An oddly simple question (If you think about it)

$5184 = 72^2$ and $7569 = 87^2$ . Since all the perfect squares of the numbers 72 to 87 inclusive are between 5184 and 7569, there are $87-72+1 = 16$ possible perfect squares.

A good technique to solve a problem like this is to consider a simpler case and move forward from that.

How many numbers are between $1$ and $5$ inclusive? Well, obviously there are 5 numbers. What about $3$ and $6$ inclusive? Well this is just $6 - 3 + 1 = 4$ . We can easily prove that between numbers $A$ and $B$ if $B > A$ , there are $B - A + 1$ numbers between them if you count the numbers defining the interval themself. Another way to look at it is that there is a function $f(n) = n$ that tells us how many numbers are between $1$ and $n$ inclusive.

So we can say that (the amount of numbers between A and B is the amount of numbers between 1 and B) - (the amount of numbers between 1 and A) + 1 (because we are subtracting A as well so we need to add 1) ) => $f(B) - f(A - 1) => B - (A) + 1 = B - A + 1$ .

So what if there is a similar function that tells us the amount of perfect squares between 1 and N, then we can apply the same logic. It turns out that $\sqrt{N}$ tells us how many perfect squares there are between 1 and N. Try it for yourself!

So using the same reasoning from above, the amount of perfect squares between A and B = g(B) - g(A - 1) =>) # of perfect squares from 1 to B - # of perfect squares from 1 to A - 1 $=> \sqrt{B} - \sqrt{A} + 1$ .

Therefore, the amount of perfect squares between 5184 and 7569 inclusive is $\sqrt{7569} - \sqrt{5184} + 1\ = 87 - 72 + 1 = 16$ .

...And this is how you overcomplicate simple arithmetic :D