Squares of Three-Digit Numbers

Multi-Digit SSQ

SSQ is applicable to any number. But SSQ becomes a little bit complicated when the digits of a number increases. The 2D.SSQ is the easiest but when the digits of numbers increases, such squaring 54,675², it involves many sub-products and many sub-totals and the process of multiplying and adding the set of numbers also become very complex. BUT DON’T WORRY, we will never to do that 5D.SSQ. But I will just give some new general rules of SSQ so that you will have an idea, how to tackle our next topics, the 3D.SSQ and 4D.SSQ.

General Rules of SSQ

General Rule No.1:

If you 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 real life, this rule seems to be not true.

12² = 144

Well in SSQ, it always agrees because we write 12² as 01’44.

12² = 01'44

General Rule No. 2

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.

In 2D.SSQ, there is only one sub-product (SP1). But be prepared, in 3D.SSQ, you will not only be dealing with one sub-product but two sub-products (SP1 and SP2). While in 4D.SSQ, you need three sub-products (SP1, SP2 and SP3), to be added to the PSL, to get the square of a four-digit number.

General Rule No. 3

The sub-totals are always less than one, the number of the sub-products


Sub-Totals

A ‘sub-total’ is simply, a temporary sum, in between, each time you add a sub-product. In squaring a two-digit number (such as, 23²), there is no sub-total because you directly add SP1 to the PSL. A sub-total is optional; you can either include or ignore it. But personally, I highly recommend you to practice including it because it is important in our study of Square Edging.

Three-Digit SSQ (3D.SSQ)

In 3.D SSQ, the same rules you learned in 2D.SSQ still work. But of course, there are new added features.

Given Problem:

743² = ?

Step 1:Create a PSL

This is the easiest part. Simply write down the index squares of 7, 4 and 3.

743² = 49’16’09

Step 2:Solve the first sub-product SP1

First Sub-Product (SP1)

By the rules in dealing with sub-products;

Rule 1: Multiply the digits 4 and 3

4x3 = 12

Rule 2: Double the product (Optional, if you wish, you can skip this one)

12 x 2 = 24

Rule 3: Take note, the last digit of SP1 must be in the tens decimal place.

4x3x2 = 24 ← PS1

Step 3: Get the sub-total by adding the first sub-product to the PSL

Sub-Total (ST1)

Take note, this sub-total is just simply like a stop-over. It is not yet our final destination.

......743² = 49’16’09 ← PSL
.+4x3x2 = .......2'4 ← SP1
................. 49’18’49 ← ST1

Step 4: Solve the second sub-product


Second Sub-Product (SP2)

Notice that here in 3D.SSQ, the two added features are the sub-total ST1 and this second sub-product (SP2). In dealing with SP2, GIVE EXTRA CARE, in cross multiplying the digits.

Again, by the rules dealing with sub-products using the R.A.R. Multiplication Pattern, the all digits to the right of 7 are 4 and 3. We should not consider 4 and 3 as individual and separate digits. The A in R.A.R. stands for the word “All”. So don’t forget this – consider ALL to the right as ONE GROUP. So the digits 4 and 3, as a group, will become 43. You must read it as forty three.

Rule 1: Multiply the reference digit 7 by 43

7 x 43 = 301 Correct

7 x 4 x 3 = 83 Incorrect

Rule 2: Just to remind you, DTP, “always double the product”

301 x 2 = 602

Rule: The last digit 2 of 602 should be in the hundreds place

7x43x2 = 602 ← SP2

Step 5: As a final step, get the total sum by adding the second sub-product (SP2) to the sub-total (ST1)

.................. 49’18’49 ← ST1

...+7x43x2 = 6'02 ← SP2 (provided that 2 of 602 aligned to 8 of the ST1)

.................. 55’20’49 ← TSum

Summary:

......743² = 49’16’09 ← PSL

........ +4x3x2 = 2'4 ← SP1

.................. 49’18’49 ← ST1

...+7x43x2 = 6'02 ← SP2

.................. 55’20’49 ← TSum