Can you easily show why 43,560 is not a perfect square?

Take the square root, it’s not an integer.

Edit, an acre is defined as “43560 square feet” but that can take any shape. It’s not a square.

A perfect square ending in zero must end with an even number of zeroes.

@Zounderkite , That is correct. Another way of saying this is that the number of zeros from the multiplication is twice the number of zeros at the end of the number being squared, which must be even, and includes the case where the last digit is not zero (zero zeros), since two times zero is zero.

Perfect squares are in the form x times x, so they form a continuous curve of increasing values. So when I look at this question (after getting the snarky “I don’t care” type reactions out of my system), my first thought is to just estimate and look and the values around it to see.

So:
200×200 = 40000
210×210 = 44100
209×209 = 43681

At this point, I am certain it’s not a perfect square, because the next closest possible number would be 208, and that’s clearly going to be a lower value, just looking at the values above.

Just for completeness:
208×208 = 43264

P.S. You check things with ChatGPT??? Wow, that seems like the opposite of mathematical rigor.

@Zaku , Look at all the work you had to put in compared to @Zounderkite. That approach is also easy to generalize.

You are correct about my use of ChatGPT, but I have good luck so far in asking it relatively straightforward questions.

From Wikipedia: “Traditionally, in the Middle Ages, an acre was conceived of as the area of land that could be ploughed by one man using a team of eight oxen in one day.[3]”

Also from Wikipedia:
Originally, an acre was understood as a strip of land sized at forty perches (660 ft, or 1 _furlong) long and four perches (66 ft) wide;[4] this may have also been understood as an approximation of the amount of land a yoke of oxen could plough in one day (a furlong being “a furrow long”).

Simple generalization of the problem: No number ending in an odd number of zeros is a perfect square.

@LostInParadise Yes, I see that now that I’ve read an explanation. My thought process actually looked at the ending zero first and thought “no, it’s not”, and then thought for a moment about the reasoning why not, exactly, but didn’t hit on it immediately, so my thoughts short-cut to the approach I used, which takes much less time than it takes to write an explanation of to other people, feels more exhaustive/satisfying to me, and also answers a more general question:

“For any positive number, is it a perfect square?”

Which is still a vanishingly rare thing for me to ever actually want to know, but works not just for numbers ending in 0. Though now I’ll probably remember this 0-based logic for a few years, during which time, I don’t expect to want to know again if any number is a perfect square.

I am now interested, though, in what uses there are for answering the question “is X a perfect square?” (outside of math classes).

Here is another example of a way of generating numbers that are not perfect squares Consider 86462. Hint: Show that the number is divisible by 2 but not by 4.