This is an interesting find. While teaching students on obtaining the square of any number, I observed an amazing patterns in the squares of the number which ends with the digits 05 like 105, 205, 1105 etc.

You already know the shortcut, how to multiply any number ending with 5. You can use the same shortcut here as well. However, you will surprised to know how the square pattern look, if the number ends with the digits 05.

Consider an example. Let’s say, we want to calculate the square of 205 i.e., 205^{2} . The steps can be illustrated in the following figure.

**Step 1 :** Break the number such that 05 is on the right and the rest of the digits ( i.e, 2 in this case) is on the left.

**Step 2 :** Obtain the square of 05 on the right to get 025. This forms the second part of the answer. Please note, a number ending with 05 will always have 025 as the last part of the answer when you take its square.

**Step 3 :** Take the rest of the digits(2) and take its square followed by the digit itself which forms the LHS part of the answer. So, 205^{2} = 2^{2}/2/025 = 42025.

**Important :** The square of any number which ends with 05 would look like a5^{2}/a/025.

Let’s take another example to make it clear. Can you guess, what is the square of 505^{2 } ?

So, 505^{2} = 5^{2}/5/025 = 255025

Very easy right

Let’s consider another example : What is 1205^{2 }?

Notice the difference from the previous example. In this case, we have 2-digits on the left. So, we need to consider the carry – over.

So, 1205^{2} = 12^{2} / 12 /025 = 144+1/2/025 = 1452025

On the similar lines, can you mentally calculate?

- 305
^{2} - 605
^{2} - 1405
^{2} - 1505
^{2}

I hope you find this tutorial useful. Please leave a comment if certain point in the tutorial is not clear.

Nice! I like it. I’m trying to think of ways to ruecrstture my approach to teaching Formal Geometry in High School. My question is this: Is this discovery an appropriate activity for HS students who may have never been asked to think this way? I don;t want to be a nay sayer because i would love to present this to my high school students but I’m afraid of how long the students will stay engaged before they just say WTF!Those are awesome thoughts!I’m going to think of to structure this so students stay engaged!