Create a new PSL

Activity 1:  To conserve time and effort  (as well as your ball pen's ink), write down at the left side of the partial sum, 23 along with the third digit 8 and underline 8. Don't forget the square sign ( _ ² ).

Activity 2: At the right of the partial sum, write down the index square for 8. 


238² = (leave this space blank)
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)

Step 3: Find the new Sub-product. Add it to the new PSL to get the T-Sum.

238² = (leave this space blank)
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)
.............   05'66'44    ← T-Sum

Step 4: As the final step, write down the actual value of T-Sum.

 Remember that we left blank the space after we wrote down 238²   
 Now, we can use this space to write down the 'actual value' of 05'66'44

238²     = 56, 644  ←  "Final Answer"  
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)
.............  05'66'44    ← T-Sum


"Double Dots" Sign

Maybe, you noticed that there are those 'double dots' after 23 in 23..² and also after 04'09 in  04’09.. . We simply use this sign to indicate that there is a following digit or digits after 23. 
As part of the rule, make it as a practice to include the 'double dots' in performing SSQ. 


 Now, it's time to level-up. You are now ready to do the squaring of numbers in more than three digits. Next topic, the multiple-digit squaring, enjoy it.