Theory:

The below table shows the squares of the first \(20\) numbers:
 
Number
Square number
Number
Square number
1
 \(1^2= 1 × 1 = 1\)
11
 \(11^2= 11 × 11 = 121\)
2
 \(2^2= 2 × 2 = 4\)
12
 \(12^2= 12 × 12 = 144\)
3
 \(3^2= 3 × 3 = 9\) 
13
 \(13^2= 13 × 13 = 169\)
4
 \(4^2= 4 × 4 = 16\)
14
 \(14^2= 14 × 14 = 196\)
5
 \(5^2= 5 × 5 = 25\)
15
 \(15^2= 15 × 15 = 225\)
6
 \(6^2= 6 × 6 = 36\)
16
 \(16^2= 16 × 16 = 256\)
7
 \(7^2= 7 × 7 = 49\)
17
 \(17^2= 17 × 17 = 289\)
8
 \(8^2= 8 × 8 = 64\)
18
 \(18^2= 18 × 18 = 324\)
9
 \(9^2= 9 × 9 = 81\)
19
 \(19^2= 19 × 19 = 361\)
10
 \(10^2= 10 × 10 = 100\)
20
 \(20^2= 20 × 20 = 400\)
 
Did you observe the above table? What are the ending digits (that is, digits in the one's place or unit digit) of the square numbers?
 
All these numbers end with \(0, 1, 4, 5, 6, 9\) (at the unit's place).
 
None of the numbers end with \(2, 3, 7, 8\) (at the unit's place).
 
And we can also find another interesting point on that table. Observe the unit's place in all the square numbers in the above table.
1. The square numbers end in \(0\), \(1\), \(4\), \(5\), \(6\) or \(9\) only.

2. If a number ends with \(1\) or \(9\), its square ends with \(1\).

3. If a number ends with \(2\) or \(8\), its square ends with \(4\).

4. If a number ends with \(3\) or \(7\), its square ends with \(9\).

5. If a number ends with \(4\) or \(6\), its square ends with \(6\).

6. If a number ends with \(5\) or \(0\), its square also ends with \(5\) or \(0\), respectively.

7. Square of an odd number is always odd, and the square of an even number is always even.

8. Numbers that end with \(2\), \(3\), \(7\) and \(8\) are not perfect squares.
Zero at the unit's place:
 
Can you guess what will happen if a squared number has zero in the unit's place?
 
Let's see few examples:
 
102=10×10=100502=50×50=500902=90×90=900
 
From the above examples, we can see that if the number with one zero in the unit's place is squared, then the square number will have two zeros.
 
Similarly, if the number has two zeros, then the square number will have four zeros.
 
Important!
Zero in the unit's place gets doubled itself if it is squared.