 This is interesting. Let's say you are selected to play who wants to be a millionaire and you have a question for say I don't know maybe one crore even if it's imaginary I mean I never thought math could help you win one crore So let's say you are going to win one crore if you answer this correctly and this question is find a square number and There are four numbers given but every number is six digit number and you have no way to identify Whether it's a square number or not. So let's say you have some lifelines like say 50-50 So if you if you select the 50-50 lifeline two of the four options just vanish and you are just left with two options And you have to select from two remaining options but still you don't really know which could be the answer here and Now we have to make a choice whether to like leave the question Get away with whatever money that you've won or just try this question out and win one crore Let's see if we can observe something about the square numbers Particularly what number appears at the end in the square numbers and let's see if it can help us answer this question You look at all the options And you notice that every number is a six digit number and you have no way to identify whether It's a perfect square or not just by looking at it But then you start looking at the digit in the unit's place and each of the number What do you notice so the numbers in the unit place are three six one and eight There is an interesting property of the square numbers only specific numbers appear at the unit's place of a square number And this property says that if there is two Three seven and eight in the unit's place. It cannot be a square number So now you definitely know that Option a is not the answer and option d is not the answer b Or c could be the answer now because you have this 50 50 lifeline You have a chance so that if you can use it for your advantage and if by chance Either b or c just gets vanished. Then you know The answer to this question and you can still win one crore. So let's take a chance If you decide to use 50 50 lifeline And two options get vanished Say option a and option b right And now because these two options are nullified after using 50 50 lifeline You definitely know d is not the answer So the only answer to this question is c which is 7,72641 And I'll just tell you the square root of this number is 879 I mean I designed this question or a scenario in a way that I could lead you to understand the properties of square numbers But again, it's it's so funny to think about the fact that maybe some questions might appear in Who wants to be a millionaire and you could still be a millionaire So let's just assume for our happiness that you really could answer this question and you won't one crore But what if instead of option a or b the system nullified option a and option d Was that a foolproof method to get the final answer? Definitely no you you could have used other methods But just by looking at the unit's place number it was not possible to gauge whether it's us It's a perfect square or not because a perfect square might have the digits one four six Nine or zero at the unit's place But that's it it doesn't guarantee that if this number appears in the unit's place The number is a perfect square, but why does this property hold? Just look at the numbers from one to nine. So I'll just quickly write one two three four five six seven eight and nine So just remember any given number will end either with these numbers or zero Right and when we square that number the digit in the unit's place is going to be exactly what we see in the square of these Numbers so square of zero is zero square of one is one Square of two is four square of three is nine Square of four is 16. So we want to focus at six Square of five is 25 We want five Square of six is 36. We want to focus on six again Square of seven is 49 So we are interested in nine square of eight is 64 That's four here and square of nine is 81 And we only see particular numbers. So we see zero then we see one Then we have four in the square nine in the square and six in the square Also five and rest of them are already written So zero one four nine six and five are the numbers which appear in the unit's place in Any square number is any number ending with these numbers say for example 44 Is not going to be a square always, right? So these are the numbers which appear in the unit's place of any square number But any number that ends with these number is not the square. So remember that now, this is where We also know that only these numbers appear at the end of the squares And that's why the rest of the digits, which is two three Seven and eight do not appear in the unit's place of a square number Let's just notice what happens when we are squaring an odd number. So this is an odd number We find that the number in the unit's place is also odd, right? This is a good observation So when you square an odd number, you find an odd number at the unit's place Let's say you you square even numbers. So what happens to the digits in the unit's place? So it's four six six again Four and zero you always find an even number in the unit's place when you are squaring the even numbers It's also noticeable that by looking at the number in the unit's place everywhere You can identify what might be the unit's place digit in the square root as well, right? So let's say the perfect square given has the digit in the unit's place as nine Then square root will have the digit in the unit's place as Either three or seven because we saw the square of three is nine and square of seven is 49 So either of these two numbers might be there in the unit's place of the square root what if The perfect square has the digit in the unit's place as say six What might be the unit's place digit in the square root? We know that the square of four is 16 and we saw square of six is also 36 Whenever there is four in the unit's place will get the square with unit's place digit as six and same with 36 So four or six have to be the digits in the unit's place when we talk about the square root When the perfect square has six in the unit's place I'll encourage you to go through the squares of all the numbers from one to 20 And see if all these properties that we discussed really hold or not