I can tell you with lot of confidence, that you donot often find many questions where you need to obtain the square of a 4-digit number. Squaring a 2 or 3 digit number is probably the most commonly asked question in any competitive examinations. However, obtaining the square of a 4-digit is very similar to obtaining square of a 2 or 3-digit number

There is a detailed article on Mathematical Shortcut for squaring numbers and Squaring any 3-digit number. Please go through it, if you haven’t already.

Assume a general 4-digit number and call it *abcd*. Let’s use the criss-cross multiplication to obtain the square of *abcd*. The steps are illustrated in the following figure:

This can be re-written as shown :

**
Step 1 : **Break the number such that you have a, b, c and d.

**Step 2 : **Compute a^{2}, b^{2}, c^{2}, d^{2}, ab, ac, ad, bc, bd and cd.

**Step 3 : **Notice the first row. You have 7 columns. You start with a^{2}, ab, ac and ad. Next, leave bc. Place bd, cd, and d^{2}. That completes the first row.

**Step 4 : **Now the second row is obtained by repeating the first row elements. We take the middle 5 terms, leaving out the end terms a^{2 }and d^{2}.

**Step 5 : **The third row, have 3-elements. So you b^{2} and c^{2 }as the end elements and bc as the middle term.

**Step 6 : **In the forth row, you just repeat the middle term bc from the third row.

So, far for populating the elements. It might look difficult. However, if you practice the steps on few random 4-digit numbers, you will understand the pattern.

Add up all the elements taking care of the carry-forward in each addition.

Let’s understand the steps with an example. Let’s use this technique to obtain the square of 1234^{2}.

The steps are illustrated in the following figure.

Let’s consider another example. Let’s use this technique to obtain the square of 3478^{2}. The steps are illustrated in the following figure.

Now, consider an example like 2222^{2}. Notice that all the numbers are same. There is a separate article on how to obtain the squares of any number if all the digits are same. You can go through the article here.

Notice, how easy it is to obtain the square of such a big number. Very easy right 🙂

Try to obtain the squares of the following 4-digit numbers:

- 3499
- 1012
- 1111
- 5678

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

Hey Megan,

I am glad you liked it. I would be interested to know about your project and like to contribute to it as well.

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Hi! I am a grade 10 student. Thanks for posting such a nice method sir. But I used to square 3-digit, 4-digit and even 5-digit numbers using another trick, which is far quicker than your method. This trick is based on the identity ->

A^2 = (A-d) (A-d) + d^2

Hi Sunny,

I am glad you liked it. I use the technique that you mentioned. However, you need to be very good with multiplication to use that shortcut. I will write about it soon.

You have an error that confused me till I recognized it was an error. That is in the squaring of a three digit number. The example of 21 squared — the the left it should be

a=2

b=1

But you have something different. When I saw your next example then I could see that there was an error in the previous example. No biggy but it is an error. I like your site!! I have some discoveries that you might be interested in. If you are interested then let me know.

That was exceptionally brilliant..keep it up

Thanks a lot Akash