Square Numbers: What Does it Mean to “Know”?

One morning after T’ai Chi practice, some of us headed to breakfast together. In our discussion, we stumbled onto the subject of knowing.
One friend said, “for example, I know 7x7 is 49” and I responded that someone might know that 7x7 = 49 to write it down, but one might not have any real understanding about it. For example, somebody could know the correct answer, but they might not know that it’s a square number. At this the others at the table immediately piped up, a little bewildered “it is? What do you mean? How do you know that?” “Well, it’s a square number because it makes a square,” I replied. Someone said, “But that’s not what they mean when they call something a square number.” “That’s exactly what they mean,” I replied. My friend then said, “No, it couldn’t make a square, because it’s not an even number.” Some of the others immediately agreed.
Taking out some pennies I demonstrated: “Okay, lets say the corner of this table has four pennies to make a square of four,” and I did so, saying, “2x2 =4 and 2
2 =4. And you’re right, it’s an even number. Now, what’s the next square number?”
“Sixteen,” came back the answer from one of the men. But one woman at the table, (who, incidentally, had said earlier that she had trouble with math in school), said “nine.” “No, it couldn’t be nine,” one of the men said, “that’s an odd number.” “Okay,” I said, “lets try it,” and I did, adding five more coins. “You see?” I asked, “it’s a picture of 3x3=9, and the numbers are all odd, but it makes a square, and so it’s a square number.”
These were all college educated adults, at least one with an advanced degree. All of them had come through the traditional US educational system. They had learned about square numbers, several times over, I’ll bet, yet none of them had a practical understanding of what a square number was.
Thinking later about my friend’s comment that “it couldn’t make a square, because it’s not an even number.” I was fascinated. We inherently seem to see a square and know a square as an “even” thing as opposed to an “odd” thing (such as a triangle). The shape itself has four sides. Four is an even (and a square) number. You can easily see how, without depth of practical understanding of the matter, one might arrive at the question: how could you make a square, which is an even thing, out of an odd number? Incidentally, this man is a twenty-year, devoted Go player, a game that takes place on a board with a 19x19 grid of squares, onto which round stones are placed. That means he deals with stones arranged in squares all the time, yet he had never made these particular connections.