You often see a number of questions on obtaining the square of 2 or 3 digit number in the competitive examinations. In this article on Mathematical shortcuts, we will see, How to obtain the square of a 3 digit numbers.

If you have read my article on Mathematical Shortcut for squaring numbers (If you have not, go through it once, you will like it đź™‚ ), you probably know how to obtain the squares of any number.

No doubt, the shortcut is very useful if you have to obtain the squares till 300. We will discuss the limitation of this technique later.

## Using 3 x 3 Multiplication

In this technique, we use the 3 x 3 criss-cross multiplication technique to derive the general formula to obtain the square of any 3-digit number.

If you do not know, please go through the article Multiply any 3-digit numbers which describe the technique of criss-cross method of multiplication.

Assume a general 3-digit number, letâ€™s call it *abc*. Letâ€™s use the criss-cross multiplication to obtain the square of abc^{2}.

The steps are illustrated in the following figure :

This can be re-written as shown :

**Step 1 :** Compute *a ^{2}, ab, ac, bc* and

*c*and write as shown in the updated formula.

^{2}**Step 2 : **Notice the second row of the formula is obtained by repeating the terms *ab, ac* and *bc*.

**Step 3 : **Now, this is different from obtaining the square of a 2-digit number. Here, just compute *b ^{2}* and place it below the middle

*ac*term.

**Step 4 : **Now, add the terms taking care of the carry-forward in each step to obtain the square of the number.

Letâ€™s understand the steps with an example. Let’s say we want to compute 713^{2}.

The entire steps are illustrated in the following figure.

**Step 1:** Break the number such that a=7, b=1 and c=3

**Step 2:** Compute a^{2}, ab, ac, bc and c^{2} and write as shown in the updated formula.

**Step 3: **Notice the second row of the formula is obtained by repeating the second, third and fourth term

**Step 4: **Compute b^{2} and place it in the 3^{rd} column below ac term.

**Step 5: **Now, add the terms taking care of the carry-forward in each step to obtain the square of the number.

So, 713^{2} = 508369

Letâ€™s take up another example. Let’s try to obtain the square of 642^{2}.

The steps are illustrated in the following figure.

Notice the carry-over in each step.

Easy đź™‚ ?

Now, consider an example like 222^{2}. Notice that all the numbers are same.

Notice, how easy it would be to obtain the squares of numbers where all the digits are same.

There is a seperate article on how to obtain the square of number when all the numbers are same. Please refer to my detailed article here.

**Exercise** : Try obtain the squares of the following 3-digit numbers:

- 342
- 101
- 121
- 111
- 675

## Using (a + b)^{2} Formula

Now, let’s go back to the limitations of the modified formula technique as discussed in Mathematical Shortcut for squaring numbers.

Let us consider the same example. How do we obtain 713^{2}. The steps are illustrated in the following figure.

** **

**Step 1 :** Break the number such that a=7, and b=13

**Step 2 :** Compute a^{2}, ab and b^{2} and write as shown in the updated formula

**Step 3 : **Notice the second row of the formula is obtained by just repeating the middle term ab of the first row.

**Important:** Notice the carry-forward in each stage. In the unitâ€™s place we carry forward only 1 instead of 16. This is a very important difference. We keep 2-digits rather than 1-digit as in squaring 2-digit numbers.

In this case, we know both a^{2} and b^{2} in this case. So we can use this technique as well.

Letâ€™s take another example. Letâ€™s say we want to compute 253^{2}. In this a=2 and b=53.

I wrote 53^{2}= 2809 directly. If you do not know how I did it, please refer to my article â€śHow to obtain the square of a number from 51 till 59â€ť.

Letâ€™s consider the square of 482^{2}, you will understand the true limitation, I am talking about.

Let’s break the number such that a=4 and b =82 ( You can also break the number as a=48, b=2).

Compute a^{2}, ab and b^{2}. However, we do not know 82^{2}. So we need to compute 82^{2} using the shortcut described in Mathematical Shortcut for squaring any 2-digit number. The rest of the steps are similar.

This method sometimes will prove really time consuming.

When you practice both the techniques, you will realize which technique to use by looking at the number itself.

**Exercise :** Can you obtain the squares of:

- 113
^{2} - 748
^{2} - 152
^{2} - 767
^{2}

Please contact me if some point in the tutorial is not clear. I would be glad to help.